Anil MadhavapeddyLawrence C. PaulsonDavid AllsoppJon LudlamJeremy Yallop2026101https://www.cl.cam.ac.uk/teaching/2627/FoundsCS/forest/index/index/teaching/2627/FoundsCS/forest/index/Foundations of Computer Science (2026–27)See also the glossary, the topic index and the collected exercises.This course has two aims. The first is to teach programming. The second is to present some fundamental principles of computer science, especially algorithm design. Most students will have some programming experience already, but there are few people whose programming cannot be improved through greater knowledge of basic principles. Please bear this point in mind if you have extensive experience and find parts of the course rather slow.The programming in this course is based on the language OCaml and mostly concerns the functional programming style. Functional programs tend to be shorter and easier to understand than their counterparts in conventional languages such as C. In the space of a few weeks, we shall cover many fundamental data structures and learn basic methods for estimating efficiency.The first thing you will notice about this course is that there is an interactive version hosted online at hub.cl.cam.ac.uk, where you can login with your Cambridge Raven identity and edit the code fragments in your browser. You are encouraged to do so — such edits will only persist in your session, and will help you to explore the world of functional programming. If you are using the web-based version, then you need to know a few concepts:The notebook consists of a sequence of textual and code snippets. The code snippets can be executed individually, and will “remember” the results of the previous snippets. To begin with, click on Cell / Run All in the menu to execute the entire notebook. You can later double click on any cell and modify its contents, and press Shift+Enter to reevaluate its contents. This will only modify the current cell, so you will have to Run All again to see the effects on the whole notebook. While editing longer snippets, you can also press Shift+Tab while typing to get more documentation hints about the code you are writing.This course is lectured by Anil Madhavapeddy and Jon Ludlam. These notes are translated from Lawrence C. Paulson’s earlier course on Standard ML, which had credits to David Allsopp, Stuart Becker, Gavin Bierman, Chloë Brown, Silas Brown, Qi Chen, David Cottingham, William Denman, Robert Harle, Daniel Hulme, Frank King, Jack Lawrence-Jones, Joseph Lord, Dimitrios Los, Farhan Mannan, James Margetson, David Morgan, Alan Mycroft, Sridhar Prabhu, Frank Stajano, Alex Trifanov, Thomas Tuerk, Xincheng Wang, Philip Withnall and Assel Zhiyenbayeva for pointing out errors. The current notes were ported to OCaml in 2019 by Anil Madhavapeddy, David Allsopp, and Jon Ludlam and subsequently edited by Jeremy Yallop. We thank Richard Sharp, Srinivasan Keshav, Ambroise Lafont, Vojtěch Tvrdík, Jeremy Yallop, Alexandra W. Chadwick, Richard Mortier and Alex Nelson for further feedback and corrections since 2020.Some books that are complementary to this course are:OCaml from the Very Beginning by John Whitington.Anil Madhavapeddyhttps://www.cl.cam.ac.uk/teaching/2627/FoundsCS/forest/focs-intro/focs-intro/teaching/2627/FoundsCS/forest/focs-intro/Introduction to ProgrammingLecturehttps://hub.cl.cam.ac.uk/Anil Madhavapeddyhttps://www.cl.cam.ac.uk/teaching/2627/FoundsCS/forest/focs-basic-concepts/focs-basic-concepts/teaching/2627/FoundsCS/forest/focs-basic-concepts/Basic Concepts in Computer ScienceComputers: a child can use them; nobody can fully understand them! We can master complexity through levels of abstraction. Focus on 2 or 3 levels at most!Recurring issues:what services to provide at each level how to implement them using lower-level services the interface that defines how the two levels should communicateA basic concept in computer science is that large systems can only be understood in levels, with each level further subdivided into functions or services of some sort. The interface to the higher level should supply the advertised services. Just as important, it should block access to the means by which those services are implemented. This abstraction barrier allows one level to be changed without affecting levels above. For example, when a manufacturer designs a faster version of a processor, it is essential that existing programs continue to run on it. Any differences between the old and new processors should be invisible to the program.Modern processors have elaborate specifications, which still sometimes leave out important details. In the old days, you then had to consult the circuit diagrams. Anil Madhavapeddyhttps://www.cl.cam.ac.uk/teaching/2627/FoundsCS/forest/focs-dates/focs-dates/teaching/2627/FoundsCS/forest/focs-dates/Example 1: DatesAbstract level: dates over a certain interval Concrete level: could be 6 characters: YYMMDD (where each character is represented by 8 bits) Date crises caused by inadequate internal formats: Digital’s PDP-10: using 12-bit dates (good for at most 11 years) 2000 crisis: 48 bits could be good for lifetime of universe!Digital Equipment Corporation’s date crisis occurred in 1975. The PDP-10 was a 36-bit mainframe computer. It represented dates using a 12-bit format designed for the tiny PDP-8. With 12 bits, one can distinguish days or 11 years.Last century, the most common industry format for dates used six characters: two for the year, two for the month and two for the day. The most common “solution” to the year 2000 crisis is to add two further characters, thereby altering file sizes. Others have noticed that the existing six characters consist of 48 bits, already sufficient to represent all dates over the projected lifetime of the universe: = days = years!Mathematicians think in terms of unbounded ranges, but the representation we choose for the computer usually imposes hard limits. A good programming language like OCaml lets one easily change the representation used in the program. But if files in the old representation exist all over the place, there will still be conversion problems. The need for compatibility with older systems causes problems across the computer industry. Anil Madhavapeddyhttps://www.cl.cam.ac.uk/teaching/2627/FoundsCS/forest/focs-floating-point/focs-floating-point/teaching/2627/FoundsCS/forest/focs-floating-point/Example 2: Floating Point NumbersComputers have integers like 1066 and floats like . A floating-point number is represented by two integers. The concept of data type involves:how a value is represented inside the computer the suite of operations given to programmers valid and invalid (or exceptional) results, such as “infinity”Computer arithmetic can yield incorrect answers!In science, numbers written with finite precision and a decimal exponent are said to be in standard form. The computational equivalent is the floating point number. These are familiar to anybody who has used a scientific calculator. Internally, a float consists of two integers.Because of its finite precision, floating-point computations are potentially inaccurate. To see an example, use your nearest electronic calculator to compute . I get ! With certain computations, the errors spiral out of control. Many programming languages fail to check whether even integer computations fall within the allowed range: you can add two positive integers and get a negative one!Most computers give us a choice of precisions. In 32-bit precision, integers typically range from (namely 2 147 483 647) to ; floats are accurate to about six decimal places and can get as large as or so. For floats, 64-bit precision is often preferred. Early languages like Fortran required variables to be declared as INTEGER, REAL or COMPLEX and barred programmers from mixing numbers in a computation. Nowadays, programs handle many different kinds of data, including text and symbols. The concept of a data type can ensure that different types of data are not combined in a senseless way.Inside the computer, all data are stored as bits. In most programming languages, the compiler uses types to generate correct machine code, and types are not stored during program execution. In this course, we focus almost entirely on programming in a high-level language: OCaml. Anil Madhavapeddyhttps://www.cl.cam.ac.uk/teaching/2627/FoundsCS/forest/focs-goals-of-programming/focs-goals-of-programming/teaching/2627/FoundsCS/forest/focs-goals-of-programming/Goals of Programmingto describe a computation so that it can be done mechanically: Expressions compute values. Commands cause effects. to do so efficiently and correctly, giving the right answers quickly to allow easy modification as needs change Through an orderly structure based on abstraction principles Such as modules or classesProgramming in-the-small concerns the writing of code to do simple, clearly defined tasks. Programs provide expressions for describing mathematical formulae and so forth. This was the original contribution of FORTRAN, the FORmula TRANslator. Commands describe how control should flow from one part of the program to the next.As we code layer upon layer, we eventually find ourselves programming in the large : joining large modules to solve some messy task. Programming languages have used various mechanisms to allow one part of the program to provide interfaces to other parts. Modules encapsulate a body of code, allowing outside access only through a programmer-defined interface. Abstract Data Types are a simpler version of this concept, which implement a single concept such as dates or floating-point numbers.Object-oriented programming is the most complicated approach to modularity. Classes define concepts, and they can be built upon other classes. Operations can be defined that work in appropriately specialised ways on a family of related classes. Objects are instances of classes and hold the data that is being manipulated.This course does not cover OCaml’s sophisticated module system, which can do many of the same things as classes. You will learn all about objects when you study Java. OCaml includes a powerful object system, although this is not used as much as its module system.Anil Madhavapeddyhttps://www.cl.cam.ac.uk/teaching/2627/FoundsCS/forest/focs-why-ocaml/focs-why-ocaml/teaching/2627/FoundsCS/forest/focs-why-ocaml/Why Program in OCaml?Why program in OCaml at all?It is interactive. It has a flexible notion of data type. It hides the underlying hardware: no crashes. Programs can easily be understood mathematically. It distinguishes naming something from updating memory. It manages storage for us.Programming languages matter. They affect the reliability, security, and efficiency of the code you write, as well as how easy it is to read, refactor, and extend. The languages you know can also change how you think, influencing the way you design software even when you’re not using them.What makes OCaml special is that it occupies a sweet spot in the space of programming language designs. It provides a combination of efficiency, expressiveness and practicality that is difficult to find matched by any other language. “ML” was originally the meta language of the LCF (Logic for Computable Functions) proof assistant released by Robin Milner in 1972 (at Stanford, and later at Cambridge). ML was turned into a compiler in order to make it easier to use LCF on different machines, and it was gradually turned into a full-fledged system of its own by the 1980s.The modern OCaml emerged in 1996, and the past twenty five years have seen OCaml attract a significant user base with language improvements being steadily added to support the growing commercial and academic codebases. OCaml is therefore the outcome of years of research into programming languages, and a good base to begin our journey into learning the foundations of computer science.Because of its connection to mathematics, OCaml programs can be designed and understood without thinking in detail about how the computer will run them. Although a program can abort, it cannot crash: it remains under the control of the OCaml system. It still achieves respectable efficiency and provides lower-level primitives for those who need them. Most other languages allow direct access to the underlying machine and even try to execute illegal operations, causing crashes.The only way to learn programming is by writing and running programs. This web notebook provides an interactive environment where you can modify the example fragments and see the results for yourself. You should also consider installing OCaml on your own computer so that you try more advanced programs locally.Anil Madhavapeddyhttps://www.cl.cam.ac.uk/teaching/2627/FoundsCS/forest/focs-first-session/focs-first-session/teaching/2627/FoundsCS/forest/focs-first-session/A first session with OCamlThe first line of this simple session is a value declaration. It makes the name pi stand for the floating point number 3.14159. (Such names are called identifiers.) OCaml echoes the name (pi) and type (float) of the declared identifier.The second line computes the area of the circle with radius 1.5 using the formula . We use pi as an abbreviation for 3.14159. Multiplication is expressed using *., which is called an infix operator because it is written between its two operands.OCaml replies with the computed value (about 7.07) and its type (again float). float = ]]>To work abstractly, we should provide the service “compute the area of a circle,” so that we no longer need to remember the formula. This sort of encapsulated computation is called a function. The third line declares the function area. Given any floating point number r, it returns another floating point number computed using the area formula; note that the function has type float -> float.The fourth line calls the function area supplying 2.0 as the argument. A circle of radius 2 has an area of about 12.6. Note that brackets around a function argument are not necessary.The function uses pi to stand for 3.14159. Unlike what you may have seen in other programming languages, pi cannot be "assigned to" or otherwise updated. Its meaning within area will persist even if we issue a new let declaration for pi afterwards.Anil Madhavapeddyhttps://www.cl.cam.ac.uk/teaching/2627/FoundsCS/forest/focs-npower/focs-npower/teaching/2627/FoundsCS/forest/focs-npower/Raising a Number to a Power int -> float = ]]>Our new npower definition can now take additional arguments, reflected in the arrows present in the type of npower; these represent parameters that can be passed to the new value being defined, with the final segment being the resulting type. Thus our npower type can be read as "pass in a float and integer to return a float".Mathematical Justification (for ):The function npower raises its float argument x to the power n, a non-negative integer. The function is recursive: it calls itself. You can spot a recursive function due to the rec keyword in the definition: this indicates that any invocation of the function name within the function body should call itself. This concept should be familiar from mathematics, since exponentiation is defined by the rules shown above. You may also have seen recursion in the product rule for differentiation: . In finding the derivative of , we recursively find the derivatives of and , combining them to obtain the desired result. The recursion is meaningful because it terminates: we reduce the problem to two smaller problems, and this cannot go on forever. The OCaml programmer uses recursion heavily. For , the equation yields an obvious computation:The equation clearly holds even for negative . However, the corresponding computation runs forever:Note that the function npower contains both an integer constant (0) and a floating point constant (1.0). The decimal point makes all the difference. OCaml will notice and ascribe different meaning to each type of constant. float = ]]>Now for a tiresome but necessary aside. In most languages, the types of arguments and results must always be specified. OCaml is unusual that it normally infers the types itself. However, sometimes it is useful to supply a hint to help you debug and develop your program. OCaml will still infer the types even if you don’t specify them, but in some cases it will use a more inefficient function than a specialised one. Some languages have just one type of number, converting automatically between different formats; this is slow and could lead to unexpected rounding errors. Type constraints are allowed almost anywhere. We can put one on any occurrence of x in the function. float = ]]>Or we can constrain the type of the function’s result: float = ]]>OCaml treats the equality and comparison test specially. Expressions like if x = y then … are allowed provided x and y have the same type and equality testing is possible for that type. (We discuss equality further in a later lecture.) Note that x <> y is OCaml for .A characteristic feature of the computer is its ability to test for conditions and act accordingly. In the early days, a program might jump to a given address depending on the sign of some number. Later, John McCarthy defined the conditional expression to satisfy if true then x else y = x and if false then x else y = y.OCaml evaluates the expression if then else by first evaluating . If the result is true then OCaml evaluates and otherwise . Only one of the two expressions and is evaluated! If both were evaluated, then recursive functions like npower above would run forever.The if-expression is governed by an expression of type bool, whose two values are true and false. In modern programming languages, tests are not built into “conditional branch” constructs but can just be part of normal expressions. Tests, or Boolean expressions, can be expressed using relational operators such as < and =. They can be combined using the Boolean operators for negation (not), conjunction (written as &&) and disjunction (written as ||). New properties can be declared as functions: here, to test whether an integer is even, for example: bool = ]]>Anil Madhavapeddyhttps://www.cl.cam.ac.uk/teaching/2627/FoundsCS/forest/focs-power/focs-power/teaching/2627/FoundsCS/forest/focs-power/Efficiently Raising a Number to a Power int -> float = ]]>Mathematical JustificationFor large n, computing powers using is too slow to be practical. The equations above are much faster. Example:Instead of n multiplications, we need at most multiplications, where is the logarithm of to the base .We use the function even, declared previously, to test whether the exponent is even. Integer division (/) truncates its result to an integer: dividing by 2 yields .A recurrence is a useful computation rule only if it is bound to terminate. If 0]]> then is smaller than both and . After enough recursive calls, the exponent will be reduced to . The equations also hold if , but the corresponding computation runs forever.Our reasoning assumes arithmetic to be exact. Fortunately, the calculation is well-behaved using floating-point.Computer numbers have a finite range, which if exceeded results in the integer wrapping around. You will understand this behaviour more as you learn about computer architecture and how modern systems represent numbers in memory.If integers and floats must be combined in a calculation, OCaml provides functions to convert between them:OCaml’s libraries are organised using “modules”, so we may use compound identifiers such as Float.of_int to refer to library functions. There are many thousands of library functions available in the OCaml ecosystem, including text-processing and operating systems functions in addition to the usual numerical ones.Anil Madhavapeddyhttps://www.cl.cam.ac.uk/teaching/2627/FoundsCS/forest/focs-ex-1/focs-ex-1/teaching/2627/FoundsCS/forest/focs-ex-1/ExercisesAnil MadhavapeddyAnil Madhavapeddyhttps://www.cl.cam.ac.uk/teaching/2627/FoundsCS/forest/focs-ex-1-1/focs-ex-1-1/teaching/2627/FoundsCS/forest/focs-ex-1-1/Two-digit year windowingExerciseOne solution to the year 2000 bug involves storing years as two digits, but interpreting them such that 50 means 1950 and 49 means 2049. Comment on the merits and demerits of this approach.Anil MadhavapeddyAnil Madhavapeddyhttps://www.cl.cam.ac.uk/teaching/2627/FoundsCS/forest/focs-ex-1-2/focs-ex-1-2/teaching/2627/FoundsCS/forest/focs-ex-1-2/Comparing and adding yearsExerciseUsing the date representation of the previous exercise, code OCaml functions to (a) compare two years (b) add/subtract some given number of years from another year.Anil MadhavapeddyAnil Madhavapeddyhttps://www.cl.cam.ac.uk/teaching/2627/FoundsCS/forest/focs-ex-1-3/focs-ex-1-3/teaching/2627/FoundsCS/forest/focs-ex-1-3/Redundant conditionalsExerciseWhy would no experienced programmer write an expression of the form if … then true else false? What about expressions of the form if … then false else true?Anil MadhavapeddyAnil Madhavapeddyhttps://www.cl.cam.ac.uk/teaching/2627/FoundsCS/forest/focs-ex-1-4/focs-ex-1-4/teaching/2627/FoundsCS/forest/focs-ex-1-4/Type inference in powerExerciseFunctions npower and power both return a float. The definition of npower returns the float value 1.0 in its base case. The definition of power does not, so how does the OCaml type checker know that power returns a float?Anil MadhavapeddyAnil Madhavapeddyhttps://www.cl.cam.ac.uk/teaching/2627/FoundsCS/forest/focs-ex-1-5/focs-ex-1-5/teaching/2627/FoundsCS/forest/focs-ex-1-5/Floating-point error by repeated additionExerciseBecause computer arithmetic is based on binary numbers, simple decimals such as 0.1 often cannot be represented exactly. Write a function mul that performs the computationwhere has type float. (It is essential to use repeated addition rather than multiplication!)The value computed with n = 10000 and x = 0.1 may print as 1000.0, which looks exact. If that happens, then evaluate the expression mul 0.1 10000 -. 1000.0An error of this type has been blamed for the failure of an American Patriot Missile battery to intercept an incoming Iraqi missile during the first Gulf War. The missile hit an American Army barracks, killing 28.Anil MadhavapeddyAnil Madhavapeddyhttps://www.cl.cam.ac.uk/teaching/2627/FoundsCS/forest/focs-ex-1-6/focs-ex-1-6/teaching/2627/FoundsCS/forest/focs-ex-1-6/Golden ratio instabilityExerciseAnother example of the inaccuracy of floating-point arithmetic takes the golden ratio as its starting point:In theory, it is easy to prove that for all 0]]>. Code this computation in OCaml and report the value of . Hint: in OCaml, is expressed as sqrt 5.0.Anil Madhavapeddyhttps://www.cl.cam.ac.uk/teaching/2627/FoundsCS/forest/focs-recursion/focs-recursion/teaching/2627/FoundsCS/forest/focs-recursion/Recursion and EfficiencyLecturehttps://hub.cl.cam.ac.uk/Anil Madhavapeddyhttps://www.cl.cam.ac.uk/teaching/2627/FoundsCS/forest/focs-expression-evaluation/focs-expression-evaluation/teaching/2627/FoundsCS/forest/focs-expression-evaluation/Expression EvaluationExpression evaluation concerns expressions and the values they return. This view of computation may seem to be too narrow. It is certainly far removed from computer hardware, but that can be seen as an advantage. For the traditional concept of computing solutions to problems, expression evaluation is entirely adequate.Starting with , the expression is reduced to until this process concludes with a value . A value is something like a number that cannot be further reduced.We write to say that is reduced to . Mathematically, they are equal: , but the computation goes from to and never the other way around.Computers also interact with the outside world. For a start, they need some means of accepting problems and delivering solutions. Many computer systems monitor and control industrial processes. This role of computers is familiar now, but was never envisaged in the early days. Computer pioneers focused on mathematical calculations. Modelling interaction and control requires a notion of states that can be observed and changed. Then we can consider updating the state by assigning to variables or performing input/output, finally arriving at conventional programs as coded in C, for instance.For now, we remain at the level of expressions, which is usually termed functional programming.Anil Madhavapeddyhttps://www.cl.cam.ac.uk/teaching/2627/FoundsCS/forest/focs-nsum/focs-nsum/teaching/2627/FoundsCS/forest/focs-nsum/Summing the first n integers int = ]]>The function call nsum n computes the sum 1 + … + n rather naively, hence the initial n in its name:The nesting of parentheses is not just an artifact of our notation; it indicates a real problem. The function gathers up a collection of numbers, but none of the additions can be performed until nsum 0 is reached. Meanwhile, the computer must store the numbers in an internal data structure, typically the stack. For large n, say nsum 10000, the computation might fail due to stack overflow.We all know that the additions can be performed as we go along. How do we make the computer do that?Anil Madhavapeddyhttps://www.cl.cam.ac.uk/teaching/2627/FoundsCS/forest/focs-summing-iter/focs-summing-iter/teaching/2627/FoundsCS/forest/focs-summing-iter/Iteratively summing the first n integers int -> int = ]]>Function summing takes an additional argument: a running total. If n is zero then it returns the running total; otherwise, summing adds to it and continues. The recursive calls do not nest; the additions are done immediately.A recursive function whose computation does not nest is called iterative or tail-recursive. Many functions can be made iterative by introducing an argument analogous to total, which is often called an accumulator.The gain in efficiency is sometimes worthwhile and sometimes not. The function power is not iterative because nesting occurs whenever the exponent is odd. Adding a third argument makes it iterative, but the change complicates the function and the gain in efficiency is minute; for 32-bit integers, the maximum possible nesting is 30 for the exponent .Anil Madhavapeddyhttps://www.cl.cam.ac.uk/teaching/2627/FoundsCS/forest/focs-recursion-vs-iteration/focs-recursion-vs-iteration/teaching/2627/FoundsCS/forest/focs-recursion-vs-iteration/Recursion vs Iteration“Iterative” normally refers to a loop, coded using while for example (see the final lecture) Tail-recursion is only efficient if the compiler detects it Mainly it saves space (memory), though iterative code can also run faster Do not make programs iterative unless the gain is worth itA classic book by Abelson and Sussman, which describes the Lisp dialect known as Scheme, used iterative to mean tail-recursive. Iterative functions produce computations resembling those that can be done using while-loops in conventional languages.Many algorithms can be expressed naturally using recursion, but only awkwardly using iteration. There is a story that Dijkstra sneaked recursion into Algol-60 by inserting the words “any other occurrence of the procedure name denotes execution of the procedure.” By not using the word “recursion”, he managed to slip this amendment past sceptical colleagues.Obsession with tail recursion leads to a coding style in which functions have many more arguments than necessary. Write straightforward code first, avoiding only gross inefficiency. If the program turns out to be too slow, tools are available for pinpointing the cause. Always remember KISS (Keep It Simple, Stupid).I hope you have all noticed by now that the summation can be done even more efficiently using the arithmetic progression formula:Anil Madhavapeddyhttps://www.cl.cam.ac.uk/teaching/2627/FoundsCS/forest/focs-sillysum/focs-sillysum/teaching/2627/FoundsCS/forest/focs-sillysum/Silly Summing the First n Integers int = ]]>The function calls itself times! Bigger inputs mean higher costs—but what’s the growth rate?Now let us consider how to estimate various costs associated with a program. Asymptotic complexity refers to how costs—usually time or space—grow with increasing inputs. Space complexity can never exceed time complexity, for it takes time to do anything with the space. Time complexity often greatly exceeds space complexity.The function sillySum calls itself twice in each recursive step. This function is contrived, but many mathematical formulas refer to a particular quantity more than once. In OCaml, we can create a local binding to a computed value using the local declaration syntax. In the following expression, y is computed once and used twice:You can read let x = e1 in e2 as assigning (or "binding") the name x with the value of e1 into e2. Any use of x within e2 will have the value of e1, and x will only be visible in subexpressions into which it has been bound.Why do we need let bindings? Fast hardware does not make good algorithms unnecessary. On the contrary, faster hardware magnifies the superiority of better algorithms. Typically, we want to handle the largest inputs possible. If we double our processing power, what do we gain? How much can we increase , the input to our function?With sillySum, we can only go from to . We are limited to this modest increase because the function’s running time is proportional to . With the function npower defined in the previous section, we can go from to : we can handle problems twice as big. With power we can do much better still, going from to .The following table (excerpted from a 50-year-old book!) illustrates the effect of various time complexities. The left-hand column (dubbed "complexity") is defined as how many milliseconds are required to process an input of size . The other entries show the maximum size of that can be processed in the given time (one second, minute or hour). complexity1 second1 minute1 hourgain 100060 0003 600 000 1404 895204 095 312441 897 1039153 91521 The table illustrates how large an input can be processed as a function of time. As we increase the computer time per input from one second to one minute and then to one hour, the size of the input increases accordingly.The top two rows (complexities and ) increase rapidly: for , by a factor of 60 per column. The bottom two start out close together, but (which grows by a factor of 3.9) pulls well away from (whose growth is only additive). If an algorithm’s complexity is exponential then it can never handle large inputs, even if it is given huge resources. On the other hand, suppose the complexity has the form , where is a constant. (We say the complexity is polynomial.) Doubling the argument then increases the cost by a constant factor. That is much better, though if 3]]> the algorithm may not be considered practical.Anil Madhavapeddyhttps://www.cl.cam.ac.uk/teaching/2627/FoundsCS/forest/focs-o-notation/focs-o-notation/teaching/2627/FoundsCS/forest/focs-o-notation/Comparing Algorithms: O NotationFormally, define provided as is bounded for some constant and all sufficiently large . Intuitively, look at the most significant term. Ignore constant factors as they seldom dominate and are often transitoryFor example: consider instead of .The cost of a program is usually a complicated formula. Often we should consider only the most significant term. If the cost is for an input of size , then the term will eventually dominate, even though is bigger for . The constant term may look big, but it is soon dominated by .Constant factors in costs can be ignored unless they are large. For one thing, they seldom make a difference: will be better than in the long run: or asymptotically to use the jargon. Moreover, constant factors are seldom stable. They depend upon details such as which hardware, operating system or programming language is being used. By ignoring constant factors, we can make comparisons between algorithms that remain valid in a broad range of circumstances.The “Big O” notation is commonly used to describe efficiency—to be precise, asymptotic complexity. It concerns the limit of a function as its argument tends to infinity. It is an abstraction that meets the informal criteria that we have just discussed. In the definition, sufficiently large means there is some constant such that for all greater than . The role of is to ignore finitely many exceptions to the bound, such as the cases when exceeds .Anil Madhavapeddyhttps://www.cl.cam.ac.uk/teaching/2627/FoundsCS/forest/focs-o-notation-facts/focs-o-notation-facts/teaching/2627/FoundsCS/forest/focs-o-notation-facts/Simple Facts About O Notation notation lets us reason about the costs of algorithms easily.Constant factors such as the in drop out: we can use with twice the value of in the definition. Because constant factors drop out, the base of logarithms is irrelevant. Insignificant terms drop out. To see that is the same as , consider that converges to 1 for increasing . In fact, for , so can double the constant factorIf and are constants (that is, they are independent of ) with then is contained in is contained in is contained in To say that is contained in means that the former gives a tighter bound than the latter. For example, if then trivially, but the converse does not hold.Anil Madhavapeddyhttps://www.cl.cam.ac.uk/teaching/2627/FoundsCS/forest/focs-complexity-classes/focs-complexity-classes/teaching/2627/FoundsCS/forest/focs-complexity-classes/Common Complexity Classes is constant is logarithmic is linear is quasi-linear is quadratic is cubic is exponential (for fixed )Logarithms grow very slowly, so complexity is excellent. Because notation ignores constant factors, the base of the logarithm is irrelevant!Under linear we might mention , which occasionally is called quasilinear and which scales up well for large .An example of quadratic complexity is matrix addition: forming the sum of two matrices obviously takes additions. Matrix multiplication is of cubic complexity, which limits the size of matrices that we can multiply in reasonable time. An algorithm exists, but it is too complicated to be of much use, even though it is theoretically better.An exponential growth rate such as restricts us to small values of . Already with the cost exceeds one million. However, the worst case might not arise in normal practice. OCaml type-checking is exponential in the worst case, but not for ordinary programs.Anil Madhavapeddyhttps://www.cl.cam.ac.uk/teaching/2627/FoundsCS/forest/focs-sample-costs/focs-sample-costs/teaching/2627/FoundsCS/forest/focs-sample-costs/Sample costs in O notationRecall that npower computes by repeated multiplication while nsum naively computes the sum . Each obviously performs arithmetic operations. Because they are not tail recursive, their use of space is also . The function summing is a version of nsum with an accumulating argument; its iterative behaviour lets it work in constant space. notation spares us from having to specify the units used to measure space. FunctionTimeSpace npower, nsumO()O() summingO()O() O()O() powerO()O() sillySumO()O() Even ignoring constant factors, the units chosen can influence the result. Multiplication may be regarded as a single unit of cost. However, the cost of multiplying two -digit numbers for large is itself an important question, especially now that public-key cryptography uses numbers hundreds of digits long.Few things can really be done in constant time or stored in constant space. Merely to store the number requires bits. If a program cost is , then we have probably assumed that certain operations it performs are also —typically because we expect never to exceed the capacity of the standard hardware arithmetic.With power, the precise number of operations depends upon in a complicated way, depending on how many odd numbers arise, so it is convenient that we can just write . An accumulating argument could reduce its space cost to .Anil Madhavapeddyhttps://www.cl.cam.ac.uk/teaching/2627/FoundsCS/forest/focs-recurrences/focs-recurrences/teaching/2627/FoundsCS/forest/focs-recurrences/Some Simple Recurrence RelationsConsider a function that has a cost we want to bound using notation. A typical base case is . Some recurrences are: EquationComplexity To analyse a function, inspect its OCaml declaration. Recurrence equations for the cost function can usually be read off. Since we ignore constant factors, we can give the base case a cost of one unit. Constant work done in the recursive step can also be given unit cost; since we only need an upper bound, this unit represents the larger of the two actual costs. We could use other constants if it simplifies the algebra.For example, recall our function nsum: int = ]]>Given , it performs a constant amount of work (an addition and subtraction) and calls itself recursively with argument . We get the recurrence equations and . The closed form is clearly , as we can easily verify by substitution. The cost is linear.This function, given , calls nsum, performing work. Again ignoring constant factors, we can say that this call takes exactly units. int = ]]>We get the recurrence equations and . It is easy to see that . The cost is quadratic.The function power divides its input into two, with the recurrence equation . Clearly , so .Anil Madhavapeddyhttps://www.cl.cam.ac.uk/teaching/2627/FoundsCS/forest/focs-ex-2/focs-ex-2/teaching/2627/FoundsCS/forest/focs-ex-2/ExercisesAnil MadhavapeddyAnil Madhavapeddyhttps://www.cl.cam.ac.uk/teaching/2627/FoundsCS/forest/focs-ex-2-1/focs-ex-2-1/teaching/2627/FoundsCS/forest/focs-ex-2-1/Iterative powerExerciseCode an iterative version of the function power.Anil MadhavapeddyAnil Madhavapeddyhttps://www.cl.cam.ac.uk/teaching/2627/FoundsCS/forest/focs-ex-2-2/focs-ex-2-2/teaching/2627/FoundsCS/forest/focs-ex-2-2/Extending the complexity tableExerciseAdd a column to the table of complexities from The Design and Analysis of Computer Algorithms with the heading 60 hours: complexity1 second1 minute1 hour60 hours 100060 0003 600 000 1404 895204 095 312441 897 1039153 91521 Anil MadhavapeddyAnil Madhavapeddyhttps://www.cl.cam.ac.uk/teaching/2627/FoundsCS/forest/focs-ex-2-3/focs-ex-2-3/teaching/2627/FoundsCS/forest/focs-ex-2-3/Simplifying sums in O notationExerciseLet , …, be functions such that for , …, and all sufficiently large .Show that if then .Anil MadhavapeddyAnil Madhavapeddyhttps://www.cl.cam.ac.uk/teaching/2627/FoundsCS/forest/focs-ex-2-4/focs-ex-2-4/teaching/2627/FoundsCS/forest/focs-ex-2-4/A tighter recurrence boundExerciseFind an upper bound for the recurrence given by and . You should be able to find a tighter bound than .Anil Madhavapeddyhttps://www.cl.cam.ac.uk/teaching/2627/FoundsCS/forest/focs-lists/focs-lists/teaching/2627/FoundsCS/forest/focs-lists/ListsLecturehttps://hub.cl.cam.ac.uk/A list is an ordered series of elements; repetitions are significant. So [3; 5; 9] differs from [5; 3; 9] and from [3; 3; 5; 9]. Elements in the list are separated with ; when constructed, as opposed to the , syntax used for fixed-length tuples.All elements of a list must have the same type. Above we see a list of integers and a list of (integer, string) pairs. One can also have lists of lists, such as [[3]; []; [5; 6]], which has type int list list.In the general case, if all have the same type (say ) then the list has type .Lists are the simplest data structure that can be used to process collections of items. Conventional languages use arrays whose elements are accessed using subscripting: for example, yields the th element of the array . Subscripting errors are a known cause of programmer grief, however, so arrays should be replaced by higher-level data structures whenever possible.The infix operator @ (also called List.append) concatenates two lists. Also built-in is List.rev, which reverses a list. These are demonstrated in the session above.Anil Madhavapeddyhttps://www.cl.cam.ac.uk/teaching/2627/FoundsCS/forest/focs-list-primitives/focs-list-primitives/teaching/2627/FoundsCS/forest/focs-list-primitives/The List PrimitivesThere are two kinds of lists:[] represents the empty list x :: l is the list with head and tail The operator :: (also called List.cons for “construct”), puts a new element on to the head of an existing list. While we should not be too preoccupied with implementation details, it is essential to know that :: is an operation. It uses constant time and space, regardless of the length of the resulting list. Lists are represented internally with a linked structure; adding a new element to a list merely hooks the new element to the front of the existing structure. Moreover, that structure continues to denote the same list as it did before; to see the new list, one must look at the new :: node (or “cons cell”) just created. We will explain the 'a notation in the next section.Here we see the element 1 being consed to the front of the list [3; 5; 9]:Given a list, taking its first element (its “head”) or its list of remaining elements (its “tail”) also takes constant time. Each operation just follows a link. In the diagram above, the first down arrow leads to the head and the leftmost right arrow leads to the tail. Once we have the tail, its head is the second element of the original list, etc.The tail is not the last element; it is the list of all elements other than the head!Anil Madhavapeddyhttps://www.cl.cam.ac.uk/teaching/2627/FoundsCS/forest/focs-head-tail/focs-head-tail/teaching/2627/FoundsCS/forest/focs-head-tail/Getting at the Head and Tail true | x :: l -> false val null : 'a list -> bool = # null [] - : bool = true # null [1; 2; 3] - : bool = false # let hd (x::l) = x Line 1, characters 8-14: Warning 8 [partial-match]: this pattern-matching is not exhaustive. Here is an example of a case that is not matched: [] val hd : 'a list -> 'a = # hd [1; 2; 3] - : int = 1 # let tl (x::l) = l Line 1, characters 8-14: Warning 8 [partial-match]: this pattern-matching is not exhaustive. Here is an example of a case that is not matched: [] val tl : 'a list -> 'a list = # tl [7; 6; 5] - : int list = [6; 5]]]>The empty list has neither head nor tail. Applying hd or tl to [] is an error—strictly speaking, an “exception”. The function null can be used to check for the empty list beforehand. Taking a list apart using combinations of hd and tl is hard to get right. Fortunately, it is seldom necessary because of pattern-matching.The declaration of null introduces a new concept known as "pattern matching", which we will explore more in subsequent lectures. For now, it is sufficient to observe that let null = function allows for matching on the two possible values that might be passed in as argument to null here: one for the empty list (for which it returns true) and one for non-empty lists (for which it returns false).The declaration of hd above has only one clause, for non-empty lists. They have the form x::l and the function returns x, which is the head. If you compile this program, OCaml also prints a warning to tell us that calling the function could raise an exception because not all possible inputs are handled, including a counter-example (in this case, the empty list []). The declaration of tl is similar to hd.These three primitive functions are polymorphic and allow flexibility in the types of their arguments and results. Note their types! bool = # hd - : 'a list -> 'a = # tl - : 'a list -> 'a list = ]]>Symbols 'a and 'b are called type variables and stand for any types. Code written using these functions is checked for type correctness at compile time. And this guarantees strong properties at run time, for example that the elements of any list all have the same type. They are usually read as their corresponding greek characters; 'a is "alpha", 'b is "beta", and so on.Anil Madhavapeddyhttps://www.cl.cam.ac.uk/teaching/2627/FoundsCS/forest/focs-nlength/focs-nlength/teaching/2627/FoundsCS/forest/focs-nlength/Computing the Length of a List 0 | x :: xs -> 1 + nlength xs val nlength : 'a list -> int = # nlength [] - : int = 0 # nlength [5; 6; 7] - : int = 3]]>Most list processing involves recursion. This is a simple example; patterns can be more complex. Observe the use of a vertical bar | to separate the function’s clauses. We have one function declaration that handles two cases. To understand its role, consider the following faulty code: int = # let rec nlength (x::xs) = 1 + nlength xs Line 1, characters 17-24: Warning 8 [partial-match]: this pattern-matching is not exhaustive. Here is an example of a case that is not matched: [] val nlength : 'a list -> int = ]]>These are two declarations, not one. First we declare nlength to be a function that handles only empty lists. Then we redeclare it to be a function that handles only non-empty lists; it can never deliver a result. We see that a second let declaration replaces any previous one rather than extending it to cover new cases.Now, let us return to our original declaration of nlength. The length function is polymorphic and applies to all lists regardless of element type! Most programming languages lack such flexibility.Unfortunately, this length computation is naive and wasteful. Like nsum earlier, it is not tail-recursive. It uses space, where is the length of its input. As usual, the solution is to add an accumulating argument.Anil Madhavapeddyhttps://www.cl.cam.ac.uk/teaching/2627/FoundsCS/forest/focs-length/focs-length/teaching/2627/FoundsCS/forest/focs-length/Efficiently Computing the Length of a List n | x::xs -> addlen (n + 1) xs val addlen : int -> 'a list -> int = # addlen 0 [5; 6; 7] - : int = 3]]>Recall that the use of function introduces an extra (unnamed) argument that is pattern matched in the subsequent clauses; in this case, to break open the list.Function addlen is again polymorphic. Its type mentions the integer accumulator.Now we may declare an efficient length function. It is simply a wrapper for addlen, supplying zero as the initial value of . int = # length [5; 6; 7; 8] - : int = 4]]>The recursive calls do not nest: this version is iterative. It takes space. Obviously its time requirement is because it takes at least steps to find the length of an -element list.Anil Madhavapeddyhttps://www.cl.cam.ac.uk/teaching/2627/FoundsCS/forest/focs-append/focs-append/teaching/2627/FoundsCS/forest/focs-append/Append: List Concatenation ys | x::xs, ys -> x :: append xs ys val append : 'a list -> 'a list -> 'a list = # append [1; 2; 3] [4] - : int list = [1; 2; 3; 4] # let (@) = append val ( @ ) : 'a list -> 'a list -> 'a list = # [1; 2; 3] @ [4] - : int list = [1; 2; 3; 4]]]>Patterns can be as complicated as we like. Here, the two patterns are [], ys and x::xs, ys.Here is how append might be declared, also noting that we have defined @ as an infix operator that is a more convenient way to call append on two lists. However, this function is also not iterative. It scans its first argument, sets up a string of cons operations (::) and finally does them.It uses space and time, where is the length of its first argument. Its costs are independent of its second argument.An accumulating argument could make it iterative, but with considerable complication. The iterative version would still require space and time because concatenation requires copying all the elements of the first list. Therefore, we cannot hope for asymptotic gains; at best we can decrease the constant factor involved in , but complicating the code is likely to increase that factor. Never add an accumulator merely out of habit.Note append’s polymorphic type. It tells us that two lists can be joined if their element types agree.Anil Madhavapeddyhttps://www.cl.cam.ac.uk/teaching/2627/FoundsCS/forest/focs-nrev/focs-nrev/teaching/2627/FoundsCS/forest/focs-nrev/Reversing a List in Let us consider one way to reverse a list. [] | x::xs -> (nrev xs) @ [x] val nrev : 'a list -> 'a list = # nrev [1; 2; 3] - : int list = [3; 2; 1]]]>This reverse function is grossly inefficient due to poor usage of append, which copies its first argument. If nrev is given a list of length 0]]>, then append makes conses to copy the reversed tail. Constructing the list [x] calls cons again, for a total of calls. Reversing the tail requires more conses, and so forth. The total number of conses is:The time complexity is therefore . Space complexity is only because the copies don’t all exist at the same time.Anil Madhavapeddyhttps://www.cl.cam.ac.uk/teaching/2627/FoundsCS/forest/focs-rev/focs-rev/teaching/2627/FoundsCS/forest/focs-rev/Reversing a List in ys | x::xs, ys -> rev_app xs (x::ys) val rev_app : 'a list -> 'a list -> 'a list = ]]>Calling rev_app xs ys reverses the elements of xs and prepends them to ys. Now we may declare 'a list = # rev [1; 2; 3] - : int list = [3; 2; 1]]]>It is easy to see that this reverse function performs just conses, given an -element list. For both reverse functions, we could count the number of conses precisely—not just up to a constant factor. notation is still useful to describe the overall running time: the time taken by a cons varies from one system to another.The accumulator makes the function iterative. But the gain in complexity arises from the removal of append. Replacing an expensive operation (append) by a series of cheap operations (cons) is called reduction in strength and is a common technique in computer science. It originated when many computers did not have a hardware multiply instruction; the series of products for , could more efficiently be computed by repeated addition. Reduction in strength can be done in various ways; we shall see many instances of removing append.Consing to an accumulator produces the result in reverse. If that forces the use of an extra list reversal then the iterative function may be much slower than the recursive one.Anil Madhavapeddyhttps://www.cl.cam.ac.uk/teaching/2627/FoundsCS/forest/focs-strings-chars/focs-strings-chars/teaching/2627/FoundsCS/forest/focs-strings-chars/Lists, Strings and CharactersStrings are provided in most programming languages to allow text processing. Strings are essential for communication with users. Even a purely numerical program formats its results ultimately as strings.In a few programming languages, strings simply are lists of characters. In OCaml they are a separate type, unrelated to lists, reflecting the fact that strings are an abstract concept in themselves.Similarly, characters are not strings of size one, but are a primitive concept. Character constants in OCaml have the form 'c', where is any character. For example, the comma character is ','.Special characters are coded in strings using escape sequences involving the backslash character; among many others, a double quote is written and the newline character is written . For example, the string represents three text lines.In addition to the operators described above, the relations <, <=, >, and >= work for strings and yield alphabetic order (more precisely, lexicographic order with respect to ASCII character codes).Anil Madhavapeddyhttps://www.cl.cam.ac.uk/teaching/2627/FoundsCS/forest/focs-ex-3/focs-ex-3/teaching/2627/FoundsCS/forest/focs-ex-3/ExercisesAnil MadhavapeddyAnil Madhavapeddyhttps://www.cl.cam.ac.uk/teaching/2627/FoundsCS/forest/focs-ex-3-1/focs-ex-3-1/teaching/2627/FoundsCS/forest/focs-ex-3-1/Summing a listExerciseCode a recursive function to compute the sum of a list’s elements. Then code an iterative version and comment on the improvement in efficiency.Anil MadhavapeddyAnil Madhavapeddyhttps://www.cl.cam.ac.uk/teaching/2627/FoundsCS/forest/focs-ex-3-2/focs-ex-3-2/teaching/2627/FoundsCS/forest/focs-ex-3-2/Last element of a listExerciseCode a function to return the last element of a non-empty list. How efficiently can this be done?Anil MadhavapeddyAnil Madhavapeddyhttps://www.cl.cam.ac.uk/teaching/2627/FoundsCS/forest/focs-ex-3-3/focs-ex-3-3/teaching/2627/FoundsCS/forest/focs-ex-3-3/Even-numbered elementsExerciseCode a function to return the list consisting of the even-numbered elements of the list given as its argument. For example, given [a; b; c; d] it should return [b; d].Anil MadhavapeddyAnil Madhavapeddyhttps://www.cl.cam.ac.uk/teaching/2627/FoundsCS/forest/focs-ex-3-4/focs-ex-3-4/teaching/2627/FoundsCS/forest/focs-ex-3-4/Polymorphic types of id and loopExerciseConsider the polymorphic types in these two function declarations: 'a = # let rec loop x = loop x val loop : 'a -> 'b = ]]>Explain why these types make logical sense, preventing run time type errors, even for expressions like id [id [id 0]] or loop true / loop 3. (/ is the integer division operator in OCaml)Anil MadhavapeddyAnil Madhavapeddyhttps://www.cl.cam.ac.uk/teaching/2627/FoundsCS/forest/focs-ex-3-5/focs-ex-3-5/teaching/2627/FoundsCS/forest/focs-ex-3-5/Tails of a listExerciseCode a function tails to return the list of the tails of its argument. For example, given [1; 2; 3] it should return [[1; 2; 3]; [2; 3]; [3]; []].Anil Madhavapeddyhttps://www.cl.cam.ac.uk/teaching/2627/FoundsCS/forest/focs-more-lists/focs-more-lists/teaching/2627/FoundsCS/forest/focs-more-lists/More on ListsLecturehttps://hub.cl.cam.ac.uk/Anil Madhavapeddyhttps://www.cl.cam.ac.uk/teaching/2627/FoundsCS/forest/focs-take-drop/focs-take-drop/teaching/2627/FoundsCS/forest/focs-take-drop/List Utilities: take and dropThis lecture examines more list utilities, illustrating more patterns of recursion, and concludes with a small program for making change.The functions take and drop divide a list into parts, returning or discarding the first elements.They can be implemented in OCaml as follows: [] | x::xs -> if i > 0 then x :: take (i - 1) xs else [] val take : int -> 'a list -> 'a list = # let rec drop i = function | [] -> [] | x::xs -> if i > 0 then drop (i-1) xs else x::xs val drop : int -> 'a list -> 'a list = ]]>Applications of take and drop will appear in future lectures. Typically, they divide a collection of items into equal parts for recursive processing.The take function is not iterative, but making it so would not improve its efficiency. The task requires copying up to list elements, which must take space and time.Function drop simply skips over list elements. This requires time but only constant space. It is iterative and much faster than take. Both functions use time, but skipping elements is faster than copying them: drop’s constant factor is smaller.Both functions take an integer and a list, returning a list of the same type. So their type is int -> 'a list -> 'a list.Anil Madhavapeddyhttps://www.cl.cam.ac.uk/teaching/2627/FoundsCS/forest/focs-linear-search/focs-linear-search/teaching/2627/FoundsCS/forest/focs-linear-search/Linear Searchfind in list by comparing with each element obviously time simple & general ordered searching needs only indexed lookup needs only Linear search is the obvious way to find a desired item in a collection: simply look through all the items, one at a time. If is in the list, then it will be found in steps on average, and even the worst case is obviously .Large collections of data are usually ordered or indexed so that items can be found in time, which is exponentially better than . Even is achievable (using a hash table), though subject to the usual proviso that machine limits are not exceeded.Efficient indexing methods are of prime importance: consider Web search engines. Nevertheless, linear search is often used to search small collections because it is so simple and general, and it is the starting point for better algorithms.Anil Madhavapeddyhttps://www.cl.cam.ac.uk/teaching/2627/FoundsCS/forest/focs-equality/focs-equality/teaching/2627/FoundsCS/forest/focs-equality/Equality Tests false | y::l -> if x = y then true else member x l val member : 'a -> 'a list -> bool = ]]>All the list functions we have encountered up to now have been “polymorphic”, working for lists of any type. Function member uses linear search to report whether or not x occurs in l.To do this generically, it uses a special feature of OCaml known as “polymorphic equality”, which manifests itself via the =, >=, <=, > and < operators. These operators inspect the structure of the values using a consistent order. Types you can legitimately compare this way include integers, strings, booleans, and tuples or lists of primitive types.More complex types can be compared this way within careful limits: recursive structures or function values will not work (we will cover function values in the Currying lecture later). For now, it is sufficient to use these magic polymorphic equality operators. As you get more familiar with OCaml and the use of higher order functions (also covered in a later lecture), you will encounter the use of explicit compare functions that are used to provide more complex equality tests.The presence of polymorphic equality is a contentious feature in OCaml. While it provides a great ease of use in smaller codebases, it starts to become more dangerous when building larger OCaml-based systems. Most large-scale users of OCaml tend towards not using it in important code, but it is just fine for our purposes while learning the beginning steps of computer science.Anil Madhavapeddyhttps://www.cl.cam.ac.uk/teaching/2627/FoundsCS/forest/focs-zip/focs-zip/teaching/2627/FoundsCS/forest/focs-zip/Building a List of Pairs (x, y) :: zip xs ys | _ -> [] val zip : 'a list -> 'b list -> ('a * 'b) list = ]]>The wildcard pattern _ matches anything. We could have written a variable such as p instead, but the wildcard reminds us that the relevant clause ignores this argument.The patterns are also tested in order of their definitions: first (x::xs, y::ys), then _.A list of pairs of the form associates each with . Conceptually, a telephone directory could be regarded as such a list, where ranges over names and over the corresponding telephone number. Linear search in such a list can find the associated with a given , or vice versa—very slowly.In other cases, the pairs might have been generated by applying a function to the elements of another list . ([], []) | (x, y)::pairs -> let xs, ys = unzip pairs in (x::xs, y::ys) val unzip : ('a * 'b) list -> 'a list * 'b list = ]]>Given a list of pairs, unzip has to build two lists of results, which is awkward using recursion. The version shown above uses the local binding let p = in , where the value of is bound to the variable p within . The let-construct counts as an expression and can be used (perhaps wrapped within parentheses) wherever an expression is expected.Note especially the phrase let xs, ys = unzip pairs which binds xs and ys to the results of the recursive call. In general, the phrase let P = E matches the pattern against the value of expression . It binds all the variables in to the corresponding values.The functions zip and unzip build and take apart lists of pairs: zip pairs up corresponding list elements and unzip inverts this operation. Their types reflect what they do: 'b list -> ('a * 'b) list = # unzip - : ('a * 'b) list -> 'a list * 'b list = ]]>If the lists are of unequal length, zip discards surplus items at the end of the longer list. Its first pattern only matches a pair of non-empty lists. The second pattern is just a wildcard and could match anything. OCaml tries the clauses in the order given, so the first pattern is tried first. The second only gets arguments where at least one of the lists is empty.Anil Madhavapeddyhttps://www.cl.cam.ac.uk/teaching/2627/FoundsCS/forest/focs-unzip/focs-unzip/teaching/2627/FoundsCS/forest/focs-unzip/Building a Pair of ResultsHere is a version of unzip that replaces the local declaration by a function conspair for taking apart the pair of lists in the recursive call. It defines the same computation as the previous version of unzip and is possibly clearer, but not every local binding can be eliminated as easily. 'a list * 'b list = # let rec unzip = function | [] -> ([], []) | xy :: pairs -> conspair (xy, unzip pairs) val unzip : ('a * 'b) list -> 'a list * 'b list = ]]>Making the function iterative yields revUnzip below, which is very simple. Iteration can construct many results at once in different argument positions. Both output lists are built in reverse order, which can be corrected by reversing the input to revUnzip. The total costs will probably exceed those of unzip despite the advantages of iteration. (xs, ys) | ((x, y)::pairs, xs, ys) -> revUnzip (pairs, x::xs, y::ys) val revUnzip : ('a * 'b) list * 'a list * 'b list -> 'a list * 'b list = ]]>Anil Madhavapeddyhttps://www.cl.cam.ac.uk/teaching/2627/FoundsCS/forest/focs-making-change/focs-making-change/teaching/2627/FoundsCS/forest/focs-making-change/An Application: Making ChangeConsider a till that has unlimited supplies of coins. The largest coins should be tried first, to avoid giving change all in pennies. The list of legal coin values, called till, is given in descending order, such as 50, 20, 10, 5, 2 and 1. (Recall that the head of a list is the element most easily reached.) The code for change is based on simple observations:Change for zero consists of no coins at all. (Note the pattern of 0 in the first clause.) For a nonzero amount, try the largest available coin. If it is small enough, use it and decrease the amount accordingly. Exclude from consideration any coins that are too large. [] | [], _ -> raise (Failure "no more coins!") | c::till, amt -> if amt < c then change till amt else c :: change (c::till) (amt - c) val change : int list -> int -> int list = ]]>Although nobody considers making change for zero, this is the simplest way to make the algorithm terminate. Most iterative procedures become simplest if, in their base case, they do nothing. A base case of one instead of zero is often a sign of a novice programmer.The recursion terminates when amt = 0. Tries the largest coin first to use large coins. The algorithm is greedy and can fail!The function can terminate either with success or failure. It fails by raising exception Failure namely if till becomes empty while amt is still nonzero. (Exceptions will be discussed later.)Unfortunately, failure can occur even when change can be made. The greedy "largest coin first" approach is to blame. Suppose we have coins of values 5 and 2, and must make change for 6; the only way is , ignoring the 5. Greedy algorithms are often effective, but not here.Anil Madhavapeddyhttps://www.cl.cam.ac.uk/teaching/2627/FoundsCS/forest/focs-change-all/focs-change-all/teaching/2627/FoundsCS/forest/focs-change-all/All Ways of Making ChangeNow we generalise the problem to return the list of all possible ways of making change, and write a new change function. [ [] ] | [] , _ -> [] | c::till , amt -> if amt < c then change till amt else let rec allc = function | [] -> [] | cs :: css -> (c::cs) :: allc css in allc (change (c::till) (amt - c)) @ change till amt val change : int list -> int -> int list list = ]]>Look at the type: the result is now a list of lists. The code will also never raise exceptions. It expresses failure by returning an empty list of solutions: it returns [] if the till is empty and the amount is nonzero.If the amount is zero, then there is only one way of making change; the result should be . This is success in the base case.In nontrivial cases, there are two sources of solutions: to use a coin (if possible) and decrease the amount accordingly, or to remove the current coin value from consideration.The function allc is declared locally in order to make use of c, the current coin. It adds an extra c to all the solutions returned by the recursive call to make change for amt - c.Observe the naming convention: cs is a list of coins, while css is a list of such lists. The trailing ‘s’ is suggestive of a plural.This complicated program, and the even trickier faster version, are included as challenges. Are you enthusiastic enough to work them out? We shall revisit the “making change” task later to illustrate exception-handling.Anil Madhavapeddyhttps://www.cl.cam.ac.uk/teaching/2627/FoundsCS/forest/focs-change-faster/focs-change-faster/teaching/2627/FoundsCS/forest/focs-change-faster/All Ways of Making Change — Faster! chg::chgs | [] , _ -> chgs | c::till , amt -> if amt < 0 then chgs else change (c::till) (amt - c) (c::chg) (change till amt chg chgs) val change : int list -> int -> int list -> int list list -> int list list = ]]>We’ve added another accumulating parameter! Repeatedly improving simple code is called stepwise refinement.Two extra arguments eliminate many :: and append operations from the all-solutions change function. The first, chg, accumulates the coins chosen so far; one evaluation of c::chg replaces many evaluations of allc. The second, chgs, accumulates the list of solutions so far; it avoids the need for append. This version runs several times faster than the previous one.Making change is still extremely slow for an obvious reason: the number of solutions grows rapidly in the amount being changed. Using 50, 20, 10, 5, 2 and 1, there are 4366 ways of expressing 99.Our three change functions illustrate a basic technique: program development by stepwise refinement. Begin by writing a very simple program and add requirements individually. Add efficiency refinements last of all. Even if the simpler program cannot be included in the next version and has to be discarded, one has learned about the task by writing it.Anil Madhavapeddyhttps://www.cl.cam.ac.uk/teaching/2627/FoundsCS/forest/focs-ex-4/focs-ex-4/teaching/2627/FoundsCS/forest/focs-ex-4/ExercisesAnil MadhavapeddyAnil Madhavapeddyhttps://www.cl.cam.ac.uk/teaching/2627/FoundsCS/forest/focs-ex-4-1/focs-ex-4-1/teaching/2627/FoundsCS/forest/focs-ex-4-1/Set union without repetitionsExerciseSets can be represented in OCaml using lists containing no duplicated items (i.e. where no item is equal to another using polymorphic comparison).Using the member function defined above, code a function to implement set union. It should avoid introducing repetitions, for example the union of the lists [4; 7; 1] and [6; 4; 7] should be [1; 6; 4; 7] (though the order does not matter).Anil MadhavapeddyAnil Madhavapeddyhttps://www.cl.cam.ac.uk/teaching/2627/FoundsCS/forest/focs-ex-4-2/focs-ex-4-2/teaching/2627/FoundsCS/forest/focs-ex-4-2/Partitioning integers by signExerciseCode a function that takes a list of integers and returns two lists, the first consisting of all non-negative numbers found in the input and the second consisting of all the negative numbers.Anil MadhavapeddyAnil Madhavapeddyhttps://www.cl.cam.ac.uk/teaching/2627/FoundsCS/forest/focs-ex-4-3/focs-ex-4-3/teaching/2627/FoundsCS/forest/focs-ex-4-3/A non-exhaustive zipExerciseHow does this version of zip differ from the one above? (x, y) :: zip xs ys | ([], []) -> []]]>Anil MadhavapeddyAnil Madhavapeddyhttps://www.cl.cam.ac.uk/teaching/2627/FoundsCS/forest/focs-ex-4-4/focs-ex-4-4/teaching/2627/FoundsCS/forest/focs-ex-4-4/Assumptions in making changeExerciseWhat assumptions do the ‘making change’ functions make about the variables till and amt? Describe what could happen if these assumptions were violated.Anil MadhavapeddyAnil Madhavapeddyhttps://www.cl.cam.ac.uk/teaching/2627/FoundsCS/forest/focs-ex-4-5/focs-ex-4-5/teaching/2627/FoundsCS/forest/focs-ex-4-5/Counting ways of making changeExerciseShow that the number of ways of making change for (ignoring order) is if there are two legal coin values. What if there are three, four, … coin values?Anil MadhavapeddyAnil Madhavapeddyhttps://www.cl.cam.ac.uk/teaching/2627/FoundsCS/forest/focs-ex-4-6/focs-ex-4-6/teaching/2627/FoundsCS/forest/focs-ex-4-6/Reasoning from polymorphic typesExerciseWe know nothing about the functions f and g other than their polymorphic types: val f : 'a * 'b -> 'b * 'a and val g : 'a -> 'a list. Suppose that f (1, true) and g 0 are evaluated and return their results. State, with reasons, what you think the resulting values will be.Anil Madhavapeddyhttps://www.cl.cam.ac.uk/teaching/2627/FoundsCS/forest/focs-sorting/focs-sorting/teaching/2627/FoundsCS/forest/focs-sorting/SortingLecturehttps://hub.cl.cam.ac.uk/A few applications for sorting and arranging items into order are:search merging duplicates inverting tables graphics algorithmsSorting is perhaps the most deeply studied aspect of algorithm design. Knuth’s series The Art of Computer Programming devotes an entire volume to sorting and searching! Sedgewick also covers sorting. Sorting has countless applications.Sorting a collection allows items to be found quickly. Recall that linear search requires steps to search among items. A sorted collection admits binary search which requires only time. The idea of binary search is to compare the item being sought with the middle item (in position ) and then to discard either the left half or the right, depending on the result of the comparison. Binary search needs arrays or trees, not lists; we shall come to binary search trees later.Two sorted files can quickly be merged to form a larger sorted file. Other applications include finding duplicates that, after sorting, are adjacent.A telephone directory is sorted alphabetically by name. The same information can instead be sorted by telephone number (useful to the police) or by street address (useful to junk-mail firms). Sorting information in different ways gives it different applications.Common sorting algorithms include insertion sort, quicksort, mergesort and heapsort. We shall consider the first three of these. Each algorithm has its advantages.As a concrete basis for comparison, runtimes are quoted for DECstation computers. These were based on the MIPS chip (an early RISC design) and are really old now, but still useful to look at relative to each other.Anil Madhavapeddyhttps://www.cl.cam.ac.uk/teaching/2627/FoundsCS/forest/focs-sorting-bounds/focs-sorting-bounds/teaching/2627/FoundsCS/forest/focs-sorting-bounds/How Fast Can We Sort?typically count comparisons there are permutations of elements each comparison eliminates half of the permutations therefore The usual measure of efficiency for sorting algorithms is the number of comparison operations required. Mergesort requires only comparisons to sort an input of items. It is straightforward to prove that this complexity is the best possible. There are permutations of elements and each comparison distinguishes two permutations. The lower bound on the number of comparisons, , is obtained by solving ; therefore .In order to compare the sorting algorithms, we use the following source of pseudo-random numbers. Never mind how this works: generating statistically good random numbers is hard. Much effort has gone into those few lines of code. float = # let rec randlist (seed, seeds) = function | 0 -> (seed, seeds) | n -> randlist (nextrandom seed, seed::seeds) (n-1) val randlist : float * float list -> int -> float * float list = ]]>We can now bind the identifier rs to a list of 10,000 random numbers.Anil Madhavapeddyhttps://www.cl.cam.ac.uk/teaching/2627/FoundsCS/forest/focs-insertion-sort/focs-insertion-sort/teaching/2627/FoundsCS/forest/focs-insertion-sort/Insertion SortAn insert operation does comparisons on average. [x] | y::ys -> if x <= y then x :: y :: ys else y :: ins x ys val ins : 'a -> 'a list -> 'a list = ]]>Insertion sort takes comparisons on average: [] | x::xs -> ins x (insort xs)]]>Items from the input are copied one at a time to the output. Each new item is inserted into the right place so that the output is always in order.We could easily write iterative versions of these functions, but to no purpose. Insertion sort is slow because it does comparisons (and a lot of list copying), not because it is recursive. Its quadratic runtime makes it nearly useless: it takes 174 seconds for our example while the next-worst figure is 1.4 seconds.Insertion sort is worth considering because it is easy to code and illustrates the concepts. Two efficient sorting algorithms, mergesort and heapsort, can be regarded as refinements of insertion sort.Anil Madhavapeddyhttps://www.cl.cam.ac.uk/teaching/2627/FoundsCS/forest/focs-quicksort-idea/focs-quicksort-idea/teaching/2627/FoundsCS/forest/focs-quicksort-idea/Quicksort: The IdeaThe Quicksort algorithm has the following flow:Choose a pivot element, Divide to partition the input into two sublists: those at most in value those exceeding Conquer using recursive calls to sort the sublists Combine the sorted lists by appending one to the otherQuicksort was invented by Sir Anthony Hoare (1934–2026), who in later life worked at Microsoft Research, Cambridge. Quicksort works by divide and conquer, a basic algorithm design principle. Quicksort chooses from the input some value , called the pivot. It partitions the remaining items into two parts: those , and those a]]>. It sorts each part recursively, then puts the smaller part before the greater.The cleverest feature of Hoare's algorithm was that the partition could be done in place by exchanging array elements. Quicksort was invented before recursion was well known, and people found it extremely hard to understand. As usual, we shall consider a list version based on functional programming.Anil Madhavapeddyhttps://www.cl.cam.ac.uk/teaching/2627/FoundsCS/forest/focs-quicksort/focs-quicksort/teaching/2627/FoundsCS/forest/focs-quicksort/Quicksort: The Code [] | [x] -> [x] | a::bs -> let rec part l r = function | [] -> (quick l) @ (a :: quick r) | x::xs -> if (x <= a) then part (x::l) r xs else part l (x::r) xs in part [] [] bs val quick : 'a list -> 'a list = ]]>Our OCaml quicksort copies the items. It is still pretty fast, and it is much easier to understand. It takes roughly 0.74 seconds to sort our list of random numbers.The function declaration consists of three clauses. The first handles the empty list; the second handles singleton lists (those of the form [x]); the third handles lists of two or more elements. Often, lists of length up to five or so are treated as special cases to boost speed.The locally declared function part partitions the input using a as the pivot. The arguments l and r accumulate items for the left () and right (a]]>) parts of the input, respectively.It is not hard to prove that quicksort does comparisons, in the average case (see page 94 of Aho). With random data, the pivot usually has an average value that divides the input in two approximately equal parts. We have the recurrence and , which is . In our example, it is about 235 times faster than insertion sort.In the worst case, quicksort's running time is quadratic! An example is when its input is almost sorted or reverse sorted. Nearly all of the items end up in one partition; work is not divided evenly. We have the recurrence and , which is . Randomising the input makes the worst case highly unlikely.Anil Madhavapeddyhttps://www.cl.cam.ac.uk/teaching/2627/FoundsCS/forest/focs-quicksort-append-free/focs-quicksort-append-free/teaching/2627/FoundsCS/forest/focs-quicksort-append-free/Append-Free Quicksort sorted | ([x], sorted) -> x::sorted | a::bs, sorted -> let rec part = function | l, r, [] -> quik (l, a :: quik (r, sorted)) | l, r, x::xs -> if x <= a then part (x::l, r, xs) else part (l, x::r, xs) in part ([], [], bs) val quik : 'a list * 'a list -> 'a list = ]]>The list sorted accumulates the result in the combine stage of the quicksort algorithm. We have again used the standard technique for eliminating append. Calling quik(xs, sorted) reverses the elements of xs and prepends them to the list sorted.Looking closely at part, observe that quik(r, sorted) is performed first. Then a is consed to this sorted list. Finally, quik is called again to sort the elements of l.The speedup is significant. An imperative quicksort coded in Pascal (taken from Sedgewick) is just slightly faster than function quik. The near-agreement is surprising because the computational overheads of lists exceed those of arrays. In realistic applications, comparisons are the dominant cost and the overheads matter even less.Anil Madhavapeddyhttps://www.cl.cam.ac.uk/teaching/2627/FoundsCS/forest/focs-merge/focs-merge/teaching/2627/FoundsCS/forest/focs-merge/Merging Two ListsMerge joins two sorted lists. ys | xs, [] -> xs | x::xs, y::ys -> if x <= y then x :: merge (xs, y::ys) else y :: merge (x::xs, ys) val merge : 'a list * 'a list -> 'a list = ]]>Generalises insert to two lists, and does at most comparisons.Merging means combining two sorted lists to form a larger sorted list. It does at most comparisons, where and are the lengths of the input lists. If and are roughly equal then we have a fast way of constructing sorted lists; if then merging degenerates to insertion, doing much work for little gain.Merging is the basis of several sorting algorithms; we look at a divide-and-conquer one. Mergesort is seldom found in conventional programming because it is hard to code for arrays; it works nicely with lists. It divides the input (if non-trivial) into two roughly equal parts, sorts them recursively, then merges them.Function merge is not iterative; the recursion is deep. An iterative version is of little benefit for the same reasons that apply to append in the earlier lecture on Lists.Anil Madhavapeddyhttps://www.cl.cam.ac.uk/teaching/2627/FoundsCS/forest/focs-merge-sort/focs-merge-sort/teaching/2627/FoundsCS/forest/focs-merge-sort/Top-down Merge sort [] | [x] -> [x] | xs -> let k = List.length xs / 2 in let l = tmergesort (take k xs) in let r = tmergesort (drop k xs) in merge (l, r) val tmergesort : 'a list -> 'a list = ]]> comparisons in worst caseMergesort’s divide stage divides the input not by choosing a pivot (as in quicksort) but by simply counting out half of the elements. The conquer stage again involves recursive calls, and the combine stage involves merging. Function tmergesort takes roughly 1.4 seconds to sort the list rs.In the worst case, mergesort does comparisons, with the same recurrence equation as in quicksort’s average case. Because take and drop divide the input in two equal parts (they differ at most by one element), we always have .Quicksort is nearly 3 times as fast in the example. But it risks a quadratic worst case! Merge sort is safe but slow. So which algorithm is best?We have seen a top-down mergesort. Bottom-up algorithms also exist. They start with a list of one-element lists and repeatedly merge adjacent lists until only one is left. A refinement, which exploits any initial order among the input, is to start with a list of increasing or decreasing runs of input items.Anil Madhavapeddyhttps://www.cl.cam.ac.uk/teaching/2627/FoundsCS/forest/focs-sorting-summary/focs-sorting-summary/teaching/2627/FoundsCS/forest/focs-sorting-summary/Summary of Sorting AlgorithmsOptimal is comparisons Insertion sort: simple to code; too slow (quadratic) [174 secs] Quicksort: fast on average; quadratic in worst case [0.53 secs] Mergesort: optimal in theory; often slower than quicksort [1.4 secs] Match the algorithm to the applicationQuicksort’s worst case cannot be ignored. For large , a complexity of is catastrophic. Mergesort has an worst case running time, which is optimal, but it is typically slower than quicksort for random data.Non-comparison sorting deserves mentioning. We can sort a large number of small integers using their radix representation in time. This result does not contradict the comparison-counting argument because comparisons are not used at all. Linear time is achievable only if the greatest integer is fixed in advance; as goes to infinity, increasingly many of the items are the same. It is a simple special case.Many other sorting algorithms exist. A few are outlined in the exercises.Anil Madhavapeddyhttps://www.cl.cam.ac.uk/teaching/2627/FoundsCS/forest/focs-ex-5/focs-ex-5/teaching/2627/FoundsCS/forest/focs-ex-5/ExercisesAnil MadhavapeddyAnil Madhavapeddyhttps://www.cl.cam.ac.uk/teaching/2627/FoundsCS/forest/focs-ex-5-1/focs-ex-5-1/teaching/2627/FoundsCS/forest/focs-ex-5-1/Complexity of selection sortExerciseAnother sorting algorithm (selection sort) consists of looking at the elements to be sorted, identifying and removing a minimal element, which is placed at the head of the result. The tail is obtained by recursively sorting the remaining elements. State, with justification, the time complexity of this approach.Anil MadhavapeddyAnil Madhavapeddyhttps://www.cl.cam.ac.uk/teaching/2627/FoundsCS/forest/focs-ex-5-2/focs-ex-5-2/teaching/2627/FoundsCS/forest/focs-ex-5-2/Implementing selection sortExerciseImplement selection sort (see previous exercise) using OCaml.Anil MadhavapeddyAnil Madhavapeddyhttps://www.cl.cam.ac.uk/teaching/2627/FoundsCS/forest/focs-ex-5-3/focs-ex-5-3/teaching/2627/FoundsCS/forest/focs-ex-5-3/Complexity of bubble sortExerciseAnother sorting algorithm (bubble sort) consists of looking at adjacent pairs of elements, exchanging them if they are out of order and repeating this process until no more exchanges are possible. State, with justification, the time complexity of this approach.Anil MadhavapeddyAnil Madhavapeddyhttps://www.cl.cam.ac.uk/teaching/2627/FoundsCS/forest/focs-ex-5-4/focs-ex-5-4/teaching/2627/FoundsCS/forest/focs-ex-5-4/Implementing bubble sortExerciseImplement bubble sort (see previous exercise) using OCaml.Anil Madhavapeddyhttps://www.cl.cam.ac.uk/teaching/2627/FoundsCS/forest/focs-datatypes/focs-datatypes/teaching/2627/FoundsCS/forest/focs-datatypes/Datatypes and TreesLecturehttps://hub.cl.cam.ac.uk/Anil Madhavapeddyhttps://www.cl.cam.ac.uk/teaching/2627/FoundsCS/forest/focs-enumeration/focs-enumeration/teaching/2627/FoundsCS/forest/focs-enumeration/An Enumeration TypeWe will now learn how to define more expressive types than the basic ones supplied with the core OCaml language.We have declared a new type named vehicle. along with four new constants. They are the constructors of the datatype.The type declaration adds a new type to our OCaml session. Type vehicle is as good as any built-in type and even admits pattern-matching (as we used with the built-in list types earlier). The four new identifiers of type vehicle are called constructors.We could represent the various vehicles by the numbers 0–3. However, the code would be hard to read and even harder to maintain. Consider adding Tricycle as a new vehicle. If we wanted to add it before Bike, then all the numbers would have to be changed. Using type, such additions are trivial and the compiler can (at least sometimes) warn us when it encounters a function declaration that doesn’t yet have a case for Tricycle.Representing vehicles by strings like "Bike", "Car", etc., is also bad. Comparing string values is slow and the compiler can’t warn us of misspellings like "MOtorbike": they will make our code fail.Most programming languages allow the declaration of types like vehicle. Because they consist of a series of identifiers, they are called enumeration types. Other common examples are days of the week or colours. The compiler chooses the integers for us; type-checking prevents us from confusing Bike with Red or Sunday.Anil Madhavapeddyhttps://www.cl.cam.ac.uk/teaching/2627/FoundsCS/forest/focs-vehicles/focs-vehicles/teaching/2627/FoundsCS/forest/focs-vehicles/Declaring a Function on Vehicles 2 | Motorbike -> 2 | Car -> 4 | Lorry -> 18 val wheels : vehicle -> int = ]]>Datatype constructors can be used in patterns. Pattern-matching is fast, even complicated nested patterns. Notice the type of the argument is vehicle, which we defined earlier.The beauty of datatype declarations is that the new types behave as if they were built into OCaml. Type-checking catches common errors, such as mixing up different datatypes in a function like wheels, as well as missing and redundant patterns.Anil Madhavapeddyhttps://www.cl.cam.ac.uk/teaching/2627/FoundsCS/forest/focs-constructors-args/focs-constructors-args/teaching/2627/FoundsCS/forest/focs-constructors-args/A Datatype whose Constructors have ArgumentsConstructors with arguments (like Lorry) are distinct values. (So Car true is distinct from Car false). Different kinds of vehicle can belong to one list: [Bike; Car true; Motorbike 450]OCaml generalises the notion of enumeration type to allow data to be associated with each constructor. The constructor Bike is a vehicle all by itself, but the other three constructors create vehicles from arguments.Since we might find it hard to remember what the various int and bool components are for, it is wise to include comments in complex declarations. In OCaml, comments are enclosed in the brackets and . Programmers should comment their code to explain design decisions and key features of the algorithms (sometimes by citing a reference work).The list [Bike; Car true; Motorbike 450] represents a bicycle, a Reliant Robin and a large motorbike. It can be almost seen as a mixed-type list containing integers and booleans. It is actually a list of vehicles; datatypes lessen the impact of the restriction that all list elements must have the same type.Anil Madhavapeddyhttps://www.cl.cam.ac.uk/teaching/2627/FoundsCS/forest/focs-wheels/focs-wheels/teaching/2627/FoundsCS/forest/focs-wheels/A Finer Wheel ComputationWe now define a wheels function to calculate the number of wheels in any vehicle. This requires pattern matching to retrieve the constructors and their associated arguments, around which we build the logic: 2 | Motorbike _ -> 2 | Car robin -> if robin then 3 else 4 | Lorry w -> w val wheels : vehicle -> int = ]]>This function consists of four clauses:A Bike has two wheels. This is a constant result. A Motorbike has two wheels. The _ signifies a "wildcard" pattern match that we discard, since the engine size of the bike is not relevant to the number of wheels. A Reliant Robin has three wheels; all other cars have four. We bind robin to the bool argument and then use it in the right hand side of the pattern match, much like a let binding in normal code. A Lorry has the number of wheels stored with its constructor, and we simply return that.There is no overlap between the Motorbike and Lorry cases. Although Motorbike and Lorry both hold an integer, OCaml takes the constructor into account and keeps any Motorbike distinct from any Lorry.Vehicles are one example of a concept consisting of several varieties with distinct features. Most programming languages can represent such concepts using something analogous to datatypes. (They are sometimes called union types or variant records whose tag fields play the role of the constructors.)A pattern may be built from the constructors of several datatypes, including lists. A pattern may also contain integer and string constants. There is no limit to the size of patterns or the number of clauses in a function declaration. OCaml performs pattern-matching efficiently (you do not need to understand the details of how it optimises pattern-matching at this stage).Anil Madhavapeddyhttps://www.cl.cam.ac.uk/teaching/2627/FoundsCS/forest/focs-exceptions/focs-exceptions/teaching/2627/FoundsCS/forest/focs-exceptions/Error Handling: ExceptionsDuring a computation, what happens if something goes wrong?Division by zero Pattern matching failureException-handling lets us recover gracefully.Raising an exception abandons the current computation. Handling the exception attempts an alternative computation. The raising and handling can be far apart in the code. Errors of different sorts can be handled separately.Exceptions are necessary because it is not always possible to tell in advance whether or not a search will lead to a dead end or whether a numerical calculation will encounter errors such as overflow or divide by zero. Rather than just crashing, programs should check whether things have gone wrong, and perhaps attempt an alternative computation (perhaps using a different algorithm or higher precision). A number of modern languages provide exception handling.Anil Madhavapeddyhttps://www.cl.cam.ac.uk/teaching/2627/FoundsCS/forest/focs-ocaml-exceptions/focs-ocaml-exceptions/teaching/2627/FoundsCS/forest/focs-ocaml-exceptions/Exceptions in OCamlEach exception declaration introduces a distinct sort of exception, which can be handled separately from others. If raises an exception, then its evaluation has failed; handling an exception means evaluating another expression and returning its value instead. One exception handler can specify separate expressions for different sorts of exceptions.Exception names are constructors of the special datatype exn. This is a peculiarity of OCaml that lets exception-handlers use pattern-matching. Note that exception Failure is just an error indication, while NoChange n carries further information: the integer . print_endline "handled a NoChange exception" Line 3, characters 5-23: Warning 21 [nonreturning-statement]: this statement never returns (or has an unsound type.) pre exception handled a NoChange exception - : unit = ()]]>The effect of raise <expr> is to jump to the most recently-encountered handler that matches <expr>. The matching handler can only be found dynamically (during execution); contrast with how OCaml associates occurrences of identifiers with their matching declarations, which does not require running the program. A handler is introduced via the try keyword, which executes the subexpression and dispatches any exceptions encountered to the corresponding pattern match for exceptions defined in the with block.This is also the first time that we have encountered the unit type. This represents a type that has no values, and is used to indicate that a block has no meaningful return value. We will come back to this when learning more about imperative programming later on. For now, it is sufficient to understand that print_endline will print out the argument to the console output, and return a unit type. The semicolon at the end of the expression is a convenient way to execute sequential statements that return the unit type.One criticism of OCaml’s exceptions is that—unlike the Java language—nothing in a function declaration indicates which exceptions it might raise. One alternative to exceptions is to instead return a value of datatype option.None signifies an error, while Some x returns the solution . This approach looks clean, but the drawback is that many places in the code would have to check for None. Despite this, there is a builtin option type in OCaml as it is so useful. We will see in later lectures how to define our own version of option using polymorphic datatype definitions.Anil Madhavapeddyhttps://www.cl.cam.ac.uk/teaching/2627/FoundsCS/forest/focs-change-exceptions/focs-change-exceptions/teaching/2627/FoundsCS/forest/focs-change-exceptions/Making Change with Exceptions [] | [], _ -> raise Change | c::till, amt -> if amt < 0 then raise Change else try c :: change (c::till) (amt - c) with Change -> change till amt exception Change val change : int list -> int -> int list = ]]>In the Lists lectures, we considered the problem of making change. The greedy algorithm presented there could not express “6 using 5 and 2” because it always took the largest coin. Returning the list of all possible solutions avoids that problem rather expensively: we only need one solution.Using exceptions, we can code a backtracking algorithm: one that can undo past decisions if it comes to a dead end. The exception Change is raised if we run out of coins (with a non-zero amount) or if the amount goes negative. We always try the largest coin, but enclose the recursive call in an exception handler, which undoes the choice if it goes wrong.Carefully observe how exceptions interact with recursion. The exception handler always undoes the most recent choice, leaving others possibly to be undone later. If making change really is impossible, then eventually exception Change will be raised with no handler to catch it, and it will be reported at top level.Anil Madhavapeddyhttps://www.cl.cam.ac.uk/teaching/2627/FoundsCS/forest/focs-change-trace/focs-change-trace/teaching/2627/FoundsCS/forest/focs-change-trace/Making Change: A TraceHere is the full execution. Observe how the exception handlers nest and how they drop away once the given expression has returned a value. change [2] 6}\\ \Rightarrow &\; \text{try 5::(try 5::change [5; 2] (-4)}\\ &\; \text{with Change -> change [2] 1)}\\ &\; \text{with Change -> change [2] 6}\\ \Rightarrow &\; \text{5::(change [2] 1)}\\ &\; \text{with Change -> change [2] 6}\\ \Rightarrow &\; \text{try 5::(try 2::change [2] (-1)}\\ &\; \text{with Change -> change [] 1)}\\ &\; \text{with Change -> change [2] 6} \\ \Rightarrow &\; \text{try 5::(change [] 1)}\\ &\; \text{with Change -> change [2] 6} \\ \Rightarrow &\; \text{change [2] 6} \\ \Rightarrow &\; \text{try 2::change [2] 4}\\ &\; \text{with Change -> change [] 6} \\ \Rightarrow &\; \text{try 2::(try 2::change [2] 2}\\ &\; \text{with Change -> change [] 4)}\\ &\; \text{with Change -> change [] 6} \\ \Rightarrow &\; \text{try 2::(try 2::(try 2::change [2] 0 }\\ &\; \text{with Change -> change [] 2)}\\ &\; \text{with Change -> change [] 4)}\\ &\; \text{with Change -> change [] 6} \\ \Rightarrow &\; \text{try 2::(try 2::[2]}\\ &\; \text{with Change -> change [] 4)}\\ &\; \text{with Change -> change [] 6} \\ \Rightarrow &\; \text{try 2::[2; 2]}\\ &\; \text{with Change -> change [] 6} \\ \Rightarrow &\; \text{[2; 2; 2]} \end{aligned}]]>Anil Madhavapeddyhttps://www.cl.cam.ac.uk/teaching/2627/FoundsCS/forest/focs-binary-trees/focs-binary-trees/teaching/2627/FoundsCS/forest/focs-binary-trees/Binary Trees, a Recursive DatatypeA data structure with multiple branching is called a “tree”. Trees can represent mathematical expressions, logical formulae, computer programs, the phrase structure of English sentences, etc.Binary trees are nearly as fundamental as lists. They can provide efficient storage and retrieval of information. In a binary tree, each node is empty (), or is a branch () with a label and two subtrees.OCaml lists are a datatype and could be declared as follows:We could even declare :: as an infix constructor. The only thing we could not define is the [...] notation, which is part of the OCaml grammar (although there does exist a mechanism to use a similar syntax for custom indexed datatypes).A recursive type does not have to be polymorphic. For example, here is a simple datatype of tree shapes with no attached data that is recursive but not polymorphic.The datatype 'a option (mentioned above) is the opposite – it is polymorphic, but not recursive.Anil Madhavapeddyhttps://www.cl.cam.ac.uk/teaching/2627/FoundsCS/forest/focs-tree-properties/focs-tree-properties/teaching/2627/FoundsCS/forest/focs-tree-properties/Basic Properties of Binary Trees 0 (* number of branch nodes *) | Br (v, t1, t2) -> 1 + count t1 + count t2 val count : 'a tree -> int = # let rec depth = function | Lf -> 0 (* length of longest path *) | Br (v, t1, t2) -> 1 + max (depth t1) (depth t2) val depth : 'a tree -> int = ]]>The invariant holds in the functions above.Functions on trees are expressed recursively using pattern-matching. Both functions above are analogous to on lists. Here is a third measure of a tree’s size: 1 | Br (v, t1, t2) -> leaves t1 + leaves t2 val leaves : 'a tree -> int = ]]>This function is redundant because of a basic fact about trees, which can be proved by induction: for every tree , we have . The invariant stated above also has an elementary proof by induction.A tree of depth 20 can store or approximately one million elements. The access paths to these elements are short, particularly when compared with a million-element list!Anil Madhavapeddyhttps://www.cl.cam.ac.uk/teaching/2627/FoundsCS/forest/focs-ex-6/focs-ex-6/teaching/2627/FoundsCS/forest/focs-ex-6/ExercisesAnil MadhavapeddyAnil Madhavapeddyhttps://www.cl.cam.ac.uk/teaching/2627/FoundsCS/forest/focs-ex-6-1/focs-ex-6-1/teaching/2627/FoundsCS/forest/focs-ex-6-1/Days of the week typeExerciseGive the declaration of an OCaml type for the days of the week. Comment on the practicality of such a type in a calendar application.Anil MadhavapeddyAnil Madhavapeddyhttps://www.cl.cam.ac.uk/teaching/2627/FoundsCS/forest/focs-ex-6-2/focs-ex-6-2/teaching/2627/FoundsCS/forest/focs-ex-6-2/Summing tree labelsExerciseWrite an OCaml function taking a binary tree labelled with integers and returning their sum.Anil MadhavapeddyAnil Madhavapeddyhttps://www.cl.cam.ac.uk/teaching/2627/FoundsCS/forest/focs-ex-6-3/focs-ex-6-3/teaching/2627/FoundsCS/forest/focs-ex-6-3/What ftree buildsExerciseUsing the definition of 'a tree from before:Examine the following function declaration. What does ftree (1, n) accomplish?Anil MadhavapeddyAnil Madhavapeddyhttps://www.cl.cam.ac.uk/teaching/2627/FoundsCS/forest/focs-ex-6-4/focs-ex-6-4/teaching/2627/FoundsCS/forest/focs-ex-6-4/A type for arithmetic expressionsExerciseGive the declaration of an OCaml type for arithmetic expressions that have the following possibilities: floating-point numbers, variables (represented by strings), or expressions of the form , , .Anil MadhavapeddyAnil Madhavapeddyhttps://www.cl.cam.ac.uk/teaching/2627/FoundsCS/forest/focs-ex-6-5/focs-ex-6-5/teaching/2627/FoundsCS/forest/focs-ex-6-5/Evaluating expressions with exceptionsExerciseContinuing the previous exercise, write a function that evaluates an expression. If the expression contains any variables, your function should raise an exception indicating the variable name.Anil Madhavapeddyhttps://www.cl.cam.ac.uk/teaching/2627/FoundsCS/forest/focs-dictionaries/focs-dictionaries/teaching/2627/FoundsCS/forest/focs-dictionaries/Dictionaries and Functional ArraysLecturehttps://hub.cl.cam.ac.uk/Anil Madhavapeddyhttps://www.cl.cam.ac.uk/teaching/2627/FoundsCS/forest/focs-dictionary-adt/focs-dictionary-adt/teaching/2627/FoundsCS/forest/focs-dictionary-adt/Dictionarieslookup: find an item in the dictionary update (insert): replace (store) an item in the dictionary delete: remove an item from the dictionary empty: the null dictionary Missing: exception for errors in lookup and deleteIdeally, an abstract type should provide these operations and hide the internal data structures.A dictionary attaches values to identifiers, called “keys”. Before choosing the internal representation for a data structure, you need to specify the full set of operations. In fact, here we only consider update (associating a value with an identifier) and lookup (retrieving such a value). Deletion is more difficult and would limit our choices. Some applications may need additional operations, such as merge (combining two dictionaries). We shall see that update can be done efficiently in a functional style, without excessive copying.An abstract type provides specified operations while hiding low-level details, such as the data structure used to represent dictionaries. Abstract types can be declared in any modern programming language. Java’s objects serve this role, as do OCaml’s modules. This course does not cover modules, and we simply declare the dictionary operations individually.An association list (a list of pairs) is the simplest dictionary representation. Lookup is by linear search, and therefore slow: . Association lists are only usable if there are few keys in use. However, they are general in that the keys do not need a concept of ordering, only equality. raise Missing | (x, y) :: pairs -> if a = x then y else lookup a pairs val lookup : 'a -> ('a * 'b) list -> 'b = # let update (l, b, y) = (b, y) :: l val update : ('a * 'b) list * 'a * 'b -> ('a * 'b) list = ]]>To enter a new (key, value) pair, simply “cons” it to the list with update. This takes constant time, which is the best we could hope for. But the space requirement is huge: linear in the number of updates, not in the number of distinct keys. Obsolete entries are never deleted: that would require first finding them, increasing the update time from to .Anil Madhavapeddyhttps://www.cl.cam.ac.uk/teaching/2627/FoundsCS/forest/focs-bst/focs-bst/teaching/2627/FoundsCS/forest/focs-bst/Binary Search TreesA dictionary associates values (here, numbers) with keys.Binary search trees are an important application of binary trees. They work for keys that have a total ordering, such as strings. Each branch of the tree carries a pair; its left subtree holds smaller keys; the right subtree holds greater keys. If the tree remains reasonably balanced, then update and lookup both take for a tree of size . These times hold in the average case; given random data, the tree is likely to remain balanced.At a given node, all keys in the left subtree are smaller (or equal) while all trees in the right subtree are greater.An unbalanced tree has a linear access time in the worst case. Examples include building a tree by repeated insertions of elements in increasing or decreasing order; there is a close resemblance to quicksort. Building a binary search tree, then converting it to inorder, yields a sorting algorithm called treesort.Self-balancing trees, such as Red-Black trees, attain in the worst case. They are complicated to implement.Anil Madhavapeddyhttps://www.cl.cam.ac.uk/teaching/2627/FoundsCS/forest/focs-bst-lookup/focs-bst-lookup/teaching/2627/FoundsCS/forest/focs-bst-lookup/Lookup: Seeks Left or Rightfocs-binary-trees if b < a then lookup b t1 else if a < b then lookup b t2 else x | Lf -> raise (Missing b) val lookup : string -> (string * 'a) tree -> 'a = ]]>This has guaranteed access time if the tree is balanced!Lookup in the binary search tree goes to the left subtree if the desired key is smaller than the current one and to the right if it is greater. It raises Missing if it encounters an empty tree.Since an ordering is involved, we have to declare the functions for a specific type, here string. Now exception Missing mentions that type: if lookup fails, the exception returns the missing key. The exception could be eliminated using type option of our earlier Datatypes lecture, using the constructor None for failure.Anil Madhavapeddyhttps://www.cl.cam.ac.uk/teaching/2627/FoundsCS/forest/focs-bst-update/focs-bst-update/teaching/2627/FoundsCS/forest/focs-bst-update/Update Br ((k, v), Lf, Lf) | Br ((a, x), t1, t2) -> if k < a then Br ((a, x), update k v t1, t2) else if a < k then Br ((a, x), t1, update k v t2) else (* a = k *) Br ((a, v), t1, t2) val update : 'a -> 'b -> ('a * 'b) tree -> ('a * 'b) tree = ]]>This is also as it copies the path only, and not whole subtrees!If you are familiar with the usual update operation for this sort of tree, you may wonder whether it can be implemented in OCaml, where there is no direct way to replace part of a data structure by something else.The update operation is a nice piece of functional programming. It searches in the same manner as lookup, but the recursive calls reconstruct a new tree around the result of the update. One subtree is updated and the other left unchanged. The internal representation of trees ensures that unchanged parts of the tree are not copied, but shared. Therefore, update copies only the path from the root to the new node. Its time and space requirements, for a reasonably balanced tree, are both .The comparison between and allows three cases:smaller: update the left subtree; share the right greater: update the right subtree; share the left equal: update the label and share both subtreesNote: in the function definition, (* a = k *) is a comment. Comments in OCaml are enclosed in the brackets and .Anil Madhavapeddyhttps://www.cl.cam.ac.uk/teaching/2627/FoundsCS/forest/focs-tree-traversal/focs-tree-traversal/teaching/2627/FoundsCS/forest/focs-tree-traversal/Aside: Traversing Trees (3 Methods) [] | Br (v, t1, t2) -> [v] @ preorder t1 @ preorder t2 val preorder : 'a tree -> 'a list = # let rec inorder = function | Lf -> [] | Br (v, t1, t2) -> inorder t1 @ [v] @ inorder t2 val inorder : 'a tree -> 'a list = # let rec postorder = function | Lf -> [] | Br (v, t1, t2) -> postorder t1 @ postorder t2 @ [v] val postorder : 'a tree -> 'a list = ]]>Tree traversal means examining each node of a tree in some order. D. E. Knuth has identified three forms of tree traversal: preorder, inorder and postorder. We can code these “visiting orders” as functions that convert trees into lists of labels. Algorithms based on these notions typically perform some action at each node; the functions above simply copy the nodes into lists. Consider the tree:preorder visits the label first (“Polish notation”), yielding ABDECFG inorder visits the label midway, yielding DBEAFCG postorder visits the label last (“Reverse Polish”), yielding DEBFGCA. You might be familiar with this concept if you own an ancient RPN calculator!What is the use of inorder? Consider applying it to a binary search tree: the result is a sorted list of pairs. We could use this, for example, to merge two binary search trees. It is not difficult to transform a sorted list of pairs into a binary search tree.Anil Madhavapeddyhttps://www.cl.cam.ac.uk/teaching/2627/FoundsCS/forest/focs-tree-traversal-efficient/focs-tree-traversal-efficient/teaching/2627/FoundsCS/forest/focs-tree-traversal-efficient/Efficiently Traversing TreesUnfortunately, the traversal functions are quadratic in the worst case: the appends in the recursive calls are inefficient. To correct that problem, we (as usual) add an accumulating argument. Observe how each function constructs its result list and compare with how appends were eliminated from quicksort in the Sorting lecture. vs | Br (v, t1, t2), vs -> v :: preord (t1, preord (t2, vs)) val preord : 'a tree * 'a list -> 'a list = # let rec inord = function | Lf, vs -> vs | Br (v, t1, t2), vs -> inord (t1, v::inord (t2, vs)) val inord : 'a tree * 'a list -> 'a list = # let rec postord = function | Lf, vs -> vs | Br (v, t1, t2), vs -> postord (t1, postord (t2, v::vs)) val postord : 'a tree * 'a list -> 'a list = ]]>One can prove equations relating each of these functions to its counterpart on the previous section. For example:These three types of tree traversal are related in that all are depth-first. They each traverse the left subtree in full before traversing the right subtree. Breadth-first search (from the Queues lecture) is another possibility. That involves going through the levels of a tree one at a time.Anil Madhavapeddyhttps://www.cl.cam.ac.uk/teaching/2627/FoundsCS/forest/focs-arrays/focs-arrays/teaching/2627/FoundsCS/forest/focs-arrays/ArraysA conventional array is an indexed storage area. It is updated in place by the command a.(k) <- x The concept is inherently imperative. A functional array is a finite map from integers to data. Updating implies copying to return update(A, k, x) The new array equals A except that A.(k) = x. Can we do updates efficiently?The elements of a list can only be reached by counting from the front. Elements of a tree are reached by following a path from the root. An array hides such structural matters; its elements are uniformly designated by number. Immediate access to arbitrary parts of a data structure is called random access.Arrays are the dominant data structure in conventional programming languages. The ingenious use of arrays is the key to many of the great classical algorithms, such as Hoare’s original quicksort (the partition step) and Warshall’s transitive-closure algorithm.The drawback is that subscripting is a chief cause of programmer error. That is why arrays play little role in this introductory course.Functional arrays are described below in order to illustrate another way of using trees to organise data. Here is a summary of basic dictionary data structures in order of decreasing generality and increasing efficiency:Linear search: Most general, needing only equality on keys, but inefficient: linear time. Binary search: Needs an ordering on keys. Logarithmic access time in the average case, but our binary search trees are linear in the worst case. Array subscripting: Least general, requiring keys to be integers, but even worst-case time is logarithmic.Anil Madhavapeddyhttps://www.cl.cam.ac.uk/teaching/2627/FoundsCS/forest/focs-functional-arrays/focs-functional-arrays/teaching/2627/FoundsCS/forest/focs-functional-arrays/Functional Arrays as Binary TreesThe path to element follows the binary code for (its “subscript”).This simple representation (credited to W. Braun) ensures that the tree is balanced. Complexity of access is always , which is optimal. For actual running time, access to conventional arrays is much faster: it requires only a few hardware instructions. Array access is often taken to be , which (as always) presumes that hardware limits are never exceeded.The lower bound for array subscripts (or “indices”) is one. The upper bound starts at zero (which signifies the empty array) and can grow without limit. Inspection of the diagram above should make it clear that these trees are always balanced: the left subtree can have at most one node more than the right subtree, recursively all the way down. (This assumes that the array is defined for subscripts with no gaps; an array defined only for odd numbers, for example, would obviously be unbalanced.)The numbers in the diagram above are not the labels of branch nodes, but indicate the positions of array elements. For example, the label corresponding to is at the position shown. The nodes of a functional array are labelled with the data we want to store, not with these integers.Anil Madhavapeddyhttps://www.cl.cam.ac.uk/teaching/2627/FoundsCS/forest/focs-funarray-lookup/focs-funarray-lookup/teaching/2627/FoundsCS/forest/focs-funarray-lookup/The Lookup Function raise Subscript (* Not found *) | Br (v, t1, t2), k -> if k = 1 then v else if k mod 2 = 0 then sub (t1, k / 2) else sub (t2, k / 2) exception Subscript val sub : 'a tree * int -> 'a = # let rec sub = function (* Alternative implementation *) | Lf, _ -> raise Subscript | Br (v, t1, t2), 1 -> v | Br (v, t1, t2), k when k mod 2 = 0 -> sub (t1, k / 2) | Br (v, t1, t2), k -> sub (t2, k / 2) val sub : 'a tree * int -> 'a = ]]>Notice that we have used a new keyword when above, which changes pattern clauses to be only matched if the expression evalutes to true. This can be equivalently expressed by moving the corresponding checks into an if clause on the right hand side of the pattern match, but is often more readable using when (as above).The lookup function sub, divides the subscript by 2 until 1 is reached. If the remainder is 0 then the function follows the left subtree, otherwise the right. If it reaches a leaf, it signals error by raising exception Subscript.Array access can also be understood in terms of the subscript’s binary code. Because the subscript must be a positive integer, in binary it has a leading one. Discard this one and reverse the remaining bits. Interpreting zero as left and one as right yields the path from the root to the subscript.Popular literature often explains the importance of binary as being led by hardware: because a circuit is either on or off. The truth is almost the opposite. Designers of digital electronics go to a lot of trouble to suppress the continuous behaviour that would naturally arise. The real reason why binary is important is its role in algorithms: an if-then-else decision leads to binary branching.Data structures, such as trees, and algorithms, such as mergesort, use binary branching in order to reduce a cost from to . Two is the smallest integer divisor that achieves this reduction. (Larger divisors are only occasionally helpful, as in the case of B-trees, where they reduce the constant factor.) The simplicity of binary arithmetic compared with decimal arithmetic is just another instance of the simplicity of algorithms based on binary choices.Anil Madhavapeddyhttps://www.cl.cam.ac.uk/teaching/2627/FoundsCS/forest/focs-funarray-update/focs-funarray-update/teaching/2627/FoundsCS/forest/focs-funarray-update/The Update Function if k = 1 then Br (w, Lf, Lf) else raise Subscript (* Gap in tree *) | Br (v, t1, t2), k, w -> if k = 1 then Br (w, t1, t2) else if k mod 2 = 0 then Br (v, update (t1, k / 2, w), t2) else Br (v, t1, update (t2, k / 2, w)) val update : 'a tree * int * 'a -> 'a tree = ]]>The update function also divides the subscript repeatedly by two. When it reaches a value of one, it has identified the element position. Then it replaces the branch node by another branch with the new label.A leaf may be replaced by a branch, extending the array, provided no intervening nodes have to be generated. This suffices for arrays without gaps in their subscripting. (The data structure can be modified to allow sparse arrays, where most subscript positions are undefined.) Exception Subscript indicates that the subscript position does not exist and cannot be created. This use of exceptions is not easily replaced by None and Some.Note that there are two tests involving . If we have reached a leaf, it returns a branch, extending the array by one. If we are still at a branch node, then the effect is to update an existing array element.A similar function can shrink an array by one.Anil Madhavapeddyhttps://www.cl.cam.ac.uk/teaching/2627/FoundsCS/forest/focs-ex-7/focs-ex-7/teaching/2627/FoundsCS/forest/focs-ex-7/ExercisesAnil MadhavapeddyAnil Madhavapeddyhttps://www.cl.cam.ac.uk/teaching/2627/FoundsCS/forest/focs-ex-7-1/focs-ex-7-1/teaching/2627/FoundsCS/forest/focs-ex-7-1/Insertion order in search treesExerciseDraw the binary search tree that arises from successively inserting the following pairs into the empty tree: ("Alice", 6), ("Tobias", 2), ("Gerald", 8), ("Lucy", 9). Then repeat this task using the order ("Gerald", 8), ("Alice", 6), ("Lucy", 9), ("Tobias", 2). Why are results different?Anil MadhavapeddyAnil Madhavapeddyhttps://www.cl.cam.ac.uk/teaching/2627/FoundsCS/forest/focs-ex-7-2/focs-ex-7-2/teaching/2627/FoundsCS/forest/focs-ex-7-2/Insertion with collision detectionExerciseCode an insertion function for binary search trees. It should resemble the existing update function except that it should raise the exception Collision if the item to be inserted is already present.Anil MadhavapeddyAnil Madhavapeddyhttps://www.cl.cam.ac.uk/teaching/2627/FoundsCS/forest/focs-ex-7-3/focs-ex-7-3/teaching/2627/FoundsCS/forest/focs-ex-7-3/Returning the old value on collisionExerciseContinuing the previous exercise, it would be natural for exceptional Collision to return the value previously stored in the dictionary. Why is that goal difficult to achieve?Anil MadhavapeddyAnil Madhavapeddyhttps://www.cl.cam.ac.uk/teaching/2627/FoundsCS/forest/focs-ex-7-4/focs-ex-7-4/teaching/2627/FoundsCS/forest/focs-ex-7-4/Deleting from a search treeExerciseDescribe an algorithm for deleting an entry from a binary search tree. Comment on the suitability of your approach.Anil MadhavapeddyAnil Madhavapeddyhttps://www.cl.cam.ac.uk/teaching/2627/FoundsCS/forest/focs-ex-7-5/focs-ex-7-5/teaching/2627/FoundsCS/forest/focs-ex-7-5/Implementing deletionExerciseCode the delete function outlined in the previous exercise.Anil MadhavapeddyAnil Madhavapeddyhttps://www.cl.cam.ac.uk/teaching/2627/FoundsCS/forest/focs-ex-7-6/focs-ex-7-6/teaching/2627/FoundsCS/forest/focs-ex-7-6/Quadratic traversalsExerciseShow that the functions preorder, inorder and postorder all require time in the worst case, where is the size of the tree.Anil MadhavapeddyAnil Madhavapeddyhttps://www.cl.cam.ac.uk/teaching/2627/FoundsCS/forest/focs-ex-7-7/focs-ex-7-7/teaching/2627/FoundsCS/forest/focs-ex-7-7/Linear traversalsExerciseShow that the functions preord, inord and postord all take linear time in the size of the tree.Anil MadhavapeddyAnil Madhavapeddyhttps://www.cl.cam.ac.uk/teaching/2627/FoundsCS/forest/focs-ex-7-8/focs-ex-7-8/teaching/2627/FoundsCS/forest/focs-ex-7-8/Shifting a functional arrayExerciseWrite a function to remove the first element from a functional array. All the other elements are to have their subscripts reduced by one. The cost of this operation should be linear in the size of the array.Anil Madhavapeddyhttps://www.cl.cam.ac.uk/teaching/2627/FoundsCS/forest/focs-functions/focs-functions/teaching/2627/FoundsCS/forest/focs-functions/Functions as ValuesLecturehttps://hub.cl.cam.ac.uk/In OCaml, functions can bepassed as arguments to other functions, returned as results, put into lists, trees, etc., but not tested for equality. n * 2); (fun n -> n * 3); (fun n -> n + 1)] - : (int -> int) list = [; ; ]]]>Progress in programming languages can be measured by what abstractions they admit. Conditional expressions (descended from conditional jumps based on the sign of some numeric variable) and parametric types such as are examples. The idea that functions could be used as values in a computation arose early, but it took some time before the idea was fully realised. For decades, many programming languages let functions be passed as arguments to other functions, but few took the trouble needed to allow functions to be returned as results, since this requires packaging a function together with the variables it refers to in a closure. Today the idea, pioneered by languages such as Scheme and ML, is everywhere: Python, JavaScript, Java (since version 8), C++ (since C++11), Rust, Swift and Kotlin all let anonymous functions be passed around and returned like any other value.In mathematics, a functional or higher-order function is a function that operates on other functions. Many functionals are familiar from mathematics: for example, the differential operator maps functions to their derivatives, which are also functions. To a mathematician, a function is typically an infinite, uncomputable object. We use OCaml functions to represent algorithms. Sometimes they represent infinite collections of data given by computation rules.Functions cannot be compared for equality. We could compare the machine addresses of the compiled code, but that would merely be a test of identity: it would regard any two separate functions as unequal even if they were compiled from identical pieces of source code. Such a low-level feature has no place in a principled language.Anil Madhavapeddyhttps://www.cl.cam.ac.uk/teaching/2627/FoundsCS/forest/focs-anonymous-functions/focs-anonymous-functions/teaching/2627/FoundsCS/forest/focs-anonymous-functions/Functions Without NamesIf functions are to be regarded as computational values, then we need a notation for them. The fun notation expresses a non-recursive function value without giving the function a name. is the function such that . The function fun n -> n*2 is a doubling function. n * 2 - : int -> int = # (fun n -> n * 2) 17 - : int = 34]]>The main purpose of fun-notation is to package up small expressions that are to be applied repeatedly using some other function. The expression fun n -> n*2 has the same value as the identifier double, declared as follows: int = ]]>The fun notation can also do pattern matching, and the function keyword adds an anonymous variable name to pattern match against. The following functions are all equivalent, with the latter definitions bound to the is_zero value and the earlier ones anonymous: match x with 0 -> true | _ -> false - : int -> bool = # function 0 -> true | _ -> false - : int -> bool = # let is_zero = fun x -> match x with 0 -> true | _ -> false val is_zero : int -> bool = # let is_zero = function 0 -> true | _ -> false val is_zero : int -> bool = ]]>Anil Madhavapeddyhttps://www.cl.cam.ac.uk/teaching/2627/FoundsCS/forest/focs-currying/focs-currying/teaching/2627/FoundsCS/forest/focs-currying/Curried FunctionsA curried function returns another function as its result. We use the string concatenation operator (^) to illustrate how this works. string -> string = # let prefix = fun a -> fun b -> a ^ b val prefix : string -> string -> string = # let promote = prefix "Senior " val promote : string -> string = # prefix "Junior " "Professor" - : string = "Junior Professor" # promote "Professor" - : string = "Senior Professor"]]>A short form for the definition of prefix is simply to pass multiple arguments to the function definition. The following two definitions are equivalent in OCaml: fun b -> a ^ b val prefix : string -> string -> string = # let prefix a b = a ^ b val prefix : string -> string -> string = ]]>Currying is the technique of expressing a function taking multiple arguments as nested functions, each taking a single argument. The fun-notation lets us package n*2 as the function fun n -> n * 2, but what if there are several variables, as in fun n -> n * 2 + k? A function of two arguments could be coded using pattern-matching on pairs, writing fun (n, k) -> n * 2 + k.Currying is an alternative, where we nest the fun-notation: fun n -> n * 2 + k - : int -> int -> int = ]]>Applying this curried function to the argument 1 yields another function, in which k has been replaced by 1: fun n -> n * 2 + k val fn : int -> int -> int = # let fn' = fn 1 (* n * 2 + 1 *) val fn' : int -> int = # fn' 3 (* 3 * 2 + 1 *) - : int = 7]]>And this function, when applied to 3, yields the result 7. The two arguments are supplied one after another.The earlier prefix example is similar and uses the expression a ^ b, where ^ is the infix operator for string concatenation. Function promote binds the first argument of prefix to "Senior "; the resulting function prefixes that title to any string to which it is applied.Anil Madhavapeddyhttps://www.cl.cam.ac.uk/teaching/2627/FoundsCS/forest/focs-curry-shorthand/focs-curry-shorthand/teaching/2627/FoundsCS/forest/focs-curry-shorthand/Shorthand for Curried FunctionsA function-returning function is just a function of two arguments.This curried function syntax is nicer than nested fun binders: string -> string = # let dub = prefix "Sir " val dub : string -> string = ]]>Curried functions allows partial application (to the first argument).In OCaml, an -argument curried function f can be declared using the syntax:and applied using the syntax:If f is not recursive, then it is equivalent to the function expressed via nesting as follows:We now have two ways of expressing functions of multiple arguments: either by passing a pair of arguments or by currying. Currying allows partial application which is useful when fixing the first argument yields a function that is interesting in its own right. An example from mathematics is the function , where fixing yields a function in alone, namely squaring. Similarly, yields cubing, while yields the identity function.Though the function hd (which returns the head of a list) is not curried, it may be used with the curried application syntax in some expressions:Here List.hd is applied to a list of functions, and the resulting function dub is then applied to the string "Hamilton". The idea of executing code stored in data structures reaches its full development in object-oriented programming, like in Java.Anil Madhavapeddyhttps://www.cl.cam.ac.uk/teaching/2627/FoundsCS/forest/focs-partial-application/focs-partial-application/teaching/2627/FoundsCS/forest/focs-partial-application/Partial Application: A Curried Insertion Sort [x] | y::ys -> if lessequal x y then x :: y :: ys else y :: ins x ys in let rec sort = function | [] -> [] | x::xs -> ins x (sort xs) in sort val insort : ('a -> 'a -> bool) -> 'a list -> 'a list = ]]>The sorting functions we discussed in earlier lectures are coded to sort floating-point numbers. They can be generalised to an arbitrary ordered type by passing the ordering predicate lessequal as an argument.Functions ins and sort are declared locally, referring to lessequal. Though it may not be obvious, insort is a curried function. Given its first argument, a predicate for comparing some particular type of items, it returns the function sort for sorting lists of that type of items.Some examples of its use:=) [5; 3; 9; 8] - : int list = [9; 8; 5; 3]]]>An obscure point: the syntax (<=) denotes the comparison operator as a function, which is then given to insort. Passing the relation for lessequal gives a decreasing sort. This is no coding trick; it is justified in mathematics, since if is a partial ordering then so is .Anil Madhavapeddyhttps://www.cl.cam.ac.uk/teaching/2627/FoundsCS/forest/focs-map/focs-map/teaching/2627/FoundsCS/forest/focs-map/map: the “Apply to All” Function [] | x::xs -> (f x) :: map f xs val map : ('a -> 'b) -> 'a list -> 'b list = # map (fun s -> s ^ "ppy") ["Hi"; "Ho"] - : string list = ["Hippy"; "Hoppy"] # map (map double) [[1]; [2; 3]] - : int list list = [[2]; [4; 6]]]]>The functional map applies a function to every element of a list, returning a list of the function’s results. “Apply to all” is a fundamental operation and we shall see several applications of it below. We again see the advantages of fun-notation, currying and map. If we did not have them, the first use of map in the above code block would require a preliminary function declaration: [] | s::ss -> (s ^ "ppy") :: sillylist ss val sillylist : string list -> string list = ]]>An expression containing several applications of functionals—such as our second example—can abbreviate a long series of declarations. Sometimes this coding style is cryptic, but it can be clear as crystal. Treating functions as values lets us capture common program structures once and for all.In the second example, double is the obvious integer doubling function we defined earlier. Note that map is a built-in OCaml function in the form of List.map. OCaml’s standard library includes, among much else, many list functions.Anil Madhavapeddyhttps://www.cl.cam.ac.uk/teaching/2627/FoundsCS/forest/focs-transpose/focs-transpose/teaching/2627/FoundsCS/forest/focs-transpose/Example: Matrix Transpose [] | rows -> (map List.hd rows) :: (transp (map List.tl rows)) val transp : 'a list list -> 'a list list = ]]>A matrix can be viewed as a list of rows, each row a list of matrix elements. This representation is not especially efficient compared with the conventional one (using arrays). Lists of lists turn up often, though, and we can see how to deal with them by taking familiar matrix operations as examples. ML for the Working Programmer goes as far as Gaussian elimination, which presents surprisingly few difficulties.The transpose of the matrix is , which in OCaml corresponds to the following transformation on lists of lists:The workings of function transp are simple. If rows is the matrix to be transposed, then map hd extracts its first column and map tl extracts its second column:A recursive call transposes the latter matrix, which is then given the column [a; d] as its first row. The two functions expressed using map would otherwise have to be declared separately.Anil Madhavapeddyhttps://www.cl.cam.ac.uk/teaching/2627/FoundsCS/forest/focs-matrix-multiplication/focs-matrix-multiplication/teaching/2627/FoundsCS/forest/focs-matrix-multiplication/Review of Matrix MultiplicationThe right side is the vector dot product . Repeat for each row of and column of .The dot product of two vectors isA simple case of matrix multiplication is when consists of a single row and consists of a single column. Provided and contain the same number of elements, multiplying them yields a matrix whose single element is the dot product shown above.If is an matrix and is a matrix then is an matrix. For each and , the element of is the dot product of row of with column of .The (1, 1) element above is computed byCoding matrix multiplication in a conventional programming language usually involves three nested loops. It is hard to avoid mistakes in the subscripting, which often runs slowly due to redundant internal calculations.Anil Madhavapeddyhttps://www.cl.cam.ac.uk/teaching/2627/FoundsCS/forest/focs-matmul-ocaml/focs-matmul-ocaml/teaching/2627/FoundsCS/forest/focs-matmul-ocaml/Matrix Multiplication in OCamlDot product of two vectors—a curried function 0.0 | x::xs, y::ys -> (x *. y) +. (dotprod xs ys) Lines 2-4, characters 5-51: Warning 8 [partial-match]: this pattern-matching is not exhaustive. Here is an example of a case that is not matched: ([], _::_) val dotprod : float list -> float list -> float = ]]>Matrix product map (dotprod row) cols) arows val matprod : float list list -> float list list -> float list list = ]]>The transp brows converts into a list of columns, yielding a list whose elements are the columns of . Each row of is obtained by multiplying a row of by the columns of .Because dotprod is curried, it can be applied to a row of . The resulting function is applied to all the columns of . We have another example of currying and partial application.The outer map applies dotprod to each row of . The inner map, using fun-notation, applies dotprod row to each column of . Compare with the version in ML for the Working Programmer (page 89) which does not use map and requires two additional function declarations.In the dot product function, the two vectors must have the same length. Otherwise, exception Match_failure is raised.Anil Madhavapeddyhttps://www.cl.cam.ac.uk/teaching/2627/FoundsCS/forest/focs-predicate-functionals/focs-predicate-functionals/teaching/2627/FoundsCS/forest/focs-predicate-functionals/List Functionals for Predicates false | x::xs -> (p x) || (exists p xs) val exists : ('a -> bool) -> 'a list -> bool = # let rec filter p = function | [] -> [] | x::xs -> if p x then x :: filter p xs else filter p xs val filter : ('a -> bool) -> 'a list -> 'a list = ]]>A predicate is a boolean-valued function.The functional exists transforms a predicate into a predicate over lists. Given a list, exists p tests whether or not some list element satisfies p (making it return true). If it finds one, it stops searching immediately, thanks to the behaviour of the lazy || operator.Dually, we have a functional to test whether all list elements satisfy the predicate. If it finds a counterexample then it, too, stops searching. true | x::xs -> (p x) && all p xs val all : ('a -> bool) -> 'a list -> bool = ]]>The filter functional, like map, transforms lists. It applies a predicate to all the list elements, but instead of returning the resulting values (which could only be true or false), it returns the list of elements satisfying the predicate.Anil Madhavapeddyhttps://www.cl.cam.ac.uk/teaching/2627/FoundsCS/forest/focs-predicate-applications/focs-predicate-applications/teaching/2627/FoundsCS/forest/focs-predicate-applications/Applications of the Predicate Functionals x=y) xs val member : 'a -> 'a list -> bool = # let inter xs ys = filter (fun x -> member x ys) xs val inter : 'a list -> 'a list -> 'a list = ]]>Testing whether two lists have no common elements all (fun y -> x<>y) ys) xs val disjoint : 'a list -> 'a list -> bool = ]]>The Lists lecture presented the function member, which tests whether a specified value can be found as a list element, and inter, which returns the “intersection” of two lists: the list of elements they have in common.But remember: the purpose of list functionals is not to replace the declarations of popular functions, which probably are available already. It is to eliminate the need for separate declarations of ad-hoc functions. When they are nested, like the calls to all in disjoint above, the inner functions are almost certainly one-offs, not worth declaring separately.Our primitives themselves can be seen as a programming language. Part of the task of programming is to extend our programming language with notation for solving the problem at hand. The levels of notation that we define should correspond to natural levels of abstraction in the problem domain.Historical Note: Alonzo Church’s -calculus gave a simple syntax, -notation, for expressing functions. It is the direct precursor of OCaml’s fun-notation. It was soon shown that his system was equivalent in computational power to Turing machines, and Church’s thesis states that this defines precisely the set of functions that can be computed effectively.The -calculus had a tremendous influence on the design of functional programming languages. McCarthy’s Lisp was something of a false start; it interpreted variable binding incorrectly, an error that stood for some 20 years. But in 1966, Peter Landin (of Queen Mary College, University of London) sketched out the main features of functional languages.Anil Madhavapeddyhttps://www.cl.cam.ac.uk/teaching/2627/FoundsCS/forest/focs-ex-8/focs-ex-8/teaching/2627/FoundsCS/forest/focs-ex-8/ExercisesAnil MadhavapeddyAnil Madhavapeddyhttps://www.cl.cam.ac.uk/teaching/2627/FoundsCS/forest/focs-ex-8-1/focs-ex-8-1/teaching/2627/FoundsCS/forest/focs-ex-8-1/The argument-swapping functionalExerciseWhat does the following function do, and what are its uses?Anil MadhavapeddyAnil Madhavapeddyhttps://www.cl.cam.ac.uk/teaching/2627/FoundsCS/forest/focs-ex-8-2/focs-ex-8-2/teaching/2627/FoundsCS/forest/focs-ex-8-2/Lexicographic orderingsExerciseThere are many ways of combining orderings. The lexicographic ordering uses two keys for comparisons. It is specified byWrite an OCaml function to lexicographically combine two orderings, supplied as functions. Explain how it allows function insort to sort a list of pairs.Anil MadhavapeddyAnil Madhavapeddyhttps://www.cl.cam.ac.uk/teaching/2627/FoundsCS/forest/focs-ex-8-3/focs-ex-8-3/teaching/2627/FoundsCS/forest/focs-ex-8-3/Mapping over lists of listsExerciseWithout using map write a function map2 such that map2 f is equivalent to map (map f). The obvious solution requires declaring two recursive functions. Try to get away with only one by exploiting nested pattern-matching.Anil MadhavapeddyAnil Madhavapeddyhttps://www.cl.cam.ac.uk/teaching/2627/FoundsCS/forest/focs-ex-8-4/focs-ex-8-4/teaching/2627/FoundsCS/forest/focs-ex-8-4/A map for optionsExerciseThe type 'a option, declared below, can be viewed as a type of lists having at most one element. (It is typically used as an alternative to exceptions.) Declare an analogue of the function map for type 'a option.Anil MadhavapeddyAnil Madhavapeddyhttps://www.cl.cam.ac.uk/teaching/2627/FoundsCS/forest/focs-ex-8-5/focs-ex-8-5/teaching/2627/FoundsCS/forest/focs-ex-8-5/Making change with mapExerciseRecall the making change function of More on Lists: [ [] ] | [] , _ -> [] | c::till , amt -> if amt < c then change till amt else let rec allc = function | [] -> [] | cs :: css -> (c::cs) :: allc css in allc (change (c::till) (amt - c)) @ change till amt val change : int list -> int -> int list list = ]]>Function allc applies the function ‘cons a c’ to every element of a list. Eliminate it by declaring a curried cons function and applying map.Anil Madhavapeddyhttps://www.cl.cam.ac.uk/teaching/2627/FoundsCS/forest/focs-sequences/focs-sequences/teaching/2627/FoundsCS/forest/focs-sequences/Sequences, or Lazy ListsLecturehttps://hub.cl.cam.ac.uk/Anil Madhavapeddyhttps://www.cl.cam.ac.uk/teaching/2627/FoundsCS/forest/focs-pipeline/focs-pipeline/teaching/2627/FoundsCS/forest/focs-pipeline/A PipelineProduce sequence of items Filter sequence in stages Consume results as needed Lazy lists join the stages togetherTwo types of program can be distinguished. A sequential program accepts a problem to solve, processes for a while, and finally terminates with its result. A typical example is the huge numerical simulations that are run on supercomputers. Most of our OCaml functions also fit this model.At the other extreme are reactive programs, whose job is to interact with the environment. They communicate constantly during their operation and run for as long as is necessary. A typical example is the software that controls many modern aircraft. Reactive programs often consist of concurrent processes running at the same time and communicating with one another.Concurrency is too difficult to consider in this course, but we can model simple pipelines such as that shown above. The Producer represents one or more sources of data, which it outputs as a stream. The Filter stages convert the input stream to an output stream, perhaps consuming several input items to yield a single output item. The Consumer takes as many elements as necessary.The Consumer drives the pipeline: nothing is computed except in response to its demand for an additional datum. Execution of the Filter stages is interleaved as required for the computation to go through. The programmer sets up the data dependencies but has no clear idea of what happens when. We have the illusion of concurrent computation.The Unix operating system provides similar ideas through its pipes that link processes together. In OCaml, we can model pipelines using lazy lists.Anil Madhavapeddyhttps://www.cl.cam.ac.uk/teaching/2627/FoundsCS/forest/focs-lazy-lists/focs-lazy-lists/teaching/2627/FoundsCS/forest/focs-lazy-lists/Lazy Lists (or Streams)Lists of possibly infinite length Elements computed upon demand Avoids waste if there are many solutions Infinite values are a useful abstractionIn OCaml, we can implement laziness by delaying evaluation of the tail of the list.Lazy lists have practical uses. Some algorithms, like making change, can yield many solutions when only a few are required. Sometimes the original problem concerns infinite series: with lazy lists, we can pretend they really exist!We are now dealing with infinite (or at least unbounded) computations. A potentially infinite source of data is processed one element at a time, upon demand. Such programs are harder to understand than terminating ones and have more ways of going wrong.Some purely functional languages, such as Haskell, use lazy evaluation everywhere. Even the if-then-else construct can be a function, and all lists are lazy. In OCaml, we can declare a type of lists such that evaluation of the tail does not occur until demanded. Delayed evaluation is weaker than lazy evaluation, but it is good enough for our purposes and often the best compromise for performance and memory usage.The traditional word “stream” is reserved in OCaml parlance for input/output channels. Let us call lazy lists sequences instead.Anil Madhavapeddyhttps://www.cl.cam.ac.uk/teaching/2627/FoundsCS/forest/focs-lazy-ocaml/focs-lazy-ocaml/teaching/2627/FoundsCS/forest/focs-lazy-ocaml/Lazy Lists in OCamlThe empty tuple () and its type unit Delayed version of is fun () -> E 'a seq) type 'a seq = Nil | Cons of 'a * (unit -> 'a seq) # let head (Cons (x, _)) = x Line 1, characters 10-23: Warning 8 [partial-match]: this pattern-matching is not exhaustive. Here is an example of a case that is not matched: Nil val head : 'a seq -> 'a = # let tail (Cons (_, xf)) = xf () Line 1, characters 10-24: Warning 8 [partial-match]: this pattern-matching is not exhaustive. Here is an example of a case that is not matched: Nil val tail : 'a seq -> 'a seq = ]]> has head and tail function The primitive OCaml type unit has one element, which is written (). This element may be regarded as a 0-tuple, and unit as the nullary Cartesian product. (Think of the connection between multiplication and the number 1.)The empty tuple serves as a placeholder in situations where no information is required. It may:appear in a data structure. For example, a unit-valued dictionary represents a set of keys. be the argument of a function, where its effect is to delay evaluation. be the argument or result of a procedure. (see the Procedural Programming section)The empty tuple, like all tuples, is a constructor and is allowed in patterns; for example: let f () = ...In particular is the function that takes an argument of type unit and returns the value of as its result. Expression is not evaluated until the function is called, even though the only possible argument is (). The function simply delays the evaluation of .Anil Madhavapeddyhttps://www.cl.cam.ac.uk/teaching/2627/FoundsCS/forest/focs-infinite-sequence/focs-infinite-sequence/teaching/2627/FoundsCS/forest/focs-infinite-sequence/The Infinite Sequence: , , , … from (k+1)) val from : int -> int seq = # let it = from 1 val it : int seq = Cons (1, ) # let it = tail it val it : int seq = Cons (2, ) # let it = tail it val it : int seq = Cons (3, )]]>Function from constructs the infinite sequence of integers starting from . Execution terminates because of the fun enclosing the recursive call. OCaml displays the tail of a sequence as fun, which stands for some function value. Each call to tail generates the next sequence element. We could do this forever.This example is of little practical value because the cost of computing a sequence element will be dominated by that of creating the dummy function. Lazy lists tend to have high overheads.Anil Madhavapeddyhttps://www.cl.cam.ac.uk/teaching/2627/FoundsCS/forest/focs-seq-consume/focs-seq-consume/teaching/2627/FoundsCS/forest/focs-seq-consume/Consuming a Sequence [] | n, Nil -> [] | n, Cons (x, xf) -> x :: get (n-1) (xf ()) val get : int -> 'a seq -> 'a list = ]]>The above code gets the first elements as a list. xf () forces evaluation.The function get converts a sequence to a list. It takes the first elements; it takes all of them if , which can terminate only if the sequence is finite.In the last line of get, the expression xf() calls the tail function, demanding evaluation of the next element. This operation is called forcing the sequence.Anil Madhavapeddyhttps://www.cl.cam.ac.uk/teaching/2627/FoundsCS/forest/focs-seq-evaluation/focs-seq-evaluation/teaching/2627/FoundsCS/forest/focs-seq-evaluation/Sample EvaluationHere we ask for two elements of the infinite sequence. In fact, three elements are computed: 6, 7 and 8. Our implementation is slightly too eager. A more complicated type declaration could avoid this problem. Another problem is that if one repeatedly examines some particular list element using forcing, that element is repeatedly evaluated. In a lazy programming language, the result of the first evaluation would be stored for later reference. To get the same effect in OCaml requires the use of references.We should be grateful that the potentially infinite computation is kept finite. The tail of the original sequence even contains the unevaluated expression 6+1.Anil Madhavapeddyhttps://www.cl.cam.ac.uk/teaching/2627/FoundsCS/forest/focs-seq-append/focs-seq-append/teaching/2627/FoundsCS/forest/focs-seq-append/Joining Two Sequences yq | Cons (x, xf) -> Cons(x, fun () -> appendq (xf ()) yq) val appendq : 'a seq -> 'a seq -> 'a seq = ]]>A more fair alternative: yq | Cons (x, xf) -> Cons (x, fun () -> interleave yq (xf ())) val interleave : 'a seq -> 'a seq -> 'a seq = ]]>Most list functions and functionals have analogues on sequences, but strange things can happen. Can an infinite list be reversed?Function appendq is precisely the same idea as append from the Lists lecture; it concatenates two sequences. If the first argument is infinite, then appendq never gets to its second argument, which is lost. Concatenation of infinite sequences is not terribly interesting.The function interleave avoids this problem by exchanging the two arguments in each recursive call. It combines the two lazy lists, losing no elements. Interleaving is the right way to combine two potentially infinite information sources into one.In both function declarations, observe that each xf () is enclosed within a . Each force is enclosed within a delay. This practice makes the functions lazy. A force not enclosed in a delay, as in get above, runs the risk of evaluating the sequence in full.Anil Madhavapeddyhttps://www.cl.cam.ac.uk/teaching/2627/FoundsCS/forest/focs-seq-functionals/focs-seq-functionals/teaching/2627/FoundsCS/forest/focs-seq-functionals/Functionals for Lazy ListsFiltering lazy lists: Nil | Cons (x, xf) -> if p x then Cons (x, fun () -> filterq p (xf ())) else filterq p (xf ()) val filterq : ('a -> bool) -> 'a seq -> 'a seq = ]]>The infinite sequence , , , … iterates f (f x)) val iterates : ('a -> 'a) -> 'a -> 'a seq = ]]>The functional filterq demands elements of xq until it finds one satisfying p. (Recall filter, the analogous operation for ordinary lists.) It contains a force not protected by a delay. If xq is infinite and contains no satisfactory element, then filterq runs forever.The functional iterates generalises from. It creates the next element not by adding one but by calling the function f.Anil Madhavapeddyhttps://www.cl.cam.ac.uk/teaching/2627/FoundsCS/forest/focs-seq-numerical/focs-seq-numerical/teaching/2627/FoundsCS/forest/focs-seq-numerical/Numerical Computations on Infinite Sequences float -> float = ]]>Close enough? match xf () with | Cons (y, yf) -> if abs_float (x -. y) <= eps then y else within eps (Cons (y, yf)) Lines 3-6, characters 7-41: Warning 8 [partial-match]: this pattern-matching is not exhaustive. Here is an example of a case that is not matched: Nil Lines 1-6, characters 22-41: Warning 8 [partial-match]: this pattern-matching is not exhaustive. Here is an example of a case that is not matched: Nil val within : float -> float seq -> float = ]]>Square Roots: float = ]]>The Newton-Raphson method is widely used for computing square roots. The infinite series converges rapidly to . The initial approximation, , is typically retrieved from a table, and is accurate enough that only a few iterations of the method are necessary. Calling iterates (next a) x0 generates the infinite series of approximations to the square root of using the Newton-Raphson method. To compute , the resulting series begins 1, 1.5, 1.41667, 1.4142157, 1.414213562, …, and this last figure is already accurate to 10 significant digits!Function within searches down the lazy list for two points whose difference is less than eps. It tests their absolute difference. Relative difference and other “close enough” tests can be coded. Such components can be used to implement other numerical functions directly as functions over sequences. The point is to build programs from small, interchangeable parts.Function root uses within, iterates and next to to apply Newton-Raphson with a tolerance of and a (poor) initial approximation of 1.0.This treatment of numerical computation has received some attention in the research literature; a recurring example is Richardson extrapolation.Anil Madhavapeddyhttps://www.cl.cam.ac.uk/teaching/2627/FoundsCS/forest/focs-ex-9/focs-ex-9/teaching/2627/FoundsCS/forest/focs-ex-9/ExercisesAnil MadhavapeddyAnil Madhavapeddyhttps://www.cl.cam.ac.uk/teaching/2627/FoundsCS/forest/focs-ex-9-1/focs-ex-9-1/teaching/2627/FoundsCS/forest/focs-ex-9-1/A map for sequencesExerciseCode an analogue of map for sequences.Anil MadhavapeddyAnil Madhavapeddyhttps://www.cl.cam.ac.uk/teaching/2627/FoundsCS/forest/focs-ex-9-2/focs-ex-9-2/teaching/2627/FoundsCS/forest/focs-ex-9-2/Concatenating infinite sequencesExerciseConsider the list function concat, which concatenates a list of lists to form a single list. Can it be generalised to concatenate a sequence of sequences? What can go wrong? [] | l::ls -> l @ concat ls]]>Anil MadhavapeddyAnil Madhavapeddyhttps://www.cl.cam.ac.uk/teaching/2627/FoundsCS/forest/focs-ex-9-3/focs-ex-9-3/teaching/2627/FoundsCS/forest/focs-ex-9-3/Making change lazilyExerciseCode a function to make change using lazy lists, delivering the sequence of all possible ways of making change. Using sequences allows us to compute solutions one at a time when there exists an astronomical number. Represent lists of coins using ordinary lists. (Hint: to benefit from laziness you may need to pass around the sequence of alternative solutions as a function of type unit -> (int list) seq.)Anil MadhavapeddyAnil Madhavapeddyhttps://www.cl.cam.ac.uk/teaching/2627/FoundsCS/forest/focs-ex-9-4/focs-ex-9-4/teaching/2627/FoundsCS/forest/focs-ex-9-4/Lazy binary treesExerciseA lazy binary tree is either empty or is a branch containing a label and two lazy binary trees, possibly to infinite depth. Present an OCaml datatype to represent lazy binary trees, along with a function that accepts a lazy binary tree and produces a lazy list that contains all of the tree’s labels. (Taken from the exam question 2008 Paper 1 Question 5.)Anil MadhavapeddyAnil Madhavapeddyhttps://www.cl.cam.ac.uk/teaching/2627/FoundsCS/forest/focs-ex-9-5/focs-ex-9-5/teaching/2627/FoundsCS/forest/focs-ex-9-5/All lists of zeroes and onesExerciseCode the lazy list whose elements are all ordinary lists of zeroes and ones, namely []; [0]; [1]; [0; 0]; [0; 1]; [1; 0]; [1; 1]; [0; 0; 0]; …. (Taken from the exam question 2003 Paper 1 Question 5.)Anil MadhavapeddyAnil Madhavapeddyhttps://www.cl.cam.ac.uk/teaching/2627/FoundsCS/forest/focs-ex-9-6/focs-ex-9-6/teaching/2627/FoundsCS/forest/focs-ex-9-6/Enumerating palindromesExercise(Continuing the previous exercise.) A palindrome is a list that equals its own reverse. Code the lazy list whose elements are all palindromes of 0s and 1s, namely []; [0]; [1]; [0; 0]; [0; 0; 0]; [0; 1; 0]; [1; 1]; [1; 0; 1]; [1; 1; 1]; [0; 0; 0; 0]; , …. You may take the reversal function List.rev as given.Anil Madhavapeddyhttps://www.cl.cam.ac.uk/teaching/2627/FoundsCS/forest/focs-queues/focs-queues/teaching/2627/FoundsCS/forest/focs-queues/Queues and Search StrategiesLecturehttps://hub.cl.cam.ac.uk/Anil Madhavapeddyhttps://www.cl.cam.ac.uk/teaching/2627/FoundsCS/forest/focs-bfs-dfs/focs-bfs-dfs/teaching/2627/FoundsCS/forest/focs-bfs-dfs/Breadth-First v Depth-First Tree Traversalbinary trees as decision trees look for solution nodes Depth-first: search one subtree in full before moving on Breadth-first: search all nodes at level before moving to finds all solutions — nearest first!Preorder, inorder and postorder tree traversals all have something in common: they are depth-first. At each node, the left subtree is entirely traversed before the right subtree. Depth-first traversals are easy to code and can be efficient, but they are ill-suited for some problems.Suppose the tree represents the possible moves in a puzzle, and the purpose of the traversal is to search for a node containing a solution. Then a depth-first traversal may find one solution node deep in the left subtree, when another solution is at the very top of the right subtree. Often we want the shortest path to a solution.Suppose the tree is infinite or simply extremely large. Depth-first search is almost useless with such trees, for if the left subtree is infinite then the search will never reach the right subtree. OCaml can represent infinite trees by the means discussed in the lecture on laziness. Another tree representation (suitable for solving solitaire, for example) is by a function next : pos -> pos list, which maps a board position to a list of the positions possible after the next move. For simplicity, the examples below use the OCaml datatype tree, which has only finite trees.A breadth-first traversal explores the nodes horizontally rather than vertically. When visiting a node, it does not traverse the subtrees until it has visited all other nodes at the current depth. This is easily implemented by keeping a list of trees to visit. Initially, this list consists of one element: the entire tree. Each iteration removes a tree from the head of the list and adds its subtrees after the end of the list.Anil Madhavapeddyhttps://www.cl.cam.ac.uk/teaching/2627/FoundsCS/forest/focs-bfs-append/focs-bfs-append/teaching/2627/FoundsCS/forest/focs-bfs-append/Breadth-First Tree Traversal — Using Append [] | Lf :: ts -> nbreadth ts | Br (v, t, u) :: ts -> v :: nbreadth (ts @ [t; u]) val nbreadth : 'a tree list -> 'a list = ]]>Keeps an enormous queue of nodes of search, and is a wasteful use of append.Breadth-first search can be inefficient, this naive implementation especially so. When the search is at depth of the tree, the list contains all the remaining trees at depth , followed by the subtrees (all at depth ) of the trees that have already been visited. At depth 10, the list could already contain 1024 elements. It requires a lot of space, and aggravates this with a gross misuse of append. Evaluating ts@[t, u] copies the long list ts just to insert two elements.Anil Madhavapeddyhttps://www.cl.cam.ac.uk/teaching/2627/FoundsCS/forest/focs-queue-adt/focs-queue-adt/teaching/2627/FoundsCS/forest/focs-queue-adt/An Abstract Data Type: Queuesqempty is the empty queue qnull tests whether a queue is empty qhd returns the element at the head of a queue deq discards the element at the head of a queue enq adds an element at the end of a queueBreadth-first search becomes much faster if we replace the lists by queues. A queue represents a sequence, allowing elements to be taken from the head and added to the tail. This is a First-In-First-Out (FIFO) discipline: the item next to be removed is the one that has been in the queue for the longest time. Lists can implement queues, but append is a poor means of adding elements to the tail.Our functional arrays are suitable, provided we augment them with a function to delete the first array element. (See ML for the Working Programmer page 156.) Each operation would take time for a queue of length .We shall describe a representation of queues that is purely functional, based upon lists, and efficient. Operations take time when “amortized”: averaged over the lifetime of a queue.A conventional programming technique is to represent a queue by an array. Two indices point to the front and back of the queue, which may wrap around the end of the array. The coding is somewhat tricky. Worse, the length of the queue must be given a fixed upper bound.Anil Madhavapeddyhttps://www.cl.cam.ac.uk/teaching/2627/FoundsCS/forest/focs-functional-queues-idea/focs-functional-queues-idea/teaching/2627/FoundsCS/forest/focs-functional-queues-idea/Efficient Functional Queues: IdeaRepresent the queue by any pair of lists Add new items to the rear list Remove items from front list and if empty move rear to front Amortized time per operation is Queues require efficient access at both ends: at the front, for removal, and at the back, for insertion. Ideally, access should take constant time, . It may appear that lists cannot provide such access. If enq(q, x) performs q@[x], then this operation will be . We could represent queues by reversed lists, implementing enq(q, x) by x::q, but then the deq and qhd operations would be . Linear time is intolerable: a series of queue operations could then require time.The solution is to represent a queue by a pair of lists, whererepresents the queue .The front part of the queue is stored in order, and the rear part is stored in reverse order. The enq operation adds elements to the rear part using cons, since this list is reversed; thus, enq takes constant time. The deq and qhd operations look at the front part, which normally takes constant time, since this list is stored in order. But sometimes deq removes the last element from the front part; when this happens, it reverses the rear part, which becomes the new front part.Amortized time refers to the cost per operation averaged over the lifetime of any complete execution. Even for the worst possible execution, the average cost per operation turns out to be constant; see the analysis below.Anil Madhavapeddyhttps://www.cl.cam.ac.uk/teaching/2627/FoundsCS/forest/focs-functional-queues/focs-functional-queues/teaching/2627/FoundsCS/forest/focs-functional-queues/Efficient Functional Queues: Code Q (List.rev tls, []) | q -> q val norm : 'a queue -> 'a queue = # let qnull = function | Q ([], []) -> true | _ -> false val qnull : 'a queue -> bool = # let enq (Q (hds, tls)) x = norm (Q (hds, x::tls)) val enq : 'a queue -> 'a -> 'a queue = # exception Empty exception Empty # let deq = function | Q (x::hds, tls) -> norm (Q (hds, tls)) | _ -> raise Empty val deq : 'a queue -> 'a queue = # let qempty = Q ([], []) val qempty : 'a queue = Q ([], []) # let qhd = function | Q (x::_, _) -> x | _ -> raise Empty val qhd : 'a queue -> 'a = ]]>The datatype of queues prevents confusion with other pairs of lists. The empty queue has both parts empty.The function norm puts a queue into normal form, ensuring that the front part is never empty unless the entire queue is empty. Functions deq and enq call norm to normalise their result.Because queues are in normal form, their head is certain to be in their front part, so qhd looks there.Let us analyse the cost of an execution comprising (in any possible order) enq operations and deq operations, starting with an empty queue. Each enq operation will perform one cons, adding an element to the rear part. Since the final queue must be empty, each element of the rear part gets transferred to the front part. The corresponding reversals perform one cons per element. Thus, the total cost of the series of queue operations is cons operations, an average of 2 per operation. The amortized time is .There is a catch. The conses need not be distributed evenly; reversing a long list could take up to of them. Unpredictable delays make the approach unsuitable for real-time programming where deadlines must be met.Anil Madhavapeddyhttps://www.cl.cam.ac.uk/teaching/2627/FoundsCS/forest/focs-bfs-queues/focs-bfs-queues/teaching/2627/FoundsCS/forest/focs-bfs-queues/Breadth-First Tree Traversal — Using Queues breadth (deq q) | Br (v, t, u) -> v :: breadth (enq (enq (deq q) t) u) val breadth : 'a tree queue -> 'a list = ]]>This function implements the same algorithm as nbreadth but uses a different data structure. It represents queues using type queue instead of type list.To compare their efficiency, I applied both functions to the full binary tree of depth 12, which contains 4095 labels. The function nbreadth took 30 seconds while breadth took only 0.15 seconds: faster by a factor of 200.For larger trees, the speedup would be greater. Choosing the right data structure pays handsomely.Anil Madhavapeddyhttps://www.cl.cam.ac.uk/teaching/2627/FoundsCS/forest/focs-iterative-deepening/focs-iterative-deepening/teaching/2627/FoundsCS/forest/focs-iterative-deepening/Iterative deepening: Another Exhaustive SearchBreadth-first search examines nodes: Recompute nodes at depth instead of storing them Time factor is if 1]]>; complexity is still Space required at depth drops from to Breadth-first search is not practical for infinite trees: it uses too much space. Large parts of the tree have to be stored. Consider the slightly more general problem of searching trees whose branching factor is (for binary trees, ). Then breadth-first search to depth examines nodes, which is , ignoring the constant factor of . Since all nodes that are examined are also stored, the space and time requirements are both .Depth-first iterative deepening combines the space efficiency of depth-first with the “nearest-first” property of breadth-first search. It performs repeated depth-first searches with increasing depth bounds, each time discarding the result of the previous search. Thus it searches to depth 1, then to depth 2, and so on until it finds a solution. We can afford to discard previous results because the number of nodes is growing exponentially. There are nodes at level ; if , this number actually exceeds the total number of nodes of all previous levels put together, namely .Korf shows that the time needed for iterative deepening to reach depth is only times that for breadth-first search, if 1]]>. This is a constant factor; both algorithms have the same time complexity, . In typical applications where the extra factor of is quite tolerable. The reduction in the space requirement is exponential, from for breadth-first to for iterative deepening. Of course, this assumes that the tree itself is not stored in memory.Anil Madhavapeddyhttps://www.cl.cam.ac.uk/teaching/2627/FoundsCS/forest/focs-stacks/focs-stacks/teaching/2627/FoundsCS/forest/focs-stacks/Another Abstract Data Type: Stacksempty is the empty stack null tests whether a stack is empty top returns the element at the top of a stack pop discards the element at the top of a stack push adds an element at the top of a stackA stack is a sequence such that items can be added or removed from the head only. A stack obeys a Last-In-First-Out (LIFO) discipline: the item next to be removed is the one that has been on the stack for the shortest time. Lists can easily implement stacks because both cons and hd affect the head. But unlike lists, stacks are often regarded as an imperative data structure: the effect of push or pop is to change an existing stack, not return a new one.In conventional programming languages, a stack is often implemented by storing the elements in an array, using a variable (the “stack pointer”) to count them. Most language processors keep track of recursive function calls using an internal stack.Anil Madhavapeddyhttps://www.cl.cam.ac.uk/teaching/2627/FoundsCS/forest/focs-search-methods/focs-search-methods/teaching/2627/FoundsCS/forest/focs-search-methods/A Survey of Search MethodsDepth-first: use a stack (efficient but incomplete) Breadth-first: use a queue (uses too much space!) Iterative deepening: use depth-first to get benefits of breadth-first (trades time for space) Best-first: use a priority queue (heuristic search)The data structure determines the search!Search procedures can be classified by the data structure used to store pending subtrees. Depth-first search stores them on a stack, which is implicit in functions like inorder, but can be made explicit. Breadth-first search stores such nodes in a queue.An important variation is to store the nodes in a priority queue, which is an ordered sequence. The priority queue applies some sort of ranking function to the nodes, placing higher-ranked nodes before lower-ranked ones. The ranking function typically estimates the distance from the node to a solution. If the estimate is good, the solution is located swiftly. This method is called best-first search.The priority queue can be kept as a sorted list, although this is slow. Binary search trees would be much better on average, and fancier data structures improve matters further.Anil Madhavapeddyhttps://www.cl.cam.ac.uk/teaching/2627/FoundsCS/forest/focs-ex-10/focs-ex-10/teaching/2627/FoundsCS/forest/focs-ex-10/ExercisesAnil MadhavapeddyAnil Madhavapeddyhttps://www.cl.cam.ac.uk/teaching/2627/FoundsCS/forest/focs-ex-10-1/focs-ex-10-1/teaching/2627/FoundsCS/forest/focs-ex-10-1/Tree-based queuesExerciseSuppose that we have an implementation of queues, based on binary trees, such that each operation takes logarithmic time in the worst case. Outline the advantages and drawbacks of such an implementation compared with one presented above.Anil MadhavapeddyAnil Madhavapeddyhttps://www.cl.cam.ac.uk/teaching/2627/FoundsCS/forest/focs-ex-10-2/focs-ex-10-2/teaching/2627/FoundsCS/forest/focs-ex-10-2/Array-based circular queuesExerciseThe traditional way to implement queues uses a fixed-length array. Two indices into the array indicate the start and end of the queue, which wraps around from the end of the array to the start. How appropriate is such a data structure for implementing breadth-first search?Anil MadhavapeddyAnil Madhavapeddyhttps://www.cl.cam.ac.uk/teaching/2627/FoundsCS/forest/focs-ex-10-3/focs-ex-10-3/teaching/2627/FoundsCS/forest/focs-ex-10-3/Rewriting breadth with letExerciseWrite a version of the function breadth using a nested let construction rather than match.Anil MadhavapeddyAnil Madhavapeddyhttps://www.cl.cam.ac.uk/teaching/2627/FoundsCS/forest/focs-ex-10-4/focs-ex-10-4/teaching/2627/FoundsCS/forest/focs-ex-10-4/Search with a branching factor near oneExerciseIterative deepening is inappropriate if , where is the branching factor. What search strategy is appropriate in this case?Anil MadhavapeddyAnil Madhavapeddyhttps://www.cl.cam.ac.uk/teaching/2627/FoundsCS/forest/focs-ex-10-5/focs-ex-10-5/teaching/2627/FoundsCS/forest/focs-ex-10-5/An implicit infinite treeExerciseConsider the following OCaml function.If we regard it as representing a tree, where the subtrees are computed from the current label, what tree does next 1 represent?Anil Madhavapeddyhttps://www.cl.cam.ac.uk/teaching/2627/FoundsCS/forest/focs-procedural/focs-procedural/teaching/2627/FoundsCS/forest/focs-procedural/Elements of Procedural ProgrammingLecturehttps://hub.cl.cam.ac.uk/Anil Madhavapeddyhttps://www.cl.cam.ac.uk/teaching/2627/FoundsCS/forest/focs-procedural-programming/focs-procedural-programming/teaching/2627/FoundsCS/forest/focs-procedural-programming/Procedural ProgrammingProcedural programs can change the machine state. They can interact with its environment They use control structures like branching, iteration and procedures.They use data abstractions of the computer’s memory:references to memory cells arrays that are blocks of memory cells linked structures such as linked listsProcedural programming is programming in the traditional sense of the word. A program state is repeatedly transformed by the execution of commands or statements. A state change might be local to the machine and consist of updating a variable or array. A state change might consist of sending data to the outside world. Even reading data counts as a state change, since this act normally removes the data from the environment.Procedural programming languages provide primitive commands and control structures for combining them. The primitive commands include assignment for updating variables, and various input/output commands for communication. Control structures include if and match constructs for conditional execution, and repetitive constructs such as while. Programmers can package up their own commands as procedures taking arguments. The need for such “subroutines” was evident from the earliest days; they represent one of the first examples of abstraction in programming languages.OCaml makes no distinction between commands and expressions. OCaml provides built-in ‘functions’ to perform assignment and communication, and these can be used in the traditional (procedural) style. OCaml programmers often follow a functional style for most internal computations and use imperative features mainly for communication with the outside world.Anil Madhavapeddyhttps://www.cl.cam.ac.uk/teaching/2627/FoundsCS/forest/focs-references/focs-references/teaching/2627/FoundsCS/forest/focs-references/OCaml Primitives for References SyntaxEffect ref Ecreate a reference with initial contents = value of E !Preturn current contents of reference P P := Eupdate contents of P to value of E The above text presents the OCaml primitives, but most languages have analogues of them, often heavily disguised. We need a means of creating references (or allocating storage), getting at the current contents of a reference cell, and updating that cell.The function ref creates references (also called “locations”). Calling ref allocates a new location in memory. Initially, this location holds the value given by expression .The function !, when applied to a reference, returns its contents. This operation is called dereferencing. Clearly ! is not a mathematical function; its result depends upon the store.The assignment P:=E evaluates expression , which must return a reference , and . It stores at address the value of . Syntactically, := is a function and P:=E is an expression, even though it updates the store. Like many functions that change the state, it returns the value () of type unit.If is some OCaml type, then ref is the type of references to cells that can hold values of . Please do not confuse the type ref with the function ref. This table of the primitive functions and their types might be useful: SyntaxOCaml Type ref'a -> 'a ref !'a ref -> 'a :='a ref -> 'a -> unit Anil Madhavapeddyhttps://www.cl.cam.ac.uk/teaching/2627/FoundsCS/forest/focs-references-examples/focs-references-examples/teaching/2627/FoundsCS/forest/focs-references-examples/Trying Out ReferencesThe first line declares p to hold a reference to an integer, initially 5. Its type is int ref, not just int, so it admits assignment. Assignment never changes let bindings: they are immutable. The identifier p will always denote the reference mentioned in its declaration unless superseded by a new usage of p. Only the contents of the reference is mutable.OCaml displays a reference value as , where value is the contents. This notation is readable but gives us no way of telling whether two references holding the same value are actually the same reference. To display a reference as a machine address has obvious drawbacks!In the first assignment, the expression !p yields the reference’s current contents, namely 5. The assignment changes the contents of p to 6. Most languages do not have an explicit dereferencing operator (like !) because of its inconvenience. Instead, by convention, occurrences of the reference on the left-hand side of the := denote locations and those on the right-hand side denote the contents. A special ‘address of’ operator may be available to override the convention and make a reference on the right-hand side to denote a location. Logically this is a mess, but it makes programs shorter.The list ps is declared to hold a new reference (initially containing 77) as well as p. Then the new reference is updated to hold 3. The assignment to hd ps does not update ps, only the contents of a reference in that list.Anil Madhavapeddyhttps://www.cl.cam.ac.uk/teaching/2627/FoundsCS/forest/focs-commands/focs-commands/teaching/2627/FoundsCS/forest/focs-commands/Commands: Expressions with EffectsBasic commands update references, write to files, etc. causes a series of expressions to be evaluated and returns the value of . A typical command returns the empty tuple: () if then else behaves like the traditional control structure if and have effects. Other OCaml constructs behave naturally with commands, including match expressions and recursive functions.We use the term command informally to refer to an expression that has an effect on the state. All expressions denote some value, but they can return (), which conveys no actual information.We need a way to execute one command after another. The construct evaluates the expressions to in the order given and returns the value of . The values of the other expressions are discarded; their only purpose is to change the state.Commands may be used with if and match much as in conventional languages. OCaml functions play the role of procedures.Other languages that combine the functional and imperative programming paradigms include Lisp (and its dialect Scheme), F#, Scala, Swift and Kotlin. Rust is a notable modern example: a systems programming language whose type system and pattern matching draw heavily on the ML family (its first compiler was written in OCaml), yet which gives the programmer precise control over mutable state.Anil Madhavapeddyhttps://www.cl.cam.ac.uk/teaching/2627/FoundsCS/forest/focs-while/focs-while/teaching/2627/FoundsCS/forest/focs-while/Iteration: the while command None | _::xs -> Some xs val tlopt : 'a list -> 'a list option = # let length xs = let lp = ref xs in (* list of uncounted elements *) let np = ref 0 in (* accumulated count *) let fin = ref false in while not !fin do match tlopt !lp with | None -> fin := true | Some xs -> lp := xs; np := 1 + !np done; !np (* the final count is returned *) val length : 'a list -> int = ]]>Once we can change the state, we need to do so repeatedly. Recursion can serve this purpose, but having to declare a procedure for every loop is clumsy, and compilers for conventional languages seldom exploit tail-recursion.Early programming languages provided little support for repetition. The programmer had to set up loops using goto commands, exiting the loop using another goto controlled by an if. Modern languages provide a confusing jumble of looping constructs, the most fundamental of which is while B do C. The boolean expression is evaluated, and if true, command is executed and the command repeats. If evaluates to false then the while command terminates, perhaps without executing even once.OCaml’s main looping construct is while, which returns the value (). The function length declares references to hold the list under examination (lp) and number of elements counted so far (np) as well as whether the end of the list has been reached (the boolean reference fin). While the list is non-empty, we skip over one more element (by setting it to its tail) and count that element.The body of the while loop first checks to see if the end of the list has been reached, in which case it sets the fin variable to true. If there is a tail value, then two assignments are executed in sequence. The lp reference is set to the tail of the list, and the np reference integer is incremented by one. When the while loop terminates due to the fin variable being set to true, the expression !np returns the computed length as the function’s result.Anil Madhavapeddyhttps://www.cl.cam.ac.uk/teaching/2627/FoundsCS/forest/focs-private-references/focs-private-references/teaching/2627/FoundsCS/forest/focs-private-references/Private, Persistent References !balance then raise (TooMuch (amt - !balance)) else begin balance := !balance - amt; !balance end in withdraw val makeAccount : int -> int -> int = ]]>As you may have noticed, OCaml’s programming style looks clumsy compared with that of languages like C. OCaml omits the defaults and abbreviations they provide to shorten programs. However, OCaml’s explicitness makes it ideal for teaching the fine points of references and arrays. OCaml’s references are more flexible than those found in other languages.The function makeAccount models a bank. Calling the function with a specified initial balance creates a new reference (balance) to maintain the account balance and returns a function (withdraw) having sole access to that reference. Calling withdraw reduces the balance by the specified amount and returns the new balance. You can pay money in by withdrawing a negative amount. The if-construct prevents the account from going overdrawn, raising an exception.Look at the sequence inside begin … end in the else part above. The first expression updates the account balance and returns the trivial value (). The second expression, !balance, returns the current balance but does not return the reference itself: that would allow unauthorised updates.This example is based on one by Dr A C Norman.Anil Madhavapeddyhttps://www.cl.cam.ac.uk/teaching/2627/FoundsCS/forest/focs-bank-accounts/focs-bank-accounts/teaching/2627/FoundsCS/forest/focs-bank-accounts/Two Bank Accounts int = # let director = makeAccount 4000000 val director : int -> int = # student 5 (* coach fare *) - : int = 495 # director 150000 (* Tesla *) - : int = 3850000 # student 500 (* oh oh *) Exception: TooMuch 5.]]>Each call to makeAccount returns a copy of withdraw holding a fresh instance of the reference balance. As with a real bank pass-book, there is no access to the account balance except via the corresponding withdraw function. If that function is discarded, the reference cell becomes unreachable; the computer will eventually reclaim it, just as banks close down dormant accounts.Here we see two people managing their accounts. For better or worse, neither can take money from the other.We could generalise makeAccount to return several functions that jointly manage information held in shared references. The functions might be packaged using OCaml records, which are not discussed in this course. Most procedural languages do not properly support the concept of private references, although object-oriented languages take them as a basic theme.Anil Madhavapeddyhttps://www.cl.cam.ac.uk/teaching/2627/FoundsCS/forest/focs-array-primitives/focs-array-primitives/teaching/2627/FoundsCS/forest/focs-array-primitives/OCaml Primitives for Arrays i * 10) (* array[5] initialised to (fun i) *) val aa : int array = [|0; 10; 20; 30; 40|] # Array.get aa 3 (* retrieve the 4th cell in the array *) - : int = 30 # Array.set aa 3 42 (* set the 4th cell's value to 42 *) - : unit = ()]]>There are many other array operations in the Array module in the OCaml standard library. 'a -> 'a array = # Array.init - : int -> (int -> 'a) -> 'a array = # Array.get - : 'a array -> int -> 'a = # Array.set - : 'a array -> int -> 'a -> unit = ]]>OCaml arrays are like references that hold several elements instead of one. The elements of an -element array are designated by the integers from 0 to . The th array element is usually written . If is a type then array is the type of arrays (of any size) with elements from .Calling Array.init n f creates an array of the size specified in by function . Initially, element holds the value of for , …, . Like ref, it allocates mutable storage to hold the specified values.Calling Array.get A i returns the contents of .Calling Array.set A i E modifies the array by storing the value of as the new contents of ; it returns () as its value.OCaml’s arrays are much safer than C’s. In C, an array is nothing more than an address indicating the start of a storage area. Nothing indicates the size of the area. Therefore C programs are vulnerable to buffer overrun attacks: an attacker sends more data than the receiving program expects, overrunning the area of storage set aside to hold it. The attack eventually overwrites the program itself, replacing it with code controlled by the attacker.Anil Madhavapeddyhttps://www.cl.cam.ac.uk/teaching/2627/FoundsCS/forest/focs-array-examples/focs-array-examples/teaching/2627/FoundsCS/forest/focs-array-examples/Array ExamplesIn the following session, the identifier ar is bound to an array of 20 elements, which are initially set to the squares of their subscripts. The array’s third element (which actually has subscript 2) is inspected and found to be four. The second call to Array.get supplies a subscript that is out of range, so OCaml rejects it. i * i) val ar : int array = [|0; 1; 4; 9; 16; 25; 36; 49; 64; 81; 100; 121; 144; 169; 196; 225; 256; 289; 324; 361|] # Array.get ar 2 - : int = 4 # Array.get ar 20 Exception: Invalid_argument "index out of bounds". # Array.set ar 2 33; ar - : int array = [|0; 1; 33; 9; 16; 25; 36; 49; 64; 81; 100; 121; 144; 169; 196; 225; 256; 289; 324; 361|]]]>By calling Array.set, we then modify the element with subscript 2. Note however that we cannot modify the array’s length. If we outgrow the array, we have to create a new one, copy the data into it, and then forget the old array. Typically the new array would be double the size of the old one, so that the cost of copying is insignificant.OCaml provides numerous operators for modifying, computing over and searching in arrays. Many are analogous to functions on lists. For example, Array.exists takes a boolean-valued function and returns true if an array element satisfies it. i > 200) ar - : bool = true # Array.exists (fun i -> i < 0) ar - : bool = false]]>Anil Madhavapeddyhttps://www.cl.cam.ac.uk/teaching/2627/FoundsCS/forest/focs-references-comparison/focs-references-comparison/teaching/2627/FoundsCS/forest/focs-references-comparison/References: OCaml vs conventional languagesWe must write !p to get the contents of p We write just p for the address of p We can store private reference cells (like balance) in functions—analogous to elements of object-oriented programming OCaml’s assignment syntax is instead of = OCaml has few control structures: while, match, if and for (the latter is not covered in this course) OCaml has syntax for updating an array via the a.(i) <- v syntax which is the same as Array.set a i v.Conventional syntax for variables and assignments has hardly changed since Fortran, the first high-level language. In conventional languages, virtually all variables can be updated. We declare something like p: int, mentioning no reference type even if the language provides them. If we do not specify an initial value, we may get whatever bits were previously at that address. Illegal values arising from uninitialised variables can cause errors that are almost impossible to diagnose.Dereferencing operators (like OCaml’s !) are especially unpopular, because they clutter the program text. Virtually all programming languages make dereferencing implicit (that is, automatic).It is generally accepted these days that a two-dimensional array is nothing but an array of arrays. An assignment to such an array is typically written something like ; in C, the syntax is A[i][j] = x. Higher dimensions are treated analogously. The corresponding OCaml code can either declare an array of arrays, or use the A.(i) syntax to calculate the linear offset into a single array.You can use the constructs we have learnt to easily create linked (mutable) lists as an alternative to arrays.It is worth mentioning that OCaml’s references fully suffice for coding the sort of linked data structures taught in algorithms courses, as the mlist type above illustrates. The programming style is a little different from the usual, but the principles are the same. OCaml also provides comprehensive input/output primitives for various types of file and operating system.OCaml’s system of modules include structures, which can be seen as encapsulated groups of declarations, and signatures, which are specifications of structures listing the name and type of each component. Finally, there are functors, which are analogous to functions that combine a number of argument structures, and which can be used to plug program components together. These primitives are useful for managing large programming projects.Anil Madhavapeddyhttps://www.cl.cam.ac.uk/teaching/2627/FoundsCS/forest/focs-ex-11/focs-ex-11/teaching/2627/FoundsCS/forest/focs-ex-11/ExercisesAnil MadhavapeddyAnil Madhavapeddyhttps://www.cl.cam.ac.uk/teaching/2627/FoundsCS/forest/focs-ex-11-1/focs-ex-11-1/teaching/2627/FoundsCS/forest/focs-ex-11-1/Lists of referencesExerciseComment, with examples, on the differences between an int ref list and an int list ref.Anil MadhavapeddyAnil Madhavapeddyhttps://www.cl.cam.ac.uk/teaching/2627/FoundsCS/forest/focs-ex-11-2/focs-ex-11-2/teaching/2627/FoundsCS/forest/focs-ex-11-2/Iterative power with whileExerciseWrite a version of function power (Lecture ) using while instead of recursion.Anil MadhavapeddyAnil Madhavapeddyhttps://www.cl.cam.ac.uk/teaching/2627/FoundsCS/forest/focs-ex-11-3/focs-ex-11-3/teaching/2627/FoundsCS/forest/focs-ex-11-3/Effects in a while conditionExerciseWhat is the effect of ?Anil MadhavapeddyAnil Madhavapeddyhttps://www.cl.cam.ac.uk/teaching/2627/FoundsCS/forest/focs-ex-11-4/focs-ex-11-4/teaching/2627/FoundsCS/forest/focs-ex-11-4/Exchanging two referencesExerciseWrite a function to exchange the values of two references, xr and yr.Anil MadhavapeddyAnil Madhavapeddyhttps://www.cl.cam.ac.uk/teaching/2627/FoundsCS/forest/focs-ex-11-5/focs-ex-11-5/teaching/2627/FoundsCS/forest/focs-ex-11-5/Matrix operations on arraysExerciseArrays of multiple dimensions are represented in OCaml by arrays of arrays. Write functions to (a) create an identity matrix, given , and (b) to transpose an matrix. Identity matrices have the following form:ReferencesContextBacklinksRelatedAnil Madhavapeddyhttps://www.cl.cam.ac.uk/teaching/2627/FoundsCS/forest/focs-all-exercises/focs-all-exercises/teaching/2627/FoundsCS/forest/focs-all-exercises/All exercisesEvery exercise in the course, collected automatically from the lecture notes.Anil MadhavapeddyAnil Madhavapeddyhttps://www.cl.cam.ac.uk/teaching/2627/FoundsCS/forest/focs-ex-8-4/focs-ex-8-4/teaching/2627/FoundsCS/forest/focs-ex-8-4/Exercises › A map for optionsExerciseThe type 'a option, declared below, can be viewed as a type of lists having at most one element. (It is typically used as an alternative to exceptions.) Declare an analogue of the function map for type 'a option.Anil MadhavapeddyAnil Madhavapeddyhttps://www.cl.cam.ac.uk/teaching/2627/FoundsCS/forest/focs-ex-9-1/focs-ex-9-1/teaching/2627/FoundsCS/forest/focs-ex-9-1/Exercises › A map for sequencesExerciseCode an analogue of map for sequences.Anil MadhavapeddyAnil Madhavapeddyhttps://www.cl.cam.ac.uk/teaching/2627/FoundsCS/forest/focs-ex-4-3/focs-ex-4-3/teaching/2627/FoundsCS/forest/focs-ex-4-3/Exercises › A non-exhaustive zipExerciseHow does this version of zip differ from the one above? (x, y) :: zip xs ys | ([], []) -> []]]>Anil MadhavapeddyAnil Madhavapeddyhttps://www.cl.cam.ac.uk/teaching/2627/FoundsCS/forest/focs-ex-2-4/focs-ex-2-4/teaching/2627/FoundsCS/forest/focs-ex-2-4/Exercises › A tighter recurrence boundExerciseFind an upper bound for the recurrence given by and . You should be able to find a tighter bound than .Anil MadhavapeddyAnil Madhavapeddyhttps://www.cl.cam.ac.uk/teaching/2627/FoundsCS/forest/focs-ex-6-4/focs-ex-6-4/teaching/2627/FoundsCS/forest/focs-ex-6-4/Exercises › A type for arithmetic expressionsExerciseGive the declaration of an OCaml type for arithmetic expressions that have the following possibilities: floating-point numbers, variables (represented by strings), or expressions of the form , , .Anil MadhavapeddyAnil Madhavapeddyhttps://www.cl.cam.ac.uk/teaching/2627/FoundsCS/forest/focs-ex-9-5/focs-ex-9-5/teaching/2627/FoundsCS/forest/focs-ex-9-5/Exercises › All lists of zeroes and onesExerciseCode the lazy list whose elements are all ordinary lists of zeroes and ones, namely []; [0]; [1]; [0; 0]; [0; 1]; [1; 0]; [1; 1]; [0; 0; 0]; …. (Taken from the exam question 2003 Paper 1 Question 5.)Anil MadhavapeddyAnil Madhavapeddyhttps://www.cl.cam.ac.uk/teaching/2627/FoundsCS/forest/focs-ex-10-5/focs-ex-10-5/teaching/2627/FoundsCS/forest/focs-ex-10-5/Exercises › An implicit infinite treeExerciseConsider the following OCaml function.If we regard it as representing a tree, where the subtrees are computed from the current label, what tree does next 1 represent?Anil MadhavapeddyAnil Madhavapeddyhttps://www.cl.cam.ac.uk/teaching/2627/FoundsCS/forest/focs-ex-10-2/focs-ex-10-2/teaching/2627/FoundsCS/forest/focs-ex-10-2/Exercises › Array-based circular queuesExerciseThe traditional way to implement queues uses a fixed-length array. Two indices into the array indicate the start and end of the queue, which wraps around from the end of the array to the start. How appropriate is such a data structure for implementing breadth-first search?Anil MadhavapeddyAnil Madhavapeddyhttps://www.cl.cam.ac.uk/teaching/2627/FoundsCS/forest/focs-ex-4-4/focs-ex-4-4/teaching/2627/FoundsCS/forest/focs-ex-4-4/Exercises › Assumptions in making changeExerciseWhat assumptions do the ‘making change’ functions make about the variables till and amt? Describe what could happen if these assumptions were violated.Anil MadhavapeddyAnil Madhavapeddyhttps://www.cl.cam.ac.uk/teaching/2627/FoundsCS/forest/focs-ex-1-2/focs-ex-1-2/teaching/2627/FoundsCS/forest/focs-ex-1-2/Exercises › Comparing and adding yearsExerciseUsing the date representation of the previous exercise, code OCaml functions to (a) compare two years (b) add/subtract some given number of years from another year.Anil MadhavapeddyAnil Madhavapeddyhttps://www.cl.cam.ac.uk/teaching/2627/FoundsCS/forest/focs-ex-5-3/focs-ex-5-3/teaching/2627/FoundsCS/forest/focs-ex-5-3/Exercises › Complexity of bubble sortExerciseAnother sorting algorithm (bubble sort) consists of looking at adjacent pairs of elements, exchanging them if they are out of order and repeating this process until no more exchanges are possible. State, with justification, the time complexity of this approach.Anil MadhavapeddyAnil Madhavapeddyhttps://www.cl.cam.ac.uk/teaching/2627/FoundsCS/forest/focs-ex-5-1/focs-ex-5-1/teaching/2627/FoundsCS/forest/focs-ex-5-1/Exercises › Complexity of selection sortExerciseAnother sorting algorithm (selection sort) consists of looking at the elements to be sorted, identifying and removing a minimal element, which is placed at the head of the result. The tail is obtained by recursively sorting the remaining elements. State, with justification, the time complexity of this approach.Anil MadhavapeddyAnil Madhavapeddyhttps://www.cl.cam.ac.uk/teaching/2627/FoundsCS/forest/focs-ex-9-2/focs-ex-9-2/teaching/2627/FoundsCS/forest/focs-ex-9-2/Exercises › Concatenating infinite sequencesExerciseConsider the list function concat, which concatenates a list of lists to form a single list. Can it be generalised to concatenate a sequence of sequences? What can go wrong? [] | l::ls -> l @ concat ls]]>Anil MadhavapeddyAnil Madhavapeddyhttps://www.cl.cam.ac.uk/teaching/2627/FoundsCS/forest/focs-ex-4-5/focs-ex-4-5/teaching/2627/FoundsCS/forest/focs-ex-4-5/Exercises › Counting ways of making changeExerciseShow that the number of ways of making change for (ignoring order) is if there are two legal coin values. What if there are three, four, … coin values?Anil MadhavapeddyAnil Madhavapeddyhttps://www.cl.cam.ac.uk/teaching/2627/FoundsCS/forest/focs-ex-6-1/focs-ex-6-1/teaching/2627/FoundsCS/forest/focs-ex-6-1/Exercises › Days of the week typeExerciseGive the declaration of an OCaml type for the days of the week. Comment on the practicality of such a type in a calendar application.Anil MadhavapeddyAnil Madhavapeddyhttps://www.cl.cam.ac.uk/teaching/2627/FoundsCS/forest/focs-ex-7-4/focs-ex-7-4/teaching/2627/FoundsCS/forest/focs-ex-7-4/Exercises › Deleting from a search treeExerciseDescribe an algorithm for deleting an entry from a binary search tree. Comment on the suitability of your approach.Anil MadhavapeddyAnil Madhavapeddyhttps://www.cl.cam.ac.uk/teaching/2627/FoundsCS/forest/focs-ex-11-3/focs-ex-11-3/teaching/2627/FoundsCS/forest/focs-ex-11-3/Exercises › Effects in a while conditionExerciseWhat is the effect of ?Anil MadhavapeddyAnil Madhavapeddyhttps://www.cl.cam.ac.uk/teaching/2627/FoundsCS/forest/focs-ex-9-6/focs-ex-9-6/teaching/2627/FoundsCS/forest/focs-ex-9-6/Exercises › Enumerating palindromesExercise(Continuing the previous exercise.) A palindrome is a list that equals its own reverse. Code the lazy list whose elements are all palindromes of 0s and 1s, namely []; [0]; [1]; [0; 0]; [0; 0; 0]; [0; 1; 0]; [1; 1]; [1; 0; 1]; [1; 1; 1]; [0; 0; 0; 0]; , …. You may take the reversal function List.rev as given.Anil MadhavapeddyAnil Madhavapeddyhttps://www.cl.cam.ac.uk/teaching/2627/FoundsCS/forest/focs-ex-6-5/focs-ex-6-5/teaching/2627/FoundsCS/forest/focs-ex-6-5/Exercises › Evaluating expressions with exceptionsExerciseContinuing the previous exercise, write a function that evaluates an expression. If the expression contains any variables, your function should raise an exception indicating the variable name.Anil MadhavapeddyAnil Madhavapeddyhttps://www.cl.cam.ac.uk/teaching/2627/FoundsCS/forest/focs-ex-3-3/focs-ex-3-3/teaching/2627/FoundsCS/forest/focs-ex-3-3/Exercises › Even-numbered elementsExerciseCode a function to return the list consisting of the even-numbered elements of the list given as its argument. For example, given [a; b; c; d] it should return [b; d].Anil MadhavapeddyAnil Madhavapeddyhttps://www.cl.cam.ac.uk/teaching/2627/FoundsCS/forest/focs-ex-11-4/focs-ex-11-4/teaching/2627/FoundsCS/forest/focs-ex-11-4/Exercises › Exchanging two referencesExerciseWrite a function to exchange the values of two references, xr and yr.Anil MadhavapeddyAnil Madhavapeddyhttps://www.cl.cam.ac.uk/teaching/2627/FoundsCS/forest/focs-ex-2-2/focs-ex-2-2/teaching/2627/FoundsCS/forest/focs-ex-2-2/Exercises › Extending the complexity tableExerciseAdd a column to the table of complexities from The Design and Analysis of Computer Algorithms with the heading 60 hours: complexity1 second1 minute1 hour60 hours 100060 0003 600 000 1404 895204 095 312441 897 1039153 91521 Anil MadhavapeddyAnil Madhavapeddyhttps://www.cl.cam.ac.uk/teaching/2627/FoundsCS/forest/focs-ex-1-5/focs-ex-1-5/teaching/2627/FoundsCS/forest/focs-ex-1-5/Exercises › Floating-point error by repeated additionExerciseBecause computer arithmetic is based on binary numbers, simple decimals such as 0.1 often cannot be represented exactly. Write a function mul that performs the computationwhere has type float. (It is essential to use repeated addition rather than multiplication!)The value computed with n = 10000 and x = 0.1 may print as 1000.0, which looks exact. If that happens, then evaluate the expression mul 0.1 10000 -. 1000.0An error of this type has been blamed for the failure of an American Patriot Missile battery to intercept an incoming Iraqi missile during the first Gulf War. The missile hit an American Army barracks, killing 28.Anil MadhavapeddyAnil Madhavapeddyhttps://www.cl.cam.ac.uk/teaching/2627/FoundsCS/forest/focs-ex-1-6/focs-ex-1-6/teaching/2627/FoundsCS/forest/focs-ex-1-6/Exercises › Golden ratio instabilityExerciseAnother example of the inaccuracy of floating-point arithmetic takes the golden ratio as its starting point:In theory, it is easy to prove that for all 0]]>. Code this computation in OCaml and report the value of . Hint: in OCaml, is expressed as sqrt 5.0.Anil MadhavapeddyAnil Madhavapeddyhttps://www.cl.cam.ac.uk/teaching/2627/FoundsCS/forest/focs-ex-5-4/focs-ex-5-4/teaching/2627/FoundsCS/forest/focs-ex-5-4/Exercises › Implementing bubble sortExerciseImplement bubble sort (see previous exercise) using OCaml.Anil MadhavapeddyAnil Madhavapeddyhttps://www.cl.cam.ac.uk/teaching/2627/FoundsCS/forest/focs-ex-7-5/focs-ex-7-5/teaching/2627/FoundsCS/forest/focs-ex-7-5/Exercises › Implementing deletionExerciseCode the delete function outlined in the previous exercise.Anil MadhavapeddyAnil Madhavapeddyhttps://www.cl.cam.ac.uk/teaching/2627/FoundsCS/forest/focs-ex-5-2/focs-ex-5-2/teaching/2627/FoundsCS/forest/focs-ex-5-2/Exercises › Implementing selection sortExerciseImplement selection sort (see previous exercise) using OCaml.Anil MadhavapeddyAnil Madhavapeddyhttps://www.cl.cam.ac.uk/teaching/2627/FoundsCS/forest/focs-ex-7-1/focs-ex-7-1/teaching/2627/FoundsCS/forest/focs-ex-7-1/Exercises › Insertion order in search treesExerciseDraw the binary search tree that arises from successively inserting the following pairs into the empty tree: ("Alice", 6), ("Tobias", 2), ("Gerald", 8), ("Lucy", 9). Then repeat this task using the order ("Gerald", 8), ("Alice", 6), ("Lucy", 9), ("Tobias", 2). Why are results different?Anil MadhavapeddyAnil Madhavapeddyhttps://www.cl.cam.ac.uk/teaching/2627/FoundsCS/forest/focs-ex-7-2/focs-ex-7-2/teaching/2627/FoundsCS/forest/focs-ex-7-2/Exercises › Insertion with collision detectionExerciseCode an insertion function for binary search trees. It should resemble the existing update function except that it should raise the exception Collision if the item to be inserted is already present.Anil MadhavapeddyAnil Madhavapeddyhttps://www.cl.cam.ac.uk/teaching/2627/FoundsCS/forest/focs-ex-2-1/focs-ex-2-1/teaching/2627/FoundsCS/forest/focs-ex-2-1/Exercises › Iterative powerExerciseCode an iterative version of the function power.Anil MadhavapeddyAnil Madhavapeddyhttps://www.cl.cam.ac.uk/teaching/2627/FoundsCS/forest/focs-ex-11-2/focs-ex-11-2/teaching/2627/FoundsCS/forest/focs-ex-11-2/Exercises › Iterative power with whileExerciseWrite a version of function power (Lecture ) using while instead of recursion.Anil MadhavapeddyAnil Madhavapeddyhttps://www.cl.cam.ac.uk/teaching/2627/FoundsCS/forest/focs-ex-3-2/focs-ex-3-2/teaching/2627/FoundsCS/forest/focs-ex-3-2/Exercises › Last element of a listExerciseCode a function to return the last element of a non-empty list. How efficiently can this be done?Anil MadhavapeddyAnil Madhavapeddyhttps://www.cl.cam.ac.uk/teaching/2627/FoundsCS/forest/focs-ex-9-4/focs-ex-9-4/teaching/2627/FoundsCS/forest/focs-ex-9-4/Exercises › Lazy binary treesExerciseA lazy binary tree is either empty or is a branch containing a label and two lazy binary trees, possibly to infinite depth. Present an OCaml datatype to represent lazy binary trees, along with a function that accepts a lazy binary tree and produces a lazy list that contains all of the tree’s labels. (Taken from the exam question 2008 Paper 1 Question 5.)Anil MadhavapeddyAnil Madhavapeddyhttps://www.cl.cam.ac.uk/teaching/2627/FoundsCS/forest/focs-ex-8-2/focs-ex-8-2/teaching/2627/FoundsCS/forest/focs-ex-8-2/Exercises › Lexicographic orderingsExerciseThere are many ways of combining orderings. The lexicographic ordering uses two keys for comparisons. It is specified byWrite an OCaml function to lexicographically combine two orderings, supplied as functions. Explain how it allows function insort to sort a list of pairs.Anil MadhavapeddyAnil Madhavapeddyhttps://www.cl.cam.ac.uk/teaching/2627/FoundsCS/forest/focs-ex-7-7/focs-ex-7-7/teaching/2627/FoundsCS/forest/focs-ex-7-7/Exercises › Linear traversalsExerciseShow that the functions preord, inord and postord all take linear time in the size of the tree.Anil MadhavapeddyAnil Madhavapeddyhttps://www.cl.cam.ac.uk/teaching/2627/FoundsCS/forest/focs-ex-11-1/focs-ex-11-1/teaching/2627/FoundsCS/forest/focs-ex-11-1/Exercises › Lists of referencesExerciseComment, with examples, on the differences between an int ref list and an int list ref.Anil MadhavapeddyAnil Madhavapeddyhttps://www.cl.cam.ac.uk/teaching/2627/FoundsCS/forest/focs-ex-9-3/focs-ex-9-3/teaching/2627/FoundsCS/forest/focs-ex-9-3/Exercises › Making change lazilyExerciseCode a function to make change using lazy lists, delivering the sequence of all possible ways of making change. Using sequences allows us to compute solutions one at a time when there exists an astronomical number. Represent lists of coins using ordinary lists. (Hint: to benefit from laziness you may need to pass around the sequence of alternative solutions as a function of type unit -> (int list) seq.)Anil MadhavapeddyAnil Madhavapeddyhttps://www.cl.cam.ac.uk/teaching/2627/FoundsCS/forest/focs-ex-8-5/focs-ex-8-5/teaching/2627/FoundsCS/forest/focs-ex-8-5/Exercises › Making change with mapExerciseRecall the making change function of More on Lists: [ [] ] | [] , _ -> [] | c::till , amt -> if amt < c then change till amt else let rec allc = function | [] -> [] | cs :: css -> (c::cs) :: allc css in allc (change (c::till) (amt - c)) @ change till amt val change : int list -> int -> int list list = ]]>Function allc applies the function ‘cons a c’ to every element of a list. Eliminate it by declaring a curried cons function and applying map.Anil MadhavapeddyAnil Madhavapeddyhttps://www.cl.cam.ac.uk/teaching/2627/FoundsCS/forest/focs-ex-8-3/focs-ex-8-3/teaching/2627/FoundsCS/forest/focs-ex-8-3/Exercises › Mapping over lists of listsExerciseWithout using map write a function map2 such that map2 f is equivalent to map (map f). The obvious solution requires declaring two recursive functions. Try to get away with only one by exploiting nested pattern-matching.Anil MadhavapeddyAnil Madhavapeddyhttps://www.cl.cam.ac.uk/teaching/2627/FoundsCS/forest/focs-ex-11-5/focs-ex-11-5/teaching/2627/FoundsCS/forest/focs-ex-11-5/Exercises › Matrix operations on arraysExerciseArrays of multiple dimensions are represented in OCaml by arrays of arrays. Write functions to (a) create an identity matrix, given , and (b) to transpose an matrix. Identity matrices have the following form:Anil MadhavapeddyAnil Madhavapeddyhttps://www.cl.cam.ac.uk/teaching/2627/FoundsCS/forest/focs-ex-4-2/focs-ex-4-2/teaching/2627/FoundsCS/forest/focs-ex-4-2/Exercises › Partitioning integers by signExerciseCode a function that takes a list of integers and returns two lists, the first consisting of all non-negative numbers found in the input and the second consisting of all the negative numbers.Anil MadhavapeddyAnil Madhavapeddyhttps://www.cl.cam.ac.uk/teaching/2627/FoundsCS/forest/focs-ex-3-4/focs-ex-3-4/teaching/2627/FoundsCS/forest/focs-ex-3-4/Exercises › Polymorphic types of id and loopExerciseConsider the polymorphic types in these two function declarations: 'a = # let rec loop x = loop x val loop : 'a -> 'b = ]]>Explain why these types make logical sense, preventing run time type errors, even for expressions like id [id [id 0]] or loop true / loop 3. (/ is the integer division operator in OCaml)Anil MadhavapeddyAnil Madhavapeddyhttps://www.cl.cam.ac.uk/teaching/2627/FoundsCS/forest/focs-ex-7-6/focs-ex-7-6/teaching/2627/FoundsCS/forest/focs-ex-7-6/Exercises › Quadratic traversalsExerciseShow that the functions preorder, inorder and postorder all require time in the worst case, where is the size of the tree.Anil MadhavapeddyAnil Madhavapeddyhttps://www.cl.cam.ac.uk/teaching/2627/FoundsCS/forest/focs-ex-4-6/focs-ex-4-6/teaching/2627/FoundsCS/forest/focs-ex-4-6/Exercises › Reasoning from polymorphic typesExerciseWe know nothing about the functions f and g other than their polymorphic types: val f : 'a * 'b -> 'b * 'a and val g : 'a -> 'a list. Suppose that f (1, true) and g 0 are evaluated and return their results. State, with reasons, what you think the resulting values will be.Anil MadhavapeddyAnil Madhavapeddyhttps://www.cl.cam.ac.uk/teaching/2627/FoundsCS/forest/focs-ex-1-3/focs-ex-1-3/teaching/2627/FoundsCS/forest/focs-ex-1-3/Exercises › Redundant conditionalsExerciseWhy would no experienced programmer write an expression of the form if … then true else false? What about expressions of the form if … then false else true?Anil MadhavapeddyAnil Madhavapeddyhttps://www.cl.cam.ac.uk/teaching/2627/FoundsCS/forest/focs-ex-7-3/focs-ex-7-3/teaching/2627/FoundsCS/forest/focs-ex-7-3/Exercises › Returning the old value on collisionExerciseContinuing the previous exercise, it would be natural for exceptional Collision to return the value previously stored in the dictionary. Why is that goal difficult to achieve?Anil MadhavapeddyAnil Madhavapeddyhttps://www.cl.cam.ac.uk/teaching/2627/FoundsCS/forest/focs-ex-10-3/focs-ex-10-3/teaching/2627/FoundsCS/forest/focs-ex-10-3/Exercises › Rewriting breadth with letExerciseWrite a version of the function breadth using a nested let construction rather than match.Anil MadhavapeddyAnil Madhavapeddyhttps://www.cl.cam.ac.uk/teaching/2627/FoundsCS/forest/focs-ex-10-4/focs-ex-10-4/teaching/2627/FoundsCS/forest/focs-ex-10-4/Exercises › Search with a branching factor near oneExerciseIterative deepening is inappropriate if , where is the branching factor. What search strategy is appropriate in this case?Anil MadhavapeddyAnil Madhavapeddyhttps://www.cl.cam.ac.uk/teaching/2627/FoundsCS/forest/focs-ex-4-1/focs-ex-4-1/teaching/2627/FoundsCS/forest/focs-ex-4-1/Exercises › Set union without repetitionsExerciseSets can be represented in OCaml using lists containing no duplicated items (i.e. where no item is equal to another using polymorphic comparison).Using the member function defined above, code a function to implement set union. It should avoid introducing repetitions, for example the union of the lists [4; 7; 1] and [6; 4; 7] should be [1; 6; 4; 7] (though the order does not matter).Anil MadhavapeddyAnil Madhavapeddyhttps://www.cl.cam.ac.uk/teaching/2627/FoundsCS/forest/focs-ex-7-8/focs-ex-7-8/teaching/2627/FoundsCS/forest/focs-ex-7-8/Exercises › Shifting a functional arrayExerciseWrite a function to remove the first element from a functional array. All the other elements are to have their subscripts reduced by one. The cost of this operation should be linear in the size of the array.Anil MadhavapeddyAnil Madhavapeddyhttps://www.cl.cam.ac.uk/teaching/2627/FoundsCS/forest/focs-ex-2-3/focs-ex-2-3/teaching/2627/FoundsCS/forest/focs-ex-2-3/Exercises › Simplifying sums in O notationExerciseLet , …, be functions such that for , …, and all sufficiently large .Show that if then .Anil MadhavapeddyAnil Madhavapeddyhttps://www.cl.cam.ac.uk/teaching/2627/FoundsCS/forest/focs-ex-3-1/focs-ex-3-1/teaching/2627/FoundsCS/forest/focs-ex-3-1/Exercises › Summing a listExerciseCode a recursive function to compute the sum of a list’s elements. Then code an iterative version and comment on the improvement in efficiency.Anil MadhavapeddyAnil Madhavapeddyhttps://www.cl.cam.ac.uk/teaching/2627/FoundsCS/forest/focs-ex-6-2/focs-ex-6-2/teaching/2627/FoundsCS/forest/focs-ex-6-2/Exercises › Summing tree labelsExerciseWrite an OCaml function taking a binary tree labelled with integers and returning their sum.Anil MadhavapeddyAnil Madhavapeddyhttps://www.cl.cam.ac.uk/teaching/2627/FoundsCS/forest/focs-ex-3-5/focs-ex-3-5/teaching/2627/FoundsCS/forest/focs-ex-3-5/Exercises › Tails of a listExerciseCode a function tails to return the list of the tails of its argument. For example, given [1; 2; 3] it should return [[1; 2; 3]; [2; 3]; [3]; []].Anil MadhavapeddyAnil Madhavapeddyhttps://www.cl.cam.ac.uk/teaching/2627/FoundsCS/forest/focs-ex-8-1/focs-ex-8-1/teaching/2627/FoundsCS/forest/focs-ex-8-1/Exercises › The argument-swapping functionalExerciseWhat does the following function do, and what are its uses?Anil MadhavapeddyAnil Madhavapeddyhttps://www.cl.cam.ac.uk/teaching/2627/FoundsCS/forest/focs-ex-10-1/focs-ex-10-1/teaching/2627/FoundsCS/forest/focs-ex-10-1/Exercises › Tree-based queuesExerciseSuppose that we have an implementation of queues, based on binary trees, such that each operation takes logarithmic time in the worst case. Outline the advantages and drawbacks of such an implementation compared with one presented above.Anil MadhavapeddyAnil Madhavapeddyhttps://www.cl.cam.ac.uk/teaching/2627/FoundsCS/forest/focs-ex-1-1/focs-ex-1-1/teaching/2627/FoundsCS/forest/focs-ex-1-1/Exercises › Two-digit year windowingExerciseOne solution to the year 2000 bug involves storing years as two digits, but interpreting them such that 50 means 1950 and 49 means 2049. Comment on the merits and demerits of this approach.Anil MadhavapeddyAnil Madhavapeddyhttps://www.cl.cam.ac.uk/teaching/2627/FoundsCS/forest/focs-ex-1-4/focs-ex-1-4/teaching/2627/FoundsCS/forest/focs-ex-1-4/Exercises › Type inference in powerExerciseFunctions npower and power both return a float. The definition of npower returns the float value 1.0 in its base case. The definition of power does not, so how does the OCaml type checker know that power returns a float?Anil MadhavapeddyAnil Madhavapeddyhttps://www.cl.cam.ac.uk/teaching/2627/FoundsCS/forest/focs-ex-6-3/focs-ex-6-3/teaching/2627/FoundsCS/forest/focs-ex-6-3/Exercises › What ftree buildsExerciseUsing the definition of 'a tree from before:Examine the following function declaration. What does ftree (1, n) accomplish?https://www.cl.cam.ac.uk/teaching/2627/FoundsCS/forest/anil-madhavapeddy/anil-madhavapeddy/teaching/2627/FoundsCS/forest/anil-madhavapeddy/Anil MadhavapeddyAnil Madhavapeddy lectures the Foundations of Computer Science course at the University of Cambridge.Anil Madhavapeddyhttps://www.cl.cam.ac.uk/teaching/2627/FoundsCS/forest/focs-glossary/focs-glossary/teaching/2627/FoundsCS/forest/focs-glossary/GlossaryDefinitions of the course’s core concepts, collected automatically.Anil Madhavapeddyhttps://www.cl.cam.ac.uk/teaching/2627/FoundsCS/forest/focs-def-abstract-data-type/focs-def-abstract-data-type/teaching/2627/FoundsCS/forest/focs-def-abstract-data-type/Abstract data typeDefinitionAn abstract type provides specified operations while hiding low-level details, such as the data structure used to represent them. Before choosing the internal representation for a data structure, you need to specify the full set of operations. Abstract types can be declared in any modern programming language: Java’s objects serve this role, as do OCaml’s modules. Introduced in Dictionaries, and applied to queues and stacks later on.Part of the course glossary.Anil Madhavapeddyhttps://www.cl.cam.ac.uk/teaching/2627/FoundsCS/forest/focs-def-abstraction-barrier/focs-def-abstraction-barrier/teaching/2627/FoundsCS/forest/focs-def-abstraction-barrier/Abstraction barrierDefinitionLarge systems can only be understood in levels: the interface to the higher level should supply the advertised services and, just as important, block access to the means by which those services are implemented. This abstraction barrier allows one level to be changed without affecting levels above. Introduced in Basic Concepts in Computer Science.Part of the course glossary.Anil Madhavapeddyhttps://www.cl.cam.ac.uk/teaching/2627/FoundsCS/forest/focs-def-algebraic-datatype/focs-def-algebraic-datatype/teaching/2627/FoundsCS/forest/focs-def-algebraic-datatype/Algebraic datatypeDefinitionOCaml generalises the notion of enumeration type to allow data to be associated with each constructor: some constructors are values all by themselves, while others create values from arguments. Constructors with arguments are distinct values, and values built from different constructors of one datatype can belong to a single list. Introduced in A Datatype whose Constructors have Arguments.Part of the course glossary.Anil Madhavapeddyhttps://www.cl.cam.ac.uk/teaching/2627/FoundsCS/forest/focs-def-binary-search-tree/focs-def-binary-search-tree/teaching/2627/FoundsCS/forest/focs-def-binary-search-tree/Binary search treeDefinitionA binary search tree works for keys that have a total ordering, such as strings. Each branch of the tree carries a pair; its left subtree holds smaller keys and its right subtree holds greater keys. If the tree remains reasonably balanced, then update and lookup both take for a tree of size . Introduced in Binary Search Trees.Part of the course glossary.Anil Madhavapeddyhttps://www.cl.cam.ac.uk/teaching/2627/FoundsCS/forest/focs-def-currying/focs-def-currying/teaching/2627/FoundsCS/forest/focs-def-currying/CurryingDefinitionCurrying is the technique of expressing a function taking multiple arguments as nested functions, each taking a single argument. A curried function returns another function as its result, so the arguments are supplied one after another. Introduced in Curried Functions.Part of the course glossary.Anil Madhavapeddyhttps://www.cl.cam.ac.uk/teaching/2627/FoundsCS/forest/focs-def-exception/focs-def-exception/teaching/2627/FoundsCS/forest/focs-def-exception/ExceptionDefinitionAn exception signals that something has gone wrong during a computation, such as division by zero or pattern matching failure. Raising an exception abandons the current computation; handling the exception attempts an alternative computation. The raising and the handling can be far apart in the code, and errors of different sorts can be handled separately. Introduced in Error Handling: Exceptions.Part of the course glossary.Anil Madhavapeddyhttps://www.cl.cam.ac.uk/teaching/2627/FoundsCS/forest/focs-def-higher-order-function/focs-def-higher-order-function/teaching/2627/FoundsCS/forest/focs-def-higher-order-function/Higher-order functionDefinitionIn mathematics, a functional or higher-order function is a function that operates on other functions. In OCaml, functions can be passed as arguments to other functions, returned as results, and put into lists, trees and so on—but not tested for equality. Introduced in Functions as Values.Part of the course glossary.Anil Madhavapeddyhttps://www.cl.cam.ac.uk/teaching/2627/FoundsCS/forest/focs-def-lazy-list/focs-def-lazy-list/teaching/2627/FoundsCS/forest/focs-def-lazy-list/Lazy listDefinitionA lazy list is a list of possibly infinite length whose elements are computed upon demand; in OCaml we implement laziness by delaying evaluation of the tail of the list. The traditional word “stream” is reserved in OCaml parlance for input/output channels, so we call lazy lists sequences instead. Introduced in Lazy Lists.Part of the course glossary.Anil Madhavapeddyhttps://www.cl.cam.ac.uk/teaching/2627/FoundsCS/forest/focs-def-mutable-reference/focs-def-mutable-reference/teaching/2627/FoundsCS/forest/focs-def-mutable-reference/Mutable referenceDefinitionA reference is a cell in the store, also called a “location”: ref E creates a reference with initial contents the value of E, !P returns the current contents of reference P (an operation called dereferencing), and P := E updates the contents of P to the value of E. If is some OCaml type, then ref is the type of references to cells that can hold values of . Introduced in OCaml Primitives for References.Part of the course glossary.Anil Madhavapeddyhttps://www.cl.cam.ac.uk/teaching/2627/FoundsCS/forest/focs-def-o-notation/focs-def-o-notation/teaching/2627/FoundsCS/forest/focs-def-o-notation/O notationDefinitionFormally, provided as , that is, for some constant and all sufficiently large . The “Big O” notation is commonly used to describe efficiency—to be precise, asymptotic complexity—by looking only at the most significant term and ignoring constant factors. Introduced in Comparing Algorithms: O Notation.Part of the course glossary.Anil Madhavapeddyhttps://www.cl.cam.ac.uk/teaching/2627/FoundsCS/forest/focs-def-pattern-matching/focs-def-pattern-matching/teaching/2627/FoundsCS/forest/focs-def-pattern-matching/Pattern matchingDefinitionPattern matching takes a value apart by matching it against the forms that value can have, letting one function declaration handle several cases, separated by a vertical bar |. Taking a list apart using combinations of hd and tl is hard to get right; fortunately, it is seldom necessary because of pattern-matching. Introduced in Getting at the Head and Tail.Part of the course glossary.Anil Madhavapeddyhttps://www.cl.cam.ac.uk/teaching/2627/FoundsCS/forest/focs-def-polymorphism/focs-def-polymorphism/teaching/2627/FoundsCS/forest/focs-def-polymorphism/PolymorphismDefinitionA polymorphic function allows flexibility in the types of its arguments and results: the list primitives null, hd and tl work whatever the element type is. Symbols 'a and 'b are called type variables and stand for any types, and code written using them is still checked for type correctness at compile time. Introduced in Getting at the Head and Tail.Part of the course glossary.Anil Madhavapeddyhttps://www.cl.cam.ac.uk/teaching/2627/FoundsCS/forest/focs-def-recursion/focs-def-recursion/teaching/2627/FoundsCS/forest/focs-def-recursion/RecursionDefinitionA function is recursive if it calls itself. You can spot a recursive function by the rec keyword in its definition: this indicates that any invocation of the function name within the function body should call itself. The recursion is meaningful because it terminates: we reduce the problem to smaller problems, and this cannot go on forever. Introduced in Raising a Number to a Power.Part of the course glossary.Anil Madhavapeddyhttps://www.cl.cam.ac.uk/teaching/2627/FoundsCS/forest/focs-def-tail-recursion/focs-def-tail-recursion/teaching/2627/FoundsCS/forest/focs-def-tail-recursion/Tail recursionDefinitionA recursive function whose computation does not nest is called iterative or tail-recursive: the recursive calls do not nest, so the work is done immediately rather than on the way back out. Many functions can be made iterative by introducing an extra argument, which is often called an accumulator. Introduced in Iteratively summing the first n integers.Part of the course glossary.Anil Madhavapeddyhttps://www.cl.cam.ac.uk/teaching/2627/FoundsCS/forest/focs-topics/focs-topics/teaching/2627/FoundsCS/forest/focs-topics/Topic indexSections and exercises grouped by topic, collected automatically from tags on each tree.Anil Madhavapeddyhttps://www.cl.cam.ac.uk/teaching/2627/FoundsCS/forest/focs-topic-recursion/focs-topic-recursion/teaching/2627/FoundsCS/forest/focs-topic-recursion/RecursionAnil Madhavapeddyhttps://www.cl.cam.ac.uk/teaching/2627/FoundsCS/forest/focs-change-all/focs-change-all/teaching/2627/FoundsCS/forest/focs-change-all/All Ways of Making ChangeNow we generalise the problem to return the list of all possible ways of making change, and write a new change function. [ [] ] | [] , _ -> [] | c::till , amt -> if amt < c then change till amt else let rec allc = function | [] -> [] | cs :: css -> (c::cs) :: allc css in allc (change (c::till) (amt - c)) @ change till amt val change : int list -> int -> int list list = ]]>Look at the type: the result is now a list of lists. The code will also never raise exceptions. It expresses failure by returning an empty list of solutions: it returns [] if the till is empty and the amount is nonzero.If the amount is zero, then there is only one way of making change; the result should be . This is success in the base case.In nontrivial cases, there are two sources of solutions: to use a coin (if possible) and decrease the amount accordingly, or to remove the current coin value from consideration.The function allc is declared locally in order to make use of c, the current coin. It adds an extra c to all the solutions returned by the recursive call to make change for amt - c.Observe the naming convention: cs is a list of coins, while css is a list of such lists. The trailing ‘s’ is suggestive of a plural.This complicated program, and the even trickier faster version, are included as challenges. Are you enthusiastic enough to work them out? We shall revisit the “making change” task later to illustrate exception-handling.Anil Madhavapeddyhttps://www.cl.cam.ac.uk/teaching/2627/FoundsCS/forest/focs-change-faster/focs-change-faster/teaching/2627/FoundsCS/forest/focs-change-faster/All Ways of Making Change — Faster! chg::chgs | [] , _ -> chgs | c::till , amt -> if amt < 0 then chgs else change (c::till) (amt - c) (c::chg) (change till amt chg chgs) val change : int list -> int -> int list -> int list list -> int list list = ]]>We’ve added another accumulating parameter! Repeatedly improving simple code is called stepwise refinement.Two extra arguments eliminate many :: and append operations from the all-solutions change function. The first, chg, accumulates the coins chosen so far; one evaluation of c::chg replaces many evaluations of allc. The second, chgs, accumulates the list of solutions so far; it avoids the need for append. This version runs several times faster than the previous one.Making change is still extremely slow for an obvious reason: the number of solutions grows rapidly in the amount being changed. Using 50, 20, 10, 5, 2 and 1, there are 4366 ways of expressing 99.Our three change functions illustrate a basic technique: program development by stepwise refinement. Begin by writing a very simple program and add requirements individually. Add efficiency refinements last of all. Even if the simpler program cannot be included in the next version and has to be discarded, one has learned about the task by writing it.Anil Madhavapeddyhttps://www.cl.cam.ac.uk/teaching/2627/FoundsCS/forest/focs-making-change/focs-making-change/teaching/2627/FoundsCS/forest/focs-making-change/An Application: Making ChangeConsider a till that has unlimited supplies of coins. The largest coins should be tried first, to avoid giving change all in pennies. The list of legal coin values, called till, is given in descending order, such as 50, 20, 10, 5, 2 and 1. (Recall that the head of a list is the element most easily reached.) The code for change is based on simple observations:Change for zero consists of no coins at all. (Note the pattern of 0 in the first clause.) For a nonzero amount, try the largest available coin. If it is small enough, use it and decrease the amount accordingly. Exclude from consideration any coins that are too large. [] | [], _ -> raise (Failure "no more coins!") | c::till, amt -> if amt < c then change till amt else c :: change (c::till) (amt - c) val change : int list -> int -> int list = ]]>Although nobody considers making change for zero, this is the simplest way to make the algorithm terminate. Most iterative procedures become simplest if, in their base case, they do nothing. A base case of one instead of zero is often a sign of a novice programmer.The recursion terminates when amt = 0. Tries the largest coin first to use large coins. The algorithm is greedy and can fail!The function can terminate either with success or failure. It fails by raising exception Failure namely if till becomes empty while amt is still nonzero. (Exceptions will be discussed later.)Unfortunately, failure can occur even when change can be made. The greedy "largest coin first" approach is to blame. Suppose we have coins of values 5 and 2, and must make change for 6; the only way is , ignoring the 5. Greedy algorithms are often effective, but not here.Anil Madhavapeddyhttps://www.cl.cam.ac.uk/teaching/2627/FoundsCS/forest/focs-quicksort-append-free/focs-quicksort-append-free/teaching/2627/FoundsCS/forest/focs-quicksort-append-free/Append-Free Quicksort sorted | ([x], sorted) -> x::sorted | a::bs, sorted -> let rec part = function | l, r, [] -> quik (l, a :: quik (r, sorted)) | l, r, x::xs -> if x <= a then part (x::l, r, xs) else part (l, x::r, xs) in part ([], [], bs) val quik : 'a list * 'a list -> 'a list = ]]>The list sorted accumulates the result in the combine stage of the quicksort algorithm. We have again used the standard technique for eliminating append. Calling quik(xs, sorted) reverses the elements of xs and prepends them to the list sorted.Looking closely at part, observe that quik(r, sorted) is performed first. Then a is consed to this sorted list. Finally, quik is called again to sort the elements of l.The speedup is significant. An imperative quicksort coded in Pascal (taken from Sedgewick) is just slightly faster than function quik. The near-agreement is surprising because the computational overheads of lists exceed those of arrays. In realistic applications, comparisons are the dominant cost and the overheads matter even less.Anil Madhavapeddyhttps://www.cl.cam.ac.uk/teaching/2627/FoundsCS/forest/focs-append/focs-append/teaching/2627/FoundsCS/forest/focs-append/Append: List Concatenation ys | x::xs, ys -> x :: append xs ys val append : 'a list -> 'a list -> 'a list = # append [1; 2; 3] [4] - : int list = [1; 2; 3; 4] # let (@) = append val ( @ ) : 'a list -> 'a list -> 'a list = # [1; 2; 3] @ [4] - : int list = [1; 2; 3; 4]]]>Patterns can be as complicated as we like. Here, the two patterns are [], ys and x::xs, ys.Here is how append might be declared, also noting that we have defined @ as an infix operator that is a more convenient way to call append on two lists. However, this function is also not iterative. It scans its first argument, sets up a string of cons operations (::) and finally does them.It uses space and time, where is the length of its first argument. Its costs are independent of its second argument.An accumulating argument could make it iterative, but with considerable complication. The iterative version would still require space and time because concatenation requires copying all the elements of the first list. Therefore, we cannot hope for asymptotic gains; at best we can decrease the constant factor involved in , but complicating the code is likely to increase that factor. Never add an accumulator merely out of habit.Note append’s polymorphic type. It tells us that two lists can be joined if their element types agree.Anil Madhavapeddyhttps://www.cl.cam.ac.uk/teaching/2627/FoundsCS/forest/focs-tree-traversal/focs-tree-traversal/teaching/2627/FoundsCS/forest/focs-tree-traversal/Aside: Traversing Trees (3 Methods) [] | Br (v, t1, t2) -> [v] @ preorder t1 @ preorder t2 val preorder : 'a tree -> 'a list = # let rec inorder = function | Lf -> [] | Br (v, t1, t2) -> inorder t1 @ [v] @ inorder t2 val inorder : 'a tree -> 'a list = # let rec postorder = function | Lf -> [] | Br (v, t1, t2) -> postorder t1 @ postorder t2 @ [v] val postorder : 'a tree -> 'a list = ]]>Tree traversal means examining each node of a tree in some order. D. E. Knuth has identified three forms of tree traversal: preorder, inorder and postorder. We can code these “visiting orders” as functions that convert trees into lists of labels. Algorithms based on these notions typically perform some action at each node; the functions above simply copy the nodes into lists. Consider the tree:preorder visits the label first (“Polish notation”), yielding ABDECFG inorder visits the label midway, yielding DBEAFCG postorder visits the label last (“Reverse Polish”), yielding DEBFGCA. You might be familiar with this concept if you own an ancient RPN calculator!What is the use of inorder? Consider applying it to a binary search tree: the result is a sorted list of pairs. We could use this, for example, to merge two binary search trees. It is not difficult to transform a sorted list of pairs into a binary search tree.Anil Madhavapeddyhttps://www.cl.cam.ac.uk/teaching/2627/FoundsCS/forest/focs-tree-properties/focs-tree-properties/teaching/2627/FoundsCS/forest/focs-tree-properties/Basic Properties of Binary Trees 0 (* number of branch nodes *) | Br (v, t1, t2) -> 1 + count t1 + count t2 val count : 'a tree -> int = # let rec depth = function | Lf -> 0 (* length of longest path *) | Br (v, t1, t2) -> 1 + max (depth t1) (depth t2) val depth : 'a tree -> int = ]]>The invariant holds in the functions above.Functions on trees are expressed recursively using pattern-matching. Both functions above are analogous to on lists. Here is a third measure of a tree’s size: 1 | Br (v, t1, t2) -> leaves t1 + leaves t2 val leaves : 'a tree -> int = ]]>This function is redundant because of a basic fact about trees, which can be proved by induction: for every tree , we have . The invariant stated above also has an elementary proof by induction.A tree of depth 20 can store or approximately one million elements. The access paths to these elements are short, particularly when compared with a million-element list!Anil Madhavapeddyhttps://www.cl.cam.ac.uk/teaching/2627/FoundsCS/forest/focs-binary-trees/focs-binary-trees/teaching/2627/FoundsCS/forest/focs-binary-trees/Binary Trees, a Recursive DatatypeA data structure with multiple branching is called a “tree”. Trees can represent mathematical expressions, logical formulae, computer programs, the phrase structure of English sentences, etc.Binary trees are nearly as fundamental as lists. They can provide efficient storage and retrieval of information. In a binary tree, each node is empty (), or is a branch () with a label and two subtrees.OCaml lists are a datatype and could be declared as follows:We could even declare :: as an infix constructor. The only thing we could not define is the [...] notation, which is part of the OCaml grammar (although there does exist a mechanism to use a similar syntax for custom indexed datatypes).A recursive type does not have to be polymorphic. For example, here is a simple datatype of tree shapes with no attached data that is recursive but not polymorphic.The datatype 'a option (mentioned above) is the opposite – it is polymorphic, but not recursive.Anil Madhavapeddyhttps://www.cl.cam.ac.uk/teaching/2627/FoundsCS/forest/focs-zip/focs-zip/teaching/2627/FoundsCS/forest/focs-zip/Building a List of Pairs (x, y) :: zip xs ys | _ -> [] val zip : 'a list -> 'b list -> ('a * 'b) list = ]]>The wildcard pattern _ matches anything. We could have written a variable such as p instead, but the wildcard reminds us that the relevant clause ignores this argument.The patterns are also tested in order of their definitions: first (x::xs, y::ys), then _.A list of pairs of the form associates each with . Conceptually, a telephone directory could be regarded as such a list, where ranges over names and over the corresponding telephone number. Linear search in such a list can find the associated with a given , or vice versa—very slowly.In other cases, the pairs might have been generated by applying a function to the elements of another list . ([], []) | (x, y)::pairs -> let xs, ys = unzip pairs in (x::xs, y::ys) val unzip : ('a * 'b) list -> 'a list * 'b list = ]]>Given a list of pairs, unzip has to build two lists of results, which is awkward using recursion. The version shown above uses the local binding let p = in , where the value of is bound to the variable p within . The let-construct counts as an expression and can be used (perhaps wrapped within parentheses) wherever an expression is expected.Note especially the phrase let xs, ys = unzip pairs which binds xs and ys to the results of the recursive call. In general, the phrase let P = E matches the pattern against the value of expression . It binds all the variables in to the corresponding values.The functions zip and unzip build and take apart lists of pairs: zip pairs up corresponding list elements and unzip inverts this operation. Their types reflect what they do: 'b list -> ('a * 'b) list = # unzip - : ('a * 'b) list -> 'a list * 'b list = ]]>If the lists are of unequal length, zip discards surplus items at the end of the longer list. Its first pattern only matches a pair of non-empty lists. The second pattern is just a wildcard and could match anything. OCaml tries the clauses in the order given, so the first pattern is tried first. The second only gets arguments where at least one of the lists is empty.Anil Madhavapeddyhttps://www.cl.cam.ac.uk/teaching/2627/FoundsCS/forest/focs-unzip/focs-unzip/teaching/2627/FoundsCS/forest/focs-unzip/Building a Pair of ResultsHere is a version of unzip that replaces the local declaration by a function conspair for taking apart the pair of lists in the recursive call. It defines the same computation as the previous version of unzip and is possibly clearer, but not every local binding can be eliminated as easily. 'a list * 'b list = # let rec unzip = function | [] -> ([], []) | xy :: pairs -> conspair (xy, unzip pairs) val unzip : ('a * 'b) list -> 'a list * 'b list = ]]>Making the function iterative yields revUnzip below, which is very simple. Iteration can construct many results at once in different argument positions. Both output lists are built in reverse order, which can be corrected by reversing the input to revUnzip. The total costs will probably exceed those of unzip despite the advantages of iteration. (xs, ys) | ((x, y)::pairs, xs, ys) -> revUnzip (pairs, x::xs, y::ys) val revUnzip : ('a * 'b) list * 'a list * 'b list -> 'a list * 'b list = ]]>Anil Madhavapeddyhttps://www.cl.cam.ac.uk/teaching/2627/FoundsCS/forest/focs-nlength/focs-nlength/teaching/2627/FoundsCS/forest/focs-nlength/Computing the Length of a List 0 | x :: xs -> 1 + nlength xs val nlength : 'a list -> int = # nlength [] - : int = 0 # nlength [5; 6; 7] - : int = 3]]>Most list processing involves recursion. This is a simple example; patterns can be more complex. Observe the use of a vertical bar | to separate the function’s clauses. We have one function declaration that handles two cases. To understand its role, consider the following faulty code: int = # let rec nlength (x::xs) = 1 + nlength xs Line 1, characters 17-24: Warning 8 [partial-match]: this pattern-matching is not exhaustive. Here is an example of a case that is not matched: [] val nlength : 'a list -> int = ]]>These are two declarations, not one. First we declare nlength to be a function that handles only empty lists. Then we redeclare it to be a function that handles only non-empty lists; it can never deliver a result. We see that a second let declaration replaces any previous one rather than extending it to cover new cases.Now, let us return to our original declaration of nlength. The length function is polymorphic and applies to all lists regardless of element type! Most programming languages lack such flexibility.Unfortunately, this length computation is naive and wasteful. Like nsum earlier, it is not tail-recursive. It uses space, where is the length of its input. As usual, the solution is to add an accumulating argument.Anil Madhavapeddyhttps://www.cl.cam.ac.uk/teaching/2627/FoundsCS/forest/focs-seq-consume/focs-seq-consume/teaching/2627/FoundsCS/forest/focs-seq-consume/Consuming a Sequence [] | n, Nil -> [] | n, Cons (x, xf) -> x :: get (n-1) (xf ()) val get : int -> 'a seq -> 'a list = ]]>The above code gets the first elements as a list. xf () forces evaluation.The function get converts a sequence to a list. It takes the first elements; it takes all of them if , which can terminate only if the sequence is finite.In the last line of get, the expression xf() calls the tail function, demanding evaluation of the next element. This operation is called forcing the sequence.Anil Madhavapeddyhttps://www.cl.cam.ac.uk/teaching/2627/FoundsCS/forest/focs-length/focs-length/teaching/2627/FoundsCS/forest/focs-length/Efficiently Computing the Length of a List n | x::xs -> addlen (n + 1) xs val addlen : int -> 'a list -> int = # addlen 0 [5; 6; 7] - : int = 3]]>Recall that the use of function introduces an extra (unnamed) argument that is pattern matched in the subsequent clauses; in this case, to break open the list.Function addlen is again polymorphic. Its type mentions the integer accumulator.Now we may declare an efficient length function. It is simply a wrapper for addlen, supplying zero as the initial value of . int = # length [5; 6; 7; 8] - : int = 4]]>The recursive calls do not nest: this version is iterative. It takes space. Obviously its time requirement is because it takes at least steps to find the length of an -element list.Anil Madhavapeddyhttps://www.cl.cam.ac.uk/teaching/2627/FoundsCS/forest/focs-power/focs-power/teaching/2627/FoundsCS/forest/focs-power/Efficiently Raising a Number to a Power int -> float = ]]>Mathematical JustificationFor large n, computing powers using is too slow to be practical. The equations above are much faster. Example:Instead of n multiplications, we need at most multiplications, where is the logarithm of to the base .We use the function even, declared previously, to test whether the exponent is even. Integer division (/) truncates its result to an integer: dividing by 2 yields .A recurrence is a useful computation rule only if it is bound to terminate. If 0]]> then is smaller than both and . After enough recursive calls, the exponent will be reduced to . The equations also hold if , but the corresponding computation runs forever.Our reasoning assumes arithmetic to be exact. Fortunately, the calculation is well-behaved using floating-point.Computer numbers have a finite range, which if exceeded results in the integer wrapping around. You will understand this behaviour more as you learn about computer architecture and how modern systems represent numbers in memory.If integers and floats must be combined in a calculation, OCaml provides functions to convert between them:OCaml’s libraries are organised using “modules”, so we may use compound identifiers such as Float.of_int to refer to library functions. There are many thousands of library functions available in the OCaml ecosystem, including text-processing and operating systems functions in addition to the usual numerical ones.Anil Madhavapeddyhttps://www.cl.cam.ac.uk/teaching/2627/FoundsCS/forest/focs-tree-traversal-efficient/focs-tree-traversal-efficient/teaching/2627/FoundsCS/forest/focs-tree-traversal-efficient/Efficiently Traversing TreesUnfortunately, the traversal functions are quadratic in the worst case: the appends in the recursive calls are inefficient. To correct that problem, we (as usual) add an accumulating argument. Observe how each function constructs its result list and compare with how appends were eliminated from quicksort in the Sorting lecture. vs | Br (v, t1, t2), vs -> v :: preord (t1, preord (t2, vs)) val preord : 'a tree * 'a list -> 'a list = # let rec inord = function | Lf, vs -> vs | Br (v, t1, t2), vs -> inord (t1, v::inord (t2, vs)) val inord : 'a tree * 'a list -> 'a list = # let rec postord = function | Lf, vs -> vs | Br (v, t1, t2), vs -> postord (t1, postord (t2, v::vs)) val postord : 'a tree * 'a list -> 'a list = ]]>One can prove equations relating each of these functions to its counterpart on the previous section. For example:These three types of tree traversal are related in that all are depth-first. They each traverse the left subtree in full before traversing the right subtree. Breadth-first search (from the Queues lecture) is another possibility. That involves going through the levels of a tree one at a time.Anil Madhavapeddyhttps://www.cl.cam.ac.uk/teaching/2627/FoundsCS/forest/focs-expression-evaluation/focs-expression-evaluation/teaching/2627/FoundsCS/forest/focs-expression-evaluation/Expression EvaluationExpression evaluation concerns expressions and the values they return. This view of computation may seem to be too narrow. It is certainly far removed from computer hardware, but that can be seen as an advantage. For the traditional concept of computing solutions to problems, expression evaluation is entirely adequate.Starting with , the expression is reduced to until this process concludes with a value . A value is something like a number that cannot be further reduced.We write to say that is reduced to . Mathematically, they are equal: , but the computation goes from to and never the other way around.Computers also interact with the outside world. For a start, they need some means of accepting problems and delivering solutions. Many computer systems monitor and control industrial processes. This role of computers is familiar now, but was never envisaged in the early days. Computer pioneers focused on mathematical calculations. Modelling interaction and control requires a notion of states that can be observed and changed. Then we can consider updating the state by assigning to variables or performing input/output, finally arriving at conventional programs as coded in C, for instance.For now, we remain at the level of expressions, which is usually termed functional programming.Anil Madhavapeddyhttps://www.cl.cam.ac.uk/teaching/2627/FoundsCS/forest/focs-insertion-sort/focs-insertion-sort/teaching/2627/FoundsCS/forest/focs-insertion-sort/Insertion SortAn insert operation does comparisons on average. [x] | y::ys -> if x <= y then x :: y :: ys else y :: ins x ys val ins : 'a -> 'a list -> 'a list = ]]>Insertion sort takes comparisons on average: [] | x::xs -> ins x (insort xs)]]>Items from the input are copied one at a time to the output. Each new item is inserted into the right place so that the output is always in order.We could easily write iterative versions of these functions, but to no purpose. Insertion sort is slow because it does comparisons (and a lot of list copying), not because it is recursive. Its quadratic runtime makes it nearly useless: it takes 174 seconds for our example while the next-worst figure is 1.4 seconds.Insertion sort is worth considering because it is easy to code and illustrates the concepts. Two efficient sorting algorithms, mergesort and heapsort, can be regarded as refinements of insertion sort.Anil Madhavapeddyhttps://www.cl.cam.ac.uk/teaching/2627/FoundsCS/forest/focs-while/focs-while/teaching/2627/FoundsCS/forest/focs-while/Iteration: the while command None | _::xs -> Some xs val tlopt : 'a list -> 'a list option = # let length xs = let lp = ref xs in (* list of uncounted elements *) let np = ref 0 in (* accumulated count *) let fin = ref false in while not !fin do match tlopt !lp with | None -> fin := true | Some xs -> lp := xs; np := 1 + !np done; !np (* the final count is returned *) val length : 'a list -> int = ]]>Once we can change the state, we need to do so repeatedly. Recursion can serve this purpose, but having to declare a procedure for every loop is clumsy, and compilers for conventional languages seldom exploit tail-recursion.Early programming languages provided little support for repetition. The programmer had to set up loops using goto commands, exiting the loop using another goto controlled by an if. Modern languages provide a confusing jumble of looping constructs, the most fundamental of which is while B do C. The boolean expression is evaluated, and if true, command is executed and the command repeats. If evaluates to false then the while command terminates, perhaps without executing even once.OCaml’s main looping construct is while, which returns the value (). The function length declares references to hold the list under examination (lp) and number of elements counted so far (np) as well as whether the end of the list has been reached (the boolean reference fin). While the list is non-empty, we skip over one more element (by setting it to its tail) and count that element.The body of the while loop first checks to see if the end of the list has been reached, in which case it sets the fin variable to true. If there is a tail value, then two assignments are executed in sequence. The lp reference is set to the tail of the list, and the np reference integer is incremented by one. When the while loop terminates due to the fin variable being set to true, the expression !np returns the computed length as the function’s result.Anil Madhavapeddyhttps://www.cl.cam.ac.uk/teaching/2627/FoundsCS/forest/focs-summing-iter/focs-summing-iter/teaching/2627/FoundsCS/forest/focs-summing-iter/Iteratively summing the first n integers int -> int = ]]>Function summing takes an additional argument: a running total. If n is zero then it returns the running total; otherwise, summing adds to it and continues. The recursive calls do not nest; the additions are done immediately.A recursive function whose computation does not nest is called iterative or tail-recursive. Many functions can be made iterative by introducing an argument analogous to total, which is often called an accumulator.The gain in efficiency is sometimes worthwhile and sometimes not. The function power is not iterative because nesting occurs whenever the exponent is odd. Adding a third argument makes it iterative, but the change complicates the function and the gain in efficiency is minute; for 32-bit integers, the maximum possible nesting is 30 for the exponent .Anil Madhavapeddyhttps://www.cl.cam.ac.uk/teaching/2627/FoundsCS/forest/focs-seq-append/focs-seq-append/teaching/2627/FoundsCS/forest/focs-seq-append/Joining Two Sequences yq | Cons (x, xf) -> Cons(x, fun () -> appendq (xf ()) yq) val appendq : 'a seq -> 'a seq -> 'a seq = ]]>A more fair alternative: yq | Cons (x, xf) -> Cons (x, fun () -> interleave yq (xf ())) val interleave : 'a seq -> 'a seq -> 'a seq = ]]>Most list functions and functionals have analogues on sequences, but strange things can happen. Can an infinite list be reversed?Function appendq is precisely the same idea as append from the Lists lecture; it concatenates two sequences. If the first argument is infinite, then appendq never gets to its second argument, which is lost. Concatenation of infinite sequences is not terribly interesting.The function interleave avoids this problem by exchanging the two arguments in each recursive call. It combines the two lazy lists, losing no elements. Interleaving is the right way to combine two potentially infinite information sources into one.In both function declarations, observe that each xf () is enclosed within a . Each force is enclosed within a delay. This practice makes the functions lazy. A force not enclosed in a delay, as in get above, runs the risk of evaluating the sequence in full.Anil Madhavapeddyhttps://www.cl.cam.ac.uk/teaching/2627/FoundsCS/forest/focs-take-drop/focs-take-drop/teaching/2627/FoundsCS/forest/focs-take-drop/List Utilities: take and dropThis lecture examines more list utilities, illustrating more patterns of recursion, and concludes with a small program for making change.The functions take and drop divide a list into parts, returning or discarding the first elements.They can be implemented in OCaml as follows: [] | x::xs -> if i > 0 then x :: take (i - 1) xs else [] val take : int -> 'a list -> 'a list = # let rec drop i = function | [] -> [] | x::xs -> if i > 0 then drop (i-1) xs else x::xs val drop : int -> 'a list -> 'a list = ]]>Applications of take and drop will appear in future lectures. Typically, they divide a collection of items into equal parts for recursive processing.The take function is not iterative, but making it so would not improve its efficiency. The task requires copying up to list elements, which must take space and time.Function drop simply skips over list elements. This requires time but only constant space. It is iterative and much faster than take. Both functions use time, but skipping elements is faster than copying them: drop’s constant factor is smaller.Both functions take an integer and a list, returning a list of the same type. So their type is int -> 'a list -> 'a list.Anil Madhavapeddyhttps://www.cl.cam.ac.uk/teaching/2627/FoundsCS/forest/focs-change-exceptions/focs-change-exceptions/teaching/2627/FoundsCS/forest/focs-change-exceptions/Making Change with Exceptions [] | [], _ -> raise Change | c::till, amt -> if amt < 0 then raise Change else try c :: change (c::till) (amt - c) with Change -> change till amt exception Change val change : int list -> int -> int list = ]]>In the Lists lectures, we considered the problem of making change. The greedy algorithm presented there could not express “6 using 5 and 2” because it always took the largest coin. Returning the list of all possible solutions avoids that problem rather expensively: we only need one solution.Using exceptions, we can code a backtracking algorithm: one that can undo past decisions if it comes to a dead end. The exception Change is raised if we run out of coins (with a non-zero amount) or if the amount goes negative. We always try the largest coin, but enclose the recursive call in an exception handler, which undoes the choice if it goes wrong.Carefully observe how exceptions interact with recursion. The exception handler always undoes the most recent choice, leaving others possibly to be undone later. If making change really is impossible, then eventually exception Change will be raised with no handler to catch it, and it will be reported at top level.Anil Madhavapeddyhttps://www.cl.cam.ac.uk/teaching/2627/FoundsCS/forest/focs-change-trace/focs-change-trace/teaching/2627/FoundsCS/forest/focs-change-trace/Making Change: A TraceHere is the full execution. Observe how the exception handlers nest and how they drop away once the given expression has returned a value. change [2] 6}\\ \Rightarrow &\; \text{try 5::(try 5::change [5; 2] (-4)}\\ &\; \text{with Change -> change [2] 1)}\\ &\; \text{with Change -> change [2] 6}\\ \Rightarrow &\; \text{5::(change [2] 1)}\\ &\; \text{with Change -> change [2] 6}\\ \Rightarrow &\; \text{try 5::(try 2::change [2] (-1)}\\ &\; \text{with Change -> change [] 1)}\\ &\; \text{with Change -> change [2] 6} \\ \Rightarrow &\; \text{try 5::(change [] 1)}\\ &\; \text{with Change -> change [2] 6} \\ \Rightarrow &\; \text{change [2] 6} \\ \Rightarrow &\; \text{try 2::change [2] 4}\\ &\; \text{with Change -> change [] 6} \\ \Rightarrow &\; \text{try 2::(try 2::change [2] 2}\\ &\; \text{with Change -> change [] 4)}\\ &\; \text{with Change -> change [] 6} \\ \Rightarrow &\; \text{try 2::(try 2::(try 2::change [2] 0 }\\ &\; \text{with Change -> change [] 2)}\\ &\; \text{with Change -> change [] 4)}\\ &\; \text{with Change -> change [] 6} \\ \Rightarrow &\; \text{try 2::(try 2::[2]}\\ &\; \text{with Change -> change [] 4)}\\ &\; \text{with Change -> change [] 6} \\ \Rightarrow &\; \text{try 2::[2; 2]}\\ &\; \text{with Change -> change [] 6} \\ \Rightarrow &\; \text{[2; 2; 2]} \end{aligned}]]>Anil Madhavapeddyhttps://www.cl.cam.ac.uk/teaching/2627/FoundsCS/forest/focs-merge/focs-merge/teaching/2627/FoundsCS/forest/focs-merge/Merging Two ListsMerge joins two sorted lists. ys | xs, [] -> xs | x::xs, y::ys -> if x <= y then x :: merge (xs, y::ys) else y :: merge (x::xs, ys) val merge : 'a list * 'a list -> 'a list = ]]>Generalises insert to two lists, and does at most comparisons.Merging means combining two sorted lists to form a larger sorted list. It does at most comparisons, where and are the lengths of the input lists. If and are roughly equal then we have a fast way of constructing sorted lists; if then merging degenerates to insertion, doing much work for little gain.Merging is the basis of several sorting algorithms; we look at a divide-and-conquer one. Mergesort is seldom found in conventional programming because it is hard to code for arrays; it works nicely with lists. It divides the input (if non-trivial) into two roughly equal parts, sorts them recursively, then merges them.Function merge is not iterative; the recursion is deep. An iterative version is of little benefit for the same reasons that apply to append in the earlier lecture on Lists.Anil Madhavapeddyhttps://www.cl.cam.ac.uk/teaching/2627/FoundsCS/forest/focs-quicksort-idea/focs-quicksort-idea/teaching/2627/FoundsCS/forest/focs-quicksort-idea/Quicksort: The IdeaThe Quicksort algorithm has the following flow:Choose a pivot element, Divide to partition the input into two sublists: those at most in value those exceeding Conquer using recursive calls to sort the sublists Combine the sorted lists by appending one to the otherQuicksort was invented by Sir Anthony Hoare (1934–2026), who in later life worked at Microsoft Research, Cambridge. Quicksort works by divide and conquer, a basic algorithm design principle. Quicksort chooses from the input some value , called the pivot. It partitions the remaining items into two parts: those , and those a]]>. It sorts each part recursively, then puts the smaller part before the greater.The cleverest feature of Hoare's algorithm was that the partition could be done in place by exchanging array elements. Quicksort was invented before recursion was well known, and people found it extremely hard to understand. As usual, we shall consider a list version based on functional programming.Anil Madhavapeddyhttps://www.cl.cam.ac.uk/teaching/2627/FoundsCS/forest/focs-npower/focs-npower/teaching/2627/FoundsCS/forest/focs-npower/Raising a Number to a Power int -> float = ]]>Our new npower definition can now take additional arguments, reflected in the arrows present in the type of npower; these represent parameters that can be passed to the new value being defined, with the final segment being the resulting type. Thus our npower type can be read as "pass in a float and integer to return a float".Mathematical Justification (for ):The function npower raises its float argument x to the power n, a non-negative integer. The function is recursive: it calls itself. You can spot a recursive function due to the rec keyword in the definition: this indicates that any invocation of the function name within the function body should call itself. This concept should be familiar from mathematics, since exponentiation is defined by the rules shown above. You may also have seen recursion in the product rule for differentiation: . In finding the derivative of , we recursively find the derivatives of and , combining them to obtain the desired result. The recursion is meaningful because it terminates: we reduce the problem to two smaller problems, and this cannot go on forever. The OCaml programmer uses recursion heavily. For , the equation yields an obvious computation:The equation clearly holds even for negative . However, the corresponding computation runs forever:Note that the function npower contains both an integer constant (0) and a floating point constant (1.0). The decimal point makes all the difference. OCaml will notice and ascribe different meaning to each type of constant. float = ]]>Now for a tiresome but necessary aside. In most languages, the types of arguments and results must always be specified. OCaml is unusual that it normally infers the types itself. However, sometimes it is useful to supply a hint to help you debug and develop your program. OCaml will still infer the types even if you don’t specify them, but in some cases it will use a more inefficient function than a specialised one. Some languages have just one type of number, converting automatically between different formats; this is slow and could lead to unexpected rounding errors. Type constraints are allowed almost anywhere. We can put one on any occurrence of x in the function. float = ]]>Or we can constrain the type of the function’s result: float = ]]>OCaml treats the equality and comparison test specially. Expressions like if x = y then … are allowed provided x and y have the same type and equality testing is possible for that type. (We discuss equality further in a later lecture.) Note that x <> y is OCaml for .A characteristic feature of the computer is its ability to test for conditions and act accordingly. In the early days, a program might jump to a given address depending on the sign of some number. Later, John McCarthy defined the conditional expression to satisfy if true then x else y = x and if false then x else y = y.OCaml evaluates the expression if then else by first evaluating . If the result is true then OCaml evaluates and otherwise . Only one of the two expressions and is evaluated! If both were evaluated, then recursive functions like npower above would run forever.The if-expression is governed by an expression of type bool, whose two values are true and false. In modern programming languages, tests are not built into “conditional branch” constructs but can just be part of normal expressions. Tests, or Boolean expressions, can be expressed using relational operators such as < and =. They can be combined using the Boolean operators for negation (not), conjunction (written as &&) and disjunction (written as ||). New properties can be declared as functions: here, to test whether an integer is even, for example: bool = ]]>Anil Madhavapeddyhttps://www.cl.cam.ac.uk/teaching/2627/FoundsCS/forest/focs-def-recursion/focs-def-recursion/teaching/2627/FoundsCS/forest/focs-def-recursion/RecursionDefinitionA function is recursive if it calls itself. You can spot a recursive function by the rec keyword in its definition: this indicates that any invocation of the function name within the function body should call itself. The recursion is meaningful because it terminates: we reduce the problem to smaller problems, and this cannot go on forever. Introduced in Raising a Number to a Power.Part of the course glossary.Anil Madhavapeddyhttps://www.cl.cam.ac.uk/teaching/2627/FoundsCS/forest/focs-recursion-vs-iteration/focs-recursion-vs-iteration/teaching/2627/FoundsCS/forest/focs-recursion-vs-iteration/Recursion vs Iteration“Iterative” normally refers to a loop, coded using while for example (see the final lecture) Tail-recursion is only efficient if the compiler detects it Mainly it saves space (memory), though iterative code can also run faster Do not make programs iterative unless the gain is worth itA classic book by Abelson and Sussman, which describes the Lisp dialect known as Scheme, used iterative to mean tail-recursive. Iterative functions produce computations resembling those that can be done using while-loops in conventional languages.Many algorithms can be expressed naturally using recursion, but only awkwardly using iteration. There is a story that Dijkstra sneaked recursion into Algol-60 by inserting the words “any other occurrence of the procedure name denotes execution of the procedure.” By not using the word “recursion”, he managed to slip this amendment past sceptical colleagues.Obsession with tail recursion leads to a coding style in which functions have many more arguments than necessary. Write straightforward code first, avoiding only gross inefficiency. If the program turns out to be too slow, tools are available for pinpointing the cause. Always remember KISS (Keep It Simple, Stupid).I hope you have all noticed by now that the summation can be done even more efficiently using the arithmetic progression formula:Anil Madhavapeddyhttps://www.cl.cam.ac.uk/teaching/2627/FoundsCS/forest/focs-rev/focs-rev/teaching/2627/FoundsCS/forest/focs-rev/Reversing a List in ys | x::xs, ys -> rev_app xs (x::ys) val rev_app : 'a list -> 'a list -> 'a list = ]]>Calling rev_app xs ys reverses the elements of xs and prepends them to ys. Now we may declare 'a list = # rev [1; 2; 3] - : int list = [3; 2; 1]]]>It is easy to see that this reverse function performs just conses, given an -element list. For both reverse functions, we could count the number of conses precisely—not just up to a constant factor. notation is still useful to describe the overall running time: the time taken by a cons varies from one system to another.The accumulator makes the function iterative. But the gain in complexity arises from the removal of append. Replacing an expensive operation (append) by a series of cheap operations (cons) is called reduction in strength and is a common technique in computer science. It originated when many computers did not have a hardware multiply instruction; the series of products for , could more efficiently be computed by repeated addition. Reduction in strength can be done in various ways; we shall see many instances of removing append.Consing to an accumulator produces the result in reverse. If that forces the use of an extra list reversal then the iterative function may be much slower than the recursive one.Anil Madhavapeddyhttps://www.cl.cam.ac.uk/teaching/2627/FoundsCS/forest/focs-nrev/focs-nrev/teaching/2627/FoundsCS/forest/focs-nrev/Reversing a List in Let us consider one way to reverse a list. [] | x::xs -> (nrev xs) @ [x] val nrev : 'a list -> 'a list = # nrev [1; 2; 3] - : int list = [3; 2; 1]]]>This reverse function is grossly inefficient due to poor usage of append, which copies its first argument. If nrev is given a list of length 0]]>, then append makes conses to copy the reversed tail. Constructing the list [x] calls cons again, for a total of calls. Reversing the tail requires more conses, and so forth. The total number of conses is:The time complexity is therefore . Space complexity is only because the copies don’t all exist at the same time.Anil Madhavapeddyhttps://www.cl.cam.ac.uk/teaching/2627/FoundsCS/forest/focs-seq-evaluation/focs-seq-evaluation/teaching/2627/FoundsCS/forest/focs-seq-evaluation/Sample EvaluationHere we ask for two elements of the infinite sequence. In fact, three elements are computed: 6, 7 and 8. Our implementation is slightly too eager. A more complicated type declaration could avoid this problem. Another problem is that if one repeatedly examines some particular list element using forcing, that element is repeatedly evaluated. In a lazy programming language, the result of the first evaluation would be stored for later reference. To get the same effect in OCaml requires the use of references.We should be grateful that the potentially infinite computation is kept finite. The tail of the original sequence even contains the unevaluated expression 6+1.Anil Madhavapeddyhttps://www.cl.cam.ac.uk/teaching/2627/FoundsCS/forest/focs-sample-costs/focs-sample-costs/teaching/2627/FoundsCS/forest/focs-sample-costs/Sample costs in O notationRecall that npower computes by repeated multiplication while nsum naively computes the sum . Each obviously performs arithmetic operations. Because they are not tail recursive, their use of space is also . The function summing is a version of nsum with an accumulating argument; its iterative behaviour lets it work in constant space. notation spares us from having to specify the units used to measure space. FunctionTimeSpace npower, nsumO()O() summingO()O() O()O() powerO()O() sillySumO()O() Even ignoring constant factors, the units chosen can influence the result. Multiplication may be regarded as a single unit of cost. However, the cost of multiplying two -digit numbers for large is itself an important question, especially now that public-key cryptography uses numbers hundreds of digits long.Few things can really be done in constant time or stored in constant space. Merely to store the number requires bits. If a program cost is , then we have probably assumed that certain operations it performs are also —typically because we expect never to exceed the capacity of the standard hardware arithmetic.With power, the precise number of operations depends upon in a complicated way, depending on how many odd numbers arise, so it is convenient that we can just write . An accumulating argument could reduce its space cost to .Anil Madhavapeddyhttps://www.cl.cam.ac.uk/teaching/2627/FoundsCS/forest/focs-sillysum/focs-sillysum/teaching/2627/FoundsCS/forest/focs-sillysum/Silly Summing the First n Integers int = ]]>The function calls itself times! Bigger inputs mean higher costs—but what’s the growth rate?Now let us consider how to estimate various costs associated with a program. Asymptotic complexity refers to how costs—usually time or space—grow with increasing inputs. Space complexity can never exceed time complexity, for it takes time to do anything with the space. Time complexity often greatly exceeds space complexity.The function sillySum calls itself twice in each recursive step. This function is contrived, but many mathematical formulas refer to a particular quantity more than once. In OCaml, we can create a local binding to a computed value using the local declaration syntax. In the following expression, y is computed once and used twice:You can read let x = e1 in e2 as assigning (or "binding") the name x with the value of e1 into e2. Any use of x within e2 will have the value of e1, and x will only be visible in subexpressions into which it has been bound.Why do we need let bindings? Fast hardware does not make good algorithms unnecessary. On the contrary, faster hardware magnifies the superiority of better algorithms. Typically, we want to handle the largest inputs possible. If we double our processing power, what do we gain? How much can we increase , the input to our function?With sillySum, we can only go from to . We are limited to this modest increase because the function’s running time is proportional to . With the function npower defined in the previous section, we can go from to : we can handle problems twice as big. With power we can do much better still, going from to .The following table (excerpted from a 50-year-old book!) illustrates the effect of various time complexities. The left-hand column (dubbed "complexity") is defined as how many milliseconds are required to process an input of size . The other entries show the maximum size of that can be processed in the given time (one second, minute or hour). complexity1 second1 minute1 hourgain 100060 0003 600 000 1404 895204 095 312441 897 1039153 91521 The table illustrates how large an input can be processed as a function of time. As we increase the computer time per input from one second to one minute and then to one hour, the size of the input increases accordingly.The top two rows (complexities and ) increase rapidly: for , by a factor of 60 per column. The bottom two start out close together, but (which grows by a factor of 3.9) pulls well away from (whose growth is only additive). If an algorithm’s complexity is exponential then it can never handle large inputs, even if it is given huge resources. On the other hand, suppose the complexity has the form , where is a constant. (We say the complexity is polynomial.) Doubling the argument then increases the cost by a constant factor. That is much better, though if 3]]> the algorithm may not be considered practical.Anil Madhavapeddyhttps://www.cl.cam.ac.uk/teaching/2627/FoundsCS/forest/focs-recurrences/focs-recurrences/teaching/2627/FoundsCS/forest/focs-recurrences/Some Simple Recurrence RelationsConsider a function that has a cost we want to bound using notation. A typical base case is . Some recurrences are: EquationComplexity To analyse a function, inspect its OCaml declaration. Recurrence equations for the cost function can usually be read off. Since we ignore constant factors, we can give the base case a cost of one unit. Constant work done in the recursive step can also be given unit cost; since we only need an upper bound, this unit represents the larger of the two actual costs. We could use other constants if it simplifies the algebra.For example, recall our function nsum: int = ]]>Given , it performs a constant amount of work (an addition and subtraction) and calls itself recursively with argument . We get the recurrence equations and . The closed form is clearly , as we can easily verify by substitution. The cost is linear.This function, given , calls nsum, performing work. Again ignoring constant factors, we can say that this call takes exactly units. int = ]]>We get the recurrence equations and . It is easy to see that . The cost is quadratic.The function power divides its input into two, with the recurrence equation . Clearly , so .Anil Madhavapeddyhttps://www.cl.cam.ac.uk/teaching/2627/FoundsCS/forest/focs-nsum/focs-nsum/teaching/2627/FoundsCS/forest/focs-nsum/Summing the first n integers int = ]]>The function call nsum n computes the sum 1 + … + n rather naively, hence the initial n in its name:The nesting of parentheses is not just an artifact of our notation; it indicates a real problem. The function gathers up a collection of numbers, but none of the additions can be performed until nsum 0 is reached. Meanwhile, the computer must store the numbers in an internal data structure, typically the stack. For large n, say nsum 10000, the computation might fail due to stack overflow.We all know that the additions can be performed as we go along. How do we make the computer do that?Anil Madhavapeddyhttps://www.cl.cam.ac.uk/teaching/2627/FoundsCS/forest/focs-def-tail-recursion/focs-def-tail-recursion/teaching/2627/FoundsCS/forest/focs-def-tail-recursion/Tail recursionDefinitionA recursive function whose computation does not nest is called iterative or tail-recursive: the recursive calls do not nest, so the work is done immediately rather than on the way back out. Many functions can be made iterative by introducing an extra argument, which is often called an accumulator. Introduced in Iteratively summing the first n integers.Part of the course glossary.Anil Madhavapeddyhttps://www.cl.cam.ac.uk/teaching/2627/FoundsCS/forest/focs-infinite-sequence/focs-infinite-sequence/teaching/2627/FoundsCS/forest/focs-infinite-sequence/The Infinite Sequence: , , , … from (k+1)) val from : int -> int seq = # let it = from 1 val it : int seq = Cons (1, ) # let it = tail it val it : int seq = Cons (2, ) # let it = tail it val it : int seq = Cons (3, )]]>Function from constructs the infinite sequence of integers starting from . Execution terminates because of the fun enclosing the recursive call. OCaml displays the tail of a sequence as fun, which stands for some function value. Each call to tail generates the next sequence element. We could do this forever.This example is of little practical value because the cost of computing a sequence element will be dominated by that of creating the dummy function. Lazy lists tend to have high overheads.Anil Madhavapeddyhttps://www.cl.cam.ac.uk/teaching/2627/FoundsCS/forest/focs-bst-update/focs-bst-update/teaching/2627/FoundsCS/forest/focs-bst-update/Update Br ((k, v), Lf, Lf) | Br ((a, x), t1, t2) -> if k < a then Br ((a, x), update k v t1, t2) else if a < k then Br ((a, x), t1, update k v t2) else (* a = k *) Br ((a, v), t1, t2) val update : 'a -> 'b -> ('a * 'b) tree -> ('a * 'b) tree = ]]>This is also as it copies the path only, and not whole subtrees!If you are familiar with the usual update operation for this sort of tree, you may wonder whether it can be implemented in OCaml, where there is no direct way to replace part of a data structure by something else.The update operation is a nice piece of functional programming. It searches in the same manner as lookup, but the recursive calls reconstruct a new tree around the result of the update. One subtree is updated and the other left unchanged. The internal representation of trees ensures that unchanged parts of the tree are not copied, but shared. Therefore, update copies only the path from the root to the new node. Its time and space requirements, for a reasonably balanced tree, are both .The comparison between and allows three cases:smaller: update the left subtree; share the right greater: update the right subtree; share the left equal: update the label and share both subtreesNote: in the function definition, (* a = k *) is a comment. Comments in OCaml are enclosed in the brackets and .Anil MadhavapeddyAnil Madhavapeddyhttps://www.cl.cam.ac.uk/teaching/2627/FoundsCS/forest/focs-ex-2-4/focs-ex-2-4/teaching/2627/FoundsCS/forest/focs-ex-2-4/Exercises › A tighter recurrence boundExerciseFind an upper bound for the recurrence given by and . You should be able to find a tighter bound than .Anil MadhavapeddyAnil Madhavapeddyhttps://www.cl.cam.ac.uk/teaching/2627/FoundsCS/forest/focs-ex-4-4/focs-ex-4-4/teaching/2627/FoundsCS/forest/focs-ex-4-4/Exercises › Assumptions in making changeExerciseWhat assumptions do the ‘making change’ functions make about the variables till and amt? Describe what could happen if these assumptions were violated.Anil MadhavapeddyAnil Madhavapeddyhttps://www.cl.cam.ac.uk/teaching/2627/FoundsCS/forest/focs-ex-4-5/focs-ex-4-5/teaching/2627/FoundsCS/forest/focs-ex-4-5/Exercises › Counting ways of making changeExerciseShow that the number of ways of making change for (ignoring order) is if there are two legal coin values. What if there are three, four, … coin values?Anil MadhavapeddyAnil Madhavapeddyhttps://www.cl.cam.ac.uk/teaching/2627/FoundsCS/forest/focs-ex-3-3/focs-ex-3-3/teaching/2627/FoundsCS/forest/focs-ex-3-3/Exercises › Even-numbered elementsExerciseCode a function to return the list consisting of the even-numbered elements of the list given as its argument. For example, given [a; b; c; d] it should return [b; d].Anil MadhavapeddyAnil Madhavapeddyhttps://www.cl.cam.ac.uk/teaching/2627/FoundsCS/forest/focs-ex-1-5/focs-ex-1-5/teaching/2627/FoundsCS/forest/focs-ex-1-5/Exercises › Floating-point error by repeated additionExerciseBecause computer arithmetic is based on binary numbers, simple decimals such as 0.1 often cannot be represented exactly. Write a function mul that performs the computationwhere has type float. (It is essential to use repeated addition rather than multiplication!)The value computed with n = 10000 and x = 0.1 may print as 1000.0, which looks exact. If that happens, then evaluate the expression mul 0.1 10000 -. 1000.0An error of this type has been blamed for the failure of an American Patriot Missile battery to intercept an incoming Iraqi missile during the first Gulf War. The missile hit an American Army barracks, killing 28.Anil MadhavapeddyAnil Madhavapeddyhttps://www.cl.cam.ac.uk/teaching/2627/FoundsCS/forest/focs-ex-1-6/focs-ex-1-6/teaching/2627/FoundsCS/forest/focs-ex-1-6/Exercises › Golden ratio instabilityExerciseAnother example of the inaccuracy of floating-point arithmetic takes the golden ratio as its starting point:In theory, it is easy to prove that for all 0]]>. Code this computation in OCaml and report the value of . Hint: in OCaml, is expressed as sqrt 5.0.Anil MadhavapeddyAnil Madhavapeddyhttps://www.cl.cam.ac.uk/teaching/2627/FoundsCS/forest/focs-ex-5-4/focs-ex-5-4/teaching/2627/FoundsCS/forest/focs-ex-5-4/Exercises › Implementing bubble sortExerciseImplement bubble sort (see previous exercise) using OCaml.Anil MadhavapeddyAnil Madhavapeddyhttps://www.cl.cam.ac.uk/teaching/2627/FoundsCS/forest/focs-ex-5-2/focs-ex-5-2/teaching/2627/FoundsCS/forest/focs-ex-5-2/Exercises › Implementing selection sortExerciseImplement selection sort (see previous exercise) using OCaml.Anil MadhavapeddyAnil Madhavapeddyhttps://www.cl.cam.ac.uk/teaching/2627/FoundsCS/forest/focs-ex-2-1/focs-ex-2-1/teaching/2627/FoundsCS/forest/focs-ex-2-1/Exercises › Iterative powerExerciseCode an iterative version of the function power.Anil MadhavapeddyAnil Madhavapeddyhttps://www.cl.cam.ac.uk/teaching/2627/FoundsCS/forest/focs-ex-11-2/focs-ex-11-2/teaching/2627/FoundsCS/forest/focs-ex-11-2/Exercises › Iterative power with whileExerciseWrite a version of function power (Lecture ) using while instead of recursion.Anil MadhavapeddyAnil Madhavapeddyhttps://www.cl.cam.ac.uk/teaching/2627/FoundsCS/forest/focs-ex-9-3/focs-ex-9-3/teaching/2627/FoundsCS/forest/focs-ex-9-3/Exercises › Making change lazilyExerciseCode a function to make change using lazy lists, delivering the sequence of all possible ways of making change. Using sequences allows us to compute solutions one at a time when there exists an astronomical number. Represent lists of coins using ordinary lists. (Hint: to benefit from laziness you may need to pass around the sequence of alternative solutions as a function of type unit -> (int list) seq.)Anil MadhavapeddyAnil Madhavapeddyhttps://www.cl.cam.ac.uk/teaching/2627/FoundsCS/forest/focs-ex-8-5/focs-ex-8-5/teaching/2627/FoundsCS/forest/focs-ex-8-5/Exercises › Making change with mapExerciseRecall the making change function of More on Lists: [ [] ] | [] , _ -> [] | c::till , amt -> if amt < c then change till amt else let rec allc = function | [] -> [] | cs :: css -> (c::cs) :: allc css in allc (change (c::till) (amt - c)) @ change till amt val change : int list -> int -> int list list = ]]>Function allc applies the function ‘cons a c’ to every element of a list. Eliminate it by declaring a curried cons function and applying map.Anil MadhavapeddyAnil Madhavapeddyhttps://www.cl.cam.ac.uk/teaching/2627/FoundsCS/forest/focs-ex-4-2/focs-ex-4-2/teaching/2627/FoundsCS/forest/focs-ex-4-2/Exercises › Partitioning integers by signExerciseCode a function that takes a list of integers and returns two lists, the first consisting of all non-negative numbers found in the input and the second consisting of all the negative numbers.Anil MadhavapeddyAnil Madhavapeddyhttps://www.cl.cam.ac.uk/teaching/2627/FoundsCS/forest/focs-ex-3-1/focs-ex-3-1/teaching/2627/FoundsCS/forest/focs-ex-3-1/Exercises › Summing a listExerciseCode a recursive function to compute the sum of a list’s elements. Then code an iterative version and comment on the improvement in efficiency.Anil MadhavapeddyAnil Madhavapeddyhttps://www.cl.cam.ac.uk/teaching/2627/FoundsCS/forest/focs-ex-6-2/focs-ex-6-2/teaching/2627/FoundsCS/forest/focs-ex-6-2/Exercises › Summing tree labelsExerciseWrite an OCaml function taking a binary tree labelled with integers and returning their sum.Anil MadhavapeddyAnil Madhavapeddyhttps://www.cl.cam.ac.uk/teaching/2627/FoundsCS/forest/focs-ex-3-5/focs-ex-3-5/teaching/2627/FoundsCS/forest/focs-ex-3-5/Exercises › Tails of a listExerciseCode a function tails to return the list of the tails of its argument. For example, given [1; 2; 3] it should return [[1; 2; 3]; [2; 3]; [3]; []].Anil MadhavapeddyAnil Madhavapeddyhttps://www.cl.cam.ac.uk/teaching/2627/FoundsCS/forest/focs-ex-1-4/focs-ex-1-4/teaching/2627/FoundsCS/forest/focs-ex-1-4/Exercises › Type inference in powerExerciseFunctions npower and power both return a float. The definition of npower returns the float value 1.0 in its base case. The definition of power does not, so how does the OCaml type checker know that power returns a float?Anil MadhavapeddyAnil Madhavapeddyhttps://www.cl.cam.ac.uk/teaching/2627/FoundsCS/forest/focs-ex-6-3/focs-ex-6-3/teaching/2627/FoundsCS/forest/focs-ex-6-3/Exercises › What ftree buildsExerciseUsing the definition of 'a tree from before:Examine the following function declaration. What does ftree (1, n) accomplish?Anil Madhavapeddyhttps://www.cl.cam.ac.uk/teaching/2627/FoundsCS/forest/focs-topic-complexity/focs-topic-complexity/teaching/2627/FoundsCS/forest/focs-topic-complexity/ComplexityAnil Madhavapeddyhttps://www.cl.cam.ac.uk/teaching/2627/FoundsCS/forest/focs-change-faster/focs-change-faster/teaching/2627/FoundsCS/forest/focs-change-faster/All Ways of Making Change — Faster! chg::chgs | [] , _ -> chgs | c::till , amt -> if amt < 0 then chgs else change (c::till) (amt - c) (c::chg) (change till amt chg chgs) val change : int list -> int -> int list -> int list list -> int list list = ]]>We’ve added another accumulating parameter! Repeatedly improving simple code is called stepwise refinement.Two extra arguments eliminate many :: and append operations from the all-solutions change function. The first, chg, accumulates the coins chosen so far; one evaluation of c::chg replaces many evaluations of allc. The second, chgs, accumulates the list of solutions so far; it avoids the need for append. This version runs several times faster than the previous one.Making change is still extremely slow for an obvious reason: the number of solutions grows rapidly in the amount being changed. Using 50, 20, 10, 5, 2 and 1, there are 4366 ways of expressing 99.Our three change functions illustrate a basic technique: program development by stepwise refinement. Begin by writing a very simple program and add requirements individually. Add efficiency refinements last of all. Even if the simpler program cannot be included in the next version and has to be discarded, one has learned about the task by writing it.Anil Madhavapeddyhttps://www.cl.cam.ac.uk/teaching/2627/FoundsCS/forest/focs-append/focs-append/teaching/2627/FoundsCS/forest/focs-append/Append: List Concatenation ys | x::xs, ys -> x :: append xs ys val append : 'a list -> 'a list -> 'a list = # append [1; 2; 3] [4] - : int list = [1; 2; 3; 4] # let (@) = append val ( @ ) : 'a list -> 'a list -> 'a list = # [1; 2; 3] @ [4] - : int list = [1; 2; 3; 4]]]>Patterns can be as complicated as we like. Here, the two patterns are [], ys and x::xs, ys.Here is how append might be declared, also noting that we have defined @ as an infix operator that is a more convenient way to call append on two lists. However, this function is also not iterative. It scans its first argument, sets up a string of cons operations (::) and finally does them.It uses space and time, where is the length of its first argument. Its costs are independent of its second argument.An accumulating argument could make it iterative, but with considerable complication. The iterative version would still require space and time because concatenation requires copying all the elements of the first list. Therefore, we cannot hope for asymptotic gains; at best we can decrease the constant factor involved in , but complicating the code is likely to increase that factor. Never add an accumulator merely out of habit.Note append’s polymorphic type. It tells us that two lists can be joined if their element types agree.Anil Madhavapeddyhttps://www.cl.cam.ac.uk/teaching/2627/FoundsCS/forest/focs-complexity-classes/focs-complexity-classes/teaching/2627/FoundsCS/forest/focs-complexity-classes/Common Complexity Classes is constant is logarithmic is linear is quasi-linear is quadratic is cubic is exponential (for fixed )Logarithms grow very slowly, so complexity is excellent. Because notation ignores constant factors, the base of the logarithm is irrelevant!Under linear we might mention , which occasionally is called quasilinear and which scales up well for large .An example of quadratic complexity is matrix addition: forming the sum of two matrices obviously takes additions. Matrix multiplication is of cubic complexity, which limits the size of matrices that we can multiply in reasonable time. An algorithm exists, but it is too complicated to be of much use, even though it is theoretically better.An exponential growth rate such as restricts us to small values of . Already with the cost exceeds one million. However, the worst case might not arise in normal practice. OCaml type-checking is exponential in the worst case, but not for ordinary programs.Anil Madhavapeddyhttps://www.cl.cam.ac.uk/teaching/2627/FoundsCS/forest/focs-o-notation/focs-o-notation/teaching/2627/FoundsCS/forest/focs-o-notation/Comparing Algorithms: O NotationFormally, define provided as is bounded for some constant and all sufficiently large . Intuitively, look at the most significant term. Ignore constant factors as they seldom dominate and are often transitoryFor example: consider instead of .The cost of a program is usually a complicated formula. Often we should consider only the most significant term. If the cost is for an input of size , then the term will eventually dominate, even though is bigger for . The constant term may look big, but it is soon dominated by .Constant factors in costs can be ignored unless they are large. For one thing, they seldom make a difference: will be better than in the long run: or asymptotically to use the jargon. Moreover, constant factors are seldom stable. They depend upon details such as which hardware, operating system or programming language is being used. By ignoring constant factors, we can make comparisons between algorithms that remain valid in a broad range of circumstances.The “Big O” notation is commonly used to describe efficiency—to be precise, asymptotic complexity. It concerns the limit of a function as its argument tends to infinity. It is an abstraction that meets the informal criteria that we have just discussed. In the definition, sufficiently large means there is some constant such that for all greater than . The role of is to ignore finitely many exceptions to the bound, such as the cases when exceeds .Anil Madhavapeddyhttps://www.cl.cam.ac.uk/teaching/2627/FoundsCS/forest/focs-functional-queues/focs-functional-queues/teaching/2627/FoundsCS/forest/focs-functional-queues/Efficient Functional Queues: Code Q (List.rev tls, []) | q -> q val norm : 'a queue -> 'a queue = # let qnull = function | Q ([], []) -> true | _ -> false val qnull : 'a queue -> bool = # let enq (Q (hds, tls)) x = norm (Q (hds, x::tls)) val enq : 'a queue -> 'a -> 'a queue = # exception Empty exception Empty # let deq = function | Q (x::hds, tls) -> norm (Q (hds, tls)) | _ -> raise Empty val deq : 'a queue -> 'a queue = # let qempty = Q ([], []) val qempty : 'a queue = Q ([], []) # let qhd = function | Q (x::_, _) -> x | _ -> raise Empty val qhd : 'a queue -> 'a = ]]>The datatype of queues prevents confusion with other pairs of lists. The empty queue has both parts empty.The function norm puts a queue into normal form, ensuring that the front part is never empty unless the entire queue is empty. Functions deq and enq call norm to normalise their result.Because queues are in normal form, their head is certain to be in their front part, so qhd looks there.Let us analyse the cost of an execution comprising (in any possible order) enq operations and deq operations, starting with an empty queue. Each enq operation will perform one cons, adding an element to the rear part. Since the final queue must be empty, each element of the rear part gets transferred to the front part. The corresponding reversals perform one cons per element. Thus, the total cost of the series of queue operations is cons operations, an average of 2 per operation. The amortized time is .There is a catch. The conses need not be distributed evenly; reversing a long list could take up to of them. Unpredictable delays make the approach unsuitable for real-time programming where deadlines must be met.Anil Madhavapeddyhttps://www.cl.cam.ac.uk/teaching/2627/FoundsCS/forest/focs-functional-queues-idea/focs-functional-queues-idea/teaching/2627/FoundsCS/forest/focs-functional-queues-idea/Efficient Functional Queues: IdeaRepresent the queue by any pair of lists Add new items to the rear list Remove items from front list and if empty move rear to front Amortized time per operation is Queues require efficient access at both ends: at the front, for removal, and at the back, for insertion. Ideally, access should take constant time, . It may appear that lists cannot provide such access. If enq(q, x) performs q@[x], then this operation will be . We could represent queues by reversed lists, implementing enq(q, x) by x::q, but then the deq and qhd operations would be . Linear time is intolerable: a series of queue operations could then require time.The solution is to represent a queue by a pair of lists, whererepresents the queue .The front part of the queue is stored in order, and the rear part is stored in reverse order. The enq operation adds elements to the rear part using cons, since this list is reversed; thus, enq takes constant time. The deq and qhd operations look at the front part, which normally takes constant time, since this list is stored in order. But sometimes deq removes the last element from the front part; when this happens, it reverses the rear part, which becomes the new front part.Amortized time refers to the cost per operation averaged over the lifetime of any complete execution. Even for the worst possible execution, the average cost per operation turns out to be constant; see the analysis below.Anil Madhavapeddyhttps://www.cl.cam.ac.uk/teaching/2627/FoundsCS/forest/focs-length/focs-length/teaching/2627/FoundsCS/forest/focs-length/Efficiently Computing the Length of a List n | x::xs -> addlen (n + 1) xs val addlen : int -> 'a list -> int = # addlen 0 [5; 6; 7] - : int = 3]]>Recall that the use of function introduces an extra (unnamed) argument that is pattern matched in the subsequent clauses; in this case, to break open the list.Function addlen is again polymorphic. Its type mentions the integer accumulator.Now we may declare an efficient length function. It is simply a wrapper for addlen, supplying zero as the initial value of . int = # length [5; 6; 7; 8] - : int = 4]]>The recursive calls do not nest: this version is iterative. It takes space. Obviously its time requirement is because it takes at least steps to find the length of an -element list.Anil Madhavapeddyhttps://www.cl.cam.ac.uk/teaching/2627/FoundsCS/forest/focs-power/focs-power/teaching/2627/FoundsCS/forest/focs-power/Efficiently Raising a Number to a Power int -> float = ]]>Mathematical JustificationFor large n, computing powers using is too slow to be practical. The equations above are much faster. Example:Instead of n multiplications, we need at most multiplications, where is the logarithm of to the base .We use the function even, declared previously, to test whether the exponent is even. Integer division (/) truncates its result to an integer: dividing by 2 yields .A recurrence is a useful computation rule only if it is bound to terminate. If 0]]> then is smaller than both and . After enough recursive calls, the exponent will be reduced to . The equations also hold if , but the corresponding computation runs forever.Our reasoning assumes arithmetic to be exact. Fortunately, the calculation is well-behaved using floating-point.Computer numbers have a finite range, which if exceeded results in the integer wrapping around. You will understand this behaviour more as you learn about computer architecture and how modern systems represent numbers in memory.If integers and floats must be combined in a calculation, OCaml provides functions to convert between them:OCaml’s libraries are organised using “modules”, so we may use compound identifiers such as Float.of_int to refer to library functions. There are many thousands of library functions available in the OCaml ecosystem, including text-processing and operating systems functions in addition to the usual numerical ones.Anil Madhavapeddyhttps://www.cl.cam.ac.uk/teaching/2627/FoundsCS/forest/focs-tree-traversal-efficient/focs-tree-traversal-efficient/teaching/2627/FoundsCS/forest/focs-tree-traversal-efficient/Efficiently Traversing TreesUnfortunately, the traversal functions are quadratic in the worst case: the appends in the recursive calls are inefficient. To correct that problem, we (as usual) add an accumulating argument. Observe how each function constructs its result list and compare with how appends were eliminated from quicksort in the Sorting lecture. vs | Br (v, t1, t2), vs -> v :: preord (t1, preord (t2, vs)) val preord : 'a tree * 'a list -> 'a list = # let rec inord = function | Lf, vs -> vs | Br (v, t1, t2), vs -> inord (t1, v::inord (t2, vs)) val inord : 'a tree * 'a list -> 'a list = # let rec postord = function | Lf, vs -> vs | Br (v, t1, t2), vs -> postord (t1, postord (t2, v::vs)) val postord : 'a tree * 'a list -> 'a list = ]]>One can prove equations relating each of these functions to its counterpart on the previous section. For example:These three types of tree traversal are related in that all are depth-first. They each traverse the left subtree in full before traversing the right subtree. Breadth-first search (from the Queues lecture) is another possibility. That involves going through the levels of a tree one at a time.Anil Madhavapeddyhttps://www.cl.cam.ac.uk/teaching/2627/FoundsCS/forest/focs-functional-arrays/focs-functional-arrays/teaching/2627/FoundsCS/forest/focs-functional-arrays/Functional Arrays as Binary TreesThe path to element follows the binary code for (its “subscript”).This simple representation (credited to W. Braun) ensures that the tree is balanced. Complexity of access is always , which is optimal. For actual running time, access to conventional arrays is much faster: it requires only a few hardware instructions. Array access is often taken to be , which (as always) presumes that hardware limits are never exceeded.The lower bound for array subscripts (or “indices”) is one. The upper bound starts at zero (which signifies the empty array) and can grow without limit. Inspection of the diagram above should make it clear that these trees are always balanced: the left subtree can have at most one node more than the right subtree, recursively all the way down. (This assumes that the array is defined for subscripts with no gaps; an array defined only for odd numbers, for example, would obviously be unbalanced.)The numbers in the diagram above are not the labels of branch nodes, but indicate the positions of array elements. For example, the label corresponding to is at the position shown. The nodes of a functional array are labelled with the data we want to store, not with these integers.Anil Madhavapeddyhttps://www.cl.cam.ac.uk/teaching/2627/FoundsCS/forest/focs-sorting-bounds/focs-sorting-bounds/teaching/2627/FoundsCS/forest/focs-sorting-bounds/How Fast Can We Sort?typically count comparisons there are permutations of elements each comparison eliminates half of the permutations therefore The usual measure of efficiency for sorting algorithms is the number of comparison operations required. Mergesort requires only comparisons to sort an input of items. It is straightforward to prove that this complexity is the best possible. There are permutations of elements and each comparison distinguishes two permutations. The lower bound on the number of comparisons, , is obtained by solving ; therefore .In order to compare the sorting algorithms, we use the following source of pseudo-random numbers. Never mind how this works: generating statistically good random numbers is hard. Much effort has gone into those few lines of code. float = # let rec randlist (seed, seeds) = function | 0 -> (seed, seeds) | n -> randlist (nextrandom seed, seed::seeds) (n-1) val randlist : float * float list -> int -> float * float list = ]]>We can now bind the identifier rs to a list of 10,000 random numbers.Anil Madhavapeddyhttps://www.cl.cam.ac.uk/teaching/2627/FoundsCS/forest/focs-iterative-deepening/focs-iterative-deepening/teaching/2627/FoundsCS/forest/focs-iterative-deepening/Iterative deepening: Another Exhaustive SearchBreadth-first search examines nodes: Recompute nodes at depth instead of storing them Time factor is if 1]]>; complexity is still Space required at depth drops from to Breadth-first search is not practical for infinite trees: it uses too much space. Large parts of the tree have to be stored. Consider the slightly more general problem of searching trees whose branching factor is (for binary trees, ). Then breadth-first search to depth examines nodes, which is , ignoring the constant factor of . Since all nodes that are examined are also stored, the space and time requirements are both .Depth-first iterative deepening combines the space efficiency of depth-first with the “nearest-first” property of breadth-first search. It performs repeated depth-first searches with increasing depth bounds, each time discarding the result of the previous search. Thus it searches to depth 1, then to depth 2, and so on until it finds a solution. We can afford to discard previous results because the number of nodes is growing exponentially. There are nodes at level ; if , this number actually exceeds the total number of nodes of all previous levels put together, namely .Korf shows that the time needed for iterative deepening to reach depth is only times that for breadth-first search, if 1]]>. This is a constant factor; both algorithms have the same time complexity, . In typical applications where the extra factor of is quite tolerable. The reduction in the space requirement is exponential, from for breadth-first to for iterative deepening. Of course, this assumes that the tree itself is not stored in memory.Anil Madhavapeddyhttps://www.cl.cam.ac.uk/teaching/2627/FoundsCS/forest/focs-linear-search/focs-linear-search/teaching/2627/FoundsCS/forest/focs-linear-search/Linear Searchfind in list by comparing with each element obviously time simple & general ordered searching needs only indexed lookup needs only Linear search is the obvious way to find a desired item in a collection: simply look through all the items, one at a time. If is in the list, then it will be found in steps on average, and even the worst case is obviously .Large collections of data are usually ordered or indexed so that items can be found in time, which is exponentially better than . Even is achievable (using a hash table), though subject to the usual proviso that machine limits are not exceeded.Efficient indexing methods are of prime importance: consider Web search engines. Nevertheless, linear search is often used to search small collections because it is so simple and general, and it is the starting point for better algorithms.Anil Madhavapeddyhttps://www.cl.cam.ac.uk/teaching/2627/FoundsCS/forest/focs-def-o-notation/focs-def-o-notation/teaching/2627/FoundsCS/forest/focs-def-o-notation/O notationDefinitionFormally, provided as , that is, for some constant and all sufficiently large . The “Big O” notation is commonly used to describe efficiency—to be precise, asymptotic complexity—by looking only at the most significant term and ignoring constant factors. Introduced in Comparing Algorithms: O Notation.Part of the course glossary.Anil Madhavapeddyhttps://www.cl.cam.ac.uk/teaching/2627/FoundsCS/forest/focs-quicksort/focs-quicksort/teaching/2627/FoundsCS/forest/focs-quicksort/Quicksort: The Code [] | [x] -> [x] | a::bs -> let rec part l r = function | [] -> (quick l) @ (a :: quick r) | x::xs -> if (x <= a) then part (x::l) r xs else part l (x::r) xs in part [] [] bs val quick : 'a list -> 'a list = ]]>Our OCaml quicksort copies the items. It is still pretty fast, and it is much easier to understand. It takes roughly 0.74 seconds to sort our list of random numbers.The function declaration consists of three clauses. The first handles the empty list; the second handles singleton lists (those of the form [x]); the third handles lists of two or more elements. Often, lists of length up to five or so are treated as special cases to boost speed.The locally declared function part partitions the input using a as the pivot. The arguments l and r accumulate items for the left () and right (a]]>) parts of the input, respectively.It is not hard to prove that quicksort does comparisons, in the average case (see page 94 of Aho). With random data, the pivot usually has an average value that divides the input in two approximately equal parts. We have the recurrence and , which is . In our example, it is about 235 times faster than insertion sort.In the worst case, quicksort's running time is quadratic! An example is when its input is almost sorted or reverse sorted. Nearly all of the items end up in one partition; work is not divided evenly. We have the recurrence and , which is . Randomising the input makes the worst case highly unlikely.Anil Madhavapeddyhttps://www.cl.cam.ac.uk/teaching/2627/FoundsCS/forest/focs-rev/focs-rev/teaching/2627/FoundsCS/forest/focs-rev/Reversing a List in ys | x::xs, ys -> rev_app xs (x::ys) val rev_app : 'a list -> 'a list -> 'a list = ]]>Calling rev_app xs ys reverses the elements of xs and prepends them to ys. Now we may declare 'a list = # rev [1; 2; 3] - : int list = [3; 2; 1]]]>It is easy to see that this reverse function performs just conses, given an -element list. For both reverse functions, we could count the number of conses precisely—not just up to a constant factor. notation is still useful to describe the overall running time: the time taken by a cons varies from one system to another.The accumulator makes the function iterative. But the gain in complexity arises from the removal of append. Replacing an expensive operation (append) by a series of cheap operations (cons) is called reduction in strength and is a common technique in computer science. It originated when many computers did not have a hardware multiply instruction; the series of products for , could more efficiently be computed by repeated addition. Reduction in strength can be done in various ways; we shall see many instances of removing append.Consing to an accumulator produces the result in reverse. If that forces the use of an extra list reversal then the iterative function may be much slower than the recursive one.Anil Madhavapeddyhttps://www.cl.cam.ac.uk/teaching/2627/FoundsCS/forest/focs-nrev/focs-nrev/teaching/2627/FoundsCS/forest/focs-nrev/Reversing a List in Let us consider one way to reverse a list. [] | x::xs -> (nrev xs) @ [x] val nrev : 'a list -> 'a list = # nrev [1; 2; 3] - : int list = [3; 2; 1]]]>This reverse function is grossly inefficient due to poor usage of append, which copies its first argument. If nrev is given a list of length 0]]>, then append makes conses to copy the reversed tail. Constructing the list [x] calls cons again, for a total of calls. Reversing the tail requires more conses, and so forth. The total number of conses is:The time complexity is therefore . Space complexity is only because the copies don’t all exist at the same time.Anil Madhavapeddyhttps://www.cl.cam.ac.uk/teaching/2627/FoundsCS/forest/focs-sample-costs/focs-sample-costs/teaching/2627/FoundsCS/forest/focs-sample-costs/Sample costs in O notationRecall that npower computes by repeated multiplication while nsum naively computes the sum . Each obviously performs arithmetic operations. Because they are not tail recursive, their use of space is also . The function summing is a version of nsum with an accumulating argument; its iterative behaviour lets it work in constant space. notation spares us from having to specify the units used to measure space. FunctionTimeSpace npower, nsumO()O() summingO()O() O()O() powerO()O() sillySumO()O() Even ignoring constant factors, the units chosen can influence the result. Multiplication may be regarded as a single unit of cost. However, the cost of multiplying two -digit numbers for large is itself an important question, especially now that public-key cryptography uses numbers hundreds of digits long.Few things can really be done in constant time or stored in constant space. Merely to store the number requires bits. If a program cost is , then we have probably assumed that certain operations it performs are also —typically because we expect never to exceed the capacity of the standard hardware arithmetic.With power, the precise number of operations depends upon in a complicated way, depending on how many odd numbers arise, so it is convenient that we can just write . An accumulating argument could reduce its space cost to .Anil Madhavapeddyhttps://www.cl.cam.ac.uk/teaching/2627/FoundsCS/forest/focs-sillysum/focs-sillysum/teaching/2627/FoundsCS/forest/focs-sillysum/Silly Summing the First n Integers int = ]]>The function calls itself times! Bigger inputs mean higher costs—but what’s the growth rate?Now let us consider how to estimate various costs associated with a program. Asymptotic complexity refers to how costs—usually time or space—grow with increasing inputs. Space complexity can never exceed time complexity, for it takes time to do anything with the space. Time complexity often greatly exceeds space complexity.The function sillySum calls itself twice in each recursive step. This function is contrived, but many mathematical formulas refer to a particular quantity more than once. In OCaml, we can create a local binding to a computed value using the local declaration syntax. In the following expression, y is computed once and used twice:You can read let x = e1 in e2 as assigning (or "binding") the name x with the value of e1 into e2. Any use of x within e2 will have the value of e1, and x will only be visible in subexpressions into which it has been bound.Why do we need let bindings? Fast hardware does not make good algorithms unnecessary. On the contrary, faster hardware magnifies the superiority of better algorithms. Typically, we want to handle the largest inputs possible. If we double our processing power, what do we gain? How much can we increase , the input to our function?With sillySum, we can only go from to . We are limited to this modest increase because the function’s running time is proportional to . With the function npower defined in the previous section, we can go from to : we can handle problems twice as big. With power we can do much better still, going from to .The following table (excerpted from a 50-year-old book!) illustrates the effect of various time complexities. The left-hand column (dubbed "complexity") is defined as how many milliseconds are required to process an input of size . The other entries show the maximum size of that can be processed in the given time (one second, minute or hour). complexity1 second1 minute1 hourgain 100060 0003 600 000 1404 895204 095 312441 897 1039153 91521 The table illustrates how large an input can be processed as a function of time. As we increase the computer time per input from one second to one minute and then to one hour, the size of the input increases accordingly.The top two rows (complexities and ) increase rapidly: for , by a factor of 60 per column. The bottom two start out close together, but (which grows by a factor of 3.9) pulls well away from (whose growth is only additive). If an algorithm’s complexity is exponential then it can never handle large inputs, even if it is given huge resources. On the other hand, suppose the complexity has the form , where is a constant. (We say the complexity is polynomial.) Doubling the argument then increases the cost by a constant factor. That is much better, though if 3]]> the algorithm may not be considered practical.Anil Madhavapeddyhttps://www.cl.cam.ac.uk/teaching/2627/FoundsCS/forest/focs-o-notation-facts/focs-o-notation-facts/teaching/2627/FoundsCS/forest/focs-o-notation-facts/Simple Facts About O Notation notation lets us reason about the costs of algorithms easily.Constant factors such as the in drop out: we can use with twice the value of in the definition. Because constant factors drop out, the base of logarithms is irrelevant. Insignificant terms drop out. To see that is the same as , consider that converges to 1 for increasing . In fact, for , so can double the constant factorIf and are constants (that is, they are independent of ) with then is contained in is contained in is contained in To say that is contained in means that the former gives a tighter bound than the latter. For example, if then trivially, but the converse does not hold.Anil Madhavapeddyhttps://www.cl.cam.ac.uk/teaching/2627/FoundsCS/forest/focs-recurrences/focs-recurrences/teaching/2627/FoundsCS/forest/focs-recurrences/Some Simple Recurrence RelationsConsider a function that has a cost we want to bound using notation. A typical base case is . Some recurrences are: EquationComplexity To analyse a function, inspect its OCaml declaration. Recurrence equations for the cost function can usually be read off. Since we ignore constant factors, we can give the base case a cost of one unit. Constant work done in the recursive step can also be given unit cost; since we only need an upper bound, this unit represents the larger of the two actual costs. We could use other constants if it simplifies the algebra.For example, recall our function nsum: int = ]]>Given , it performs a constant amount of work (an addition and subtraction) and calls itself recursively with argument . We get the recurrence equations and . The closed form is clearly , as we can easily verify by substitution. The cost is linear.This function, given , calls nsum, performing work. Again ignoring constant factors, we can say that this call takes exactly units. int = ]]>We get the recurrence equations and . It is easy to see that . The cost is quadratic.The function power divides its input into two, with the recurrence equation . Clearly , so .Anil Madhavapeddyhttps://www.cl.cam.ac.uk/teaching/2627/FoundsCS/forest/focs-sorting-summary/focs-sorting-summary/teaching/2627/FoundsCS/forest/focs-sorting-summary/Summary of Sorting AlgorithmsOptimal is comparisons Insertion sort: simple to code; too slow (quadratic) [174 secs] Quicksort: fast on average; quadratic in worst case [0.53 secs] Mergesort: optimal in theory; often slower than quicksort [1.4 secs] Match the algorithm to the applicationQuicksort’s worst case cannot be ignored. For large , a complexity of is catastrophic. Mergesort has an worst case running time, which is optimal, but it is typically slower than quicksort for random data.Non-comparison sorting deserves mentioning. We can sort a large number of small integers using their radix representation in time. This result does not contradict the comparison-counting argument because comparisons are not used at all. Linear time is achievable only if the greatest integer is fixed in advance; as goes to infinity, increasingly many of the items are the same. It is a simple special case.Many other sorting algorithms exist. A few are outlined in the exercises.Anil Madhavapeddyhttps://www.cl.cam.ac.uk/teaching/2627/FoundsCS/forest/focs-merge-sort/focs-merge-sort/teaching/2627/FoundsCS/forest/focs-merge-sort/Top-down Merge sort [] | [x] -> [x] | xs -> let k = List.length xs / 2 in let l = tmergesort (take k xs) in let r = tmergesort (drop k xs) in merge (l, r) val tmergesort : 'a list -> 'a list = ]]> comparisons in worst caseMergesort’s divide stage divides the input not by choosing a pivot (as in quicksort) but by simply counting out half of the elements. The conquer stage again involves recursive calls, and the combine stage involves merging. Function tmergesort takes roughly 1.4 seconds to sort the list rs.In the worst case, mergesort does comparisons, with the same recurrence equation as in quicksort’s average case. Because take and drop divide the input in two equal parts (they differ at most by one element), we always have .Quicksort is nearly 3 times as fast in the example. But it risks a quadratic worst case! Merge sort is safe but slow. So which algorithm is best?We have seen a top-down mergesort. Bottom-up algorithms also exist. They start with a list of one-element lists and repeatedly merge adjacent lists until only one is left. A refinement, which exploits any initial order among the input, is to start with a list of increasing or decreasing runs of input items.Anil MadhavapeddyAnil Madhavapeddyhttps://www.cl.cam.ac.uk/teaching/2627/FoundsCS/forest/focs-ex-2-4/focs-ex-2-4/teaching/2627/FoundsCS/forest/focs-ex-2-4/Exercises › A tighter recurrence boundExerciseFind an upper bound for the recurrence given by and . You should be able to find a tighter bound than .Anil MadhavapeddyAnil Madhavapeddyhttps://www.cl.cam.ac.uk/teaching/2627/FoundsCS/forest/focs-ex-5-3/focs-ex-5-3/teaching/2627/FoundsCS/forest/focs-ex-5-3/Exercises › Complexity of bubble sortExerciseAnother sorting algorithm (bubble sort) consists of looking at adjacent pairs of elements, exchanging them if they are out of order and repeating this process until no more exchanges are possible. State, with justification, the time complexity of this approach.Anil MadhavapeddyAnil Madhavapeddyhttps://www.cl.cam.ac.uk/teaching/2627/FoundsCS/forest/focs-ex-5-1/focs-ex-5-1/teaching/2627/FoundsCS/forest/focs-ex-5-1/Exercises › Complexity of selection sortExerciseAnother sorting algorithm (selection sort) consists of looking at the elements to be sorted, identifying and removing a minimal element, which is placed at the head of the result. The tail is obtained by recursively sorting the remaining elements. State, with justification, the time complexity of this approach.Anil MadhavapeddyAnil Madhavapeddyhttps://www.cl.cam.ac.uk/teaching/2627/FoundsCS/forest/focs-ex-4-5/focs-ex-4-5/teaching/2627/FoundsCS/forest/focs-ex-4-5/Exercises › Counting ways of making changeExerciseShow that the number of ways of making change for (ignoring order) is if there are two legal coin values. What if there are three, four, … coin values?Anil MadhavapeddyAnil Madhavapeddyhttps://www.cl.cam.ac.uk/teaching/2627/FoundsCS/forest/focs-ex-2-2/focs-ex-2-2/teaching/2627/FoundsCS/forest/focs-ex-2-2/Exercises › Extending the complexity tableExerciseAdd a column to the table of complexities from The Design and Analysis of Computer Algorithms with the heading 60 hours: complexity1 second1 minute1 hour60 hours 100060 0003 600 000 1404 895204 095 312441 897 1039153 91521 Anil MadhavapeddyAnil Madhavapeddyhttps://www.cl.cam.ac.uk/teaching/2627/FoundsCS/forest/focs-ex-3-2/focs-ex-3-2/teaching/2627/FoundsCS/forest/focs-ex-3-2/Exercises › Last element of a listExerciseCode a function to return the last element of a non-empty list. How efficiently can this be done?Anil MadhavapeddyAnil Madhavapeddyhttps://www.cl.cam.ac.uk/teaching/2627/FoundsCS/forest/focs-ex-7-7/focs-ex-7-7/teaching/2627/FoundsCS/forest/focs-ex-7-7/Exercises › Linear traversalsExerciseShow that the functions preord, inord and postord all take linear time in the size of the tree.Anil MadhavapeddyAnil Madhavapeddyhttps://www.cl.cam.ac.uk/teaching/2627/FoundsCS/forest/focs-ex-7-6/focs-ex-7-6/teaching/2627/FoundsCS/forest/focs-ex-7-6/Exercises › Quadratic traversalsExerciseShow that the functions preorder, inorder and postorder all require time in the worst case, where is the size of the tree.Anil MadhavapeddyAnil Madhavapeddyhttps://www.cl.cam.ac.uk/teaching/2627/FoundsCS/forest/focs-ex-10-4/focs-ex-10-4/teaching/2627/FoundsCS/forest/focs-ex-10-4/Exercises › Search with a branching factor near oneExerciseIterative deepening is inappropriate if , where is the branching factor. What search strategy is appropriate in this case?Anil MadhavapeddyAnil Madhavapeddyhttps://www.cl.cam.ac.uk/teaching/2627/FoundsCS/forest/focs-ex-2-3/focs-ex-2-3/teaching/2627/FoundsCS/forest/focs-ex-2-3/Exercises › Simplifying sums in O notationExerciseLet , …, be functions such that for , …, and all sufficiently large .Show that if then .Anil MadhavapeddyAnil Madhavapeddyhttps://www.cl.cam.ac.uk/teaching/2627/FoundsCS/forest/focs-ex-10-1/focs-ex-10-1/teaching/2627/FoundsCS/forest/focs-ex-10-1/Exercises › Tree-based queuesExerciseSuppose that we have an implementation of queues, based on binary trees, such that each operation takes logarithmic time in the worst case. Outline the advantages and drawbacks of such an implementation compared with one presented above.Anil Madhavapeddyhttps://www.cl.cam.ac.uk/teaching/2627/FoundsCS/forest/focs-topic-lists/focs-topic-lists/teaching/2627/FoundsCS/forest/focs-topic-lists/ListsAnil Madhavapeddyhttps://www.cl.cam.ac.uk/teaching/2627/FoundsCS/forest/focs-change-all/focs-change-all/teaching/2627/FoundsCS/forest/focs-change-all/All Ways of Making ChangeNow we generalise the problem to return the list of all possible ways of making change, and write a new change function. [ [] ] | [] , _ -> [] | c::till , amt -> if amt < c then change till amt else let rec allc = function | [] -> [] | cs :: css -> (c::cs) :: allc css in allc (change (c::till) (amt - c)) @ change till amt val change : int list -> int -> int list list = ]]>Look at the type: the result is now a list of lists. The code will also never raise exceptions. It expresses failure by returning an empty list of solutions: it returns [] if the till is empty and the amount is nonzero.If the amount is zero, then there is only one way of making change; the result should be . This is success in the base case.In nontrivial cases, there are two sources of solutions: to use a coin (if possible) and decrease the amount accordingly, or to remove the current coin value from consideration.The function allc is declared locally in order to make use of c, the current coin. It adds an extra c to all the solutions returned by the recursive call to make change for amt - c.Observe the naming convention: cs is a list of coins, while css is a list of such lists. The trailing ‘s’ is suggestive of a plural.This complicated program, and the even trickier faster version, are included as challenges. Are you enthusiastic enough to work them out? We shall revisit the “making change” task later to illustrate exception-handling.Anil Madhavapeddyhttps://www.cl.cam.ac.uk/teaching/2627/FoundsCS/forest/focs-change-faster/focs-change-faster/teaching/2627/FoundsCS/forest/focs-change-faster/All Ways of Making Change — Faster! chg::chgs | [] , _ -> chgs | c::till , amt -> if amt < 0 then chgs else change (c::till) (amt - c) (c::chg) (change till amt chg chgs) val change : int list -> int -> int list -> int list list -> int list list = ]]>We’ve added another accumulating parameter! Repeatedly improving simple code is called stepwise refinement.Two extra arguments eliminate many :: and append operations from the all-solutions change function. The first, chg, accumulates the coins chosen so far; one evaluation of c::chg replaces many evaluations of allc. The second, chgs, accumulates the list of solutions so far; it avoids the need for append. This version runs several times faster than the previous one.Making change is still extremely slow for an obvious reason: the number of solutions grows rapidly in the amount being changed. Using 50, 20, 10, 5, 2 and 1, there are 4366 ways of expressing 99.Our three change functions illustrate a basic technique: program development by stepwise refinement. Begin by writing a very simple program and add requirements individually. Add efficiency refinements last of all. Even if the simpler program cannot be included in the next version and has to be discarded, one has learned about the task by writing it.Anil Madhavapeddyhttps://www.cl.cam.ac.uk/teaching/2627/FoundsCS/forest/focs-making-change/focs-making-change/teaching/2627/FoundsCS/forest/focs-making-change/An Application: Making ChangeConsider a till that has unlimited supplies of coins. The largest coins should be tried first, to avoid giving change all in pennies. The list of legal coin values, called till, is given in descending order, such as 50, 20, 10, 5, 2 and 1. (Recall that the head of a list is the element most easily reached.) The code for change is based on simple observations:Change for zero consists of no coins at all. (Note the pattern of 0 in the first clause.) For a nonzero amount, try the largest available coin. If it is small enough, use it and decrease the amount accordingly. Exclude from consideration any coins that are too large. [] | [], _ -> raise (Failure "no more coins!") | c::till, amt -> if amt < c then change till amt else c :: change (c::till) (amt - c) val change : int list -> int -> int list = ]]>Although nobody considers making change for zero, this is the simplest way to make the algorithm terminate. Most iterative procedures become simplest if, in their base case, they do nothing. A base case of one instead of zero is often a sign of a novice programmer.The recursion terminates when amt = 0. Tries the largest coin first to use large coins. The algorithm is greedy and can fail!The function can terminate either with success or failure. It fails by raising exception Failure namely if till becomes empty while amt is still nonzero. (Exceptions will be discussed later.)Unfortunately, failure can occur even when change can be made. The greedy "largest coin first" approach is to blame. Suppose we have coins of values 5 and 2, and must make change for 6; the only way is , ignoring the 5. Greedy algorithms are often effective, but not here.Anil Madhavapeddyhttps://www.cl.cam.ac.uk/teaching/2627/FoundsCS/forest/focs-quicksort-append-free/focs-quicksort-append-free/teaching/2627/FoundsCS/forest/focs-quicksort-append-free/Append-Free Quicksort sorted | ([x], sorted) -> x::sorted | a::bs, sorted -> let rec part = function | l, r, [] -> quik (l, a :: quik (r, sorted)) | l, r, x::xs -> if x <= a then part (x::l, r, xs) else part (l, x::r, xs) in part ([], [], bs) val quik : 'a list * 'a list -> 'a list = ]]>The list sorted accumulates the result in the combine stage of the quicksort algorithm. We have again used the standard technique for eliminating append. Calling quik(xs, sorted) reverses the elements of xs and prepends them to the list sorted.Looking closely at part, observe that quik(r, sorted) is performed first. Then a is consed to this sorted list. Finally, quik is called again to sort the elements of l.The speedup is significant. An imperative quicksort coded in Pascal (taken from Sedgewick) is just slightly faster than function quik. The near-agreement is surprising because the computational overheads of lists exceed those of arrays. In realistic applications, comparisons are the dominant cost and the overheads matter even less.Anil Madhavapeddyhttps://www.cl.cam.ac.uk/teaching/2627/FoundsCS/forest/focs-append/focs-append/teaching/2627/FoundsCS/forest/focs-append/Append: List Concatenation ys | x::xs, ys -> x :: append xs ys val append : 'a list -> 'a list -> 'a list = # append [1; 2; 3] [4] - : int list = [1; 2; 3; 4] # let (@) = append val ( @ ) : 'a list -> 'a list -> 'a list = # [1; 2; 3] @ [4] - : int list = [1; 2; 3; 4]]]>Patterns can be as complicated as we like. Here, the two patterns are [], ys and x::xs, ys.Here is how append might be declared, also noting that we have defined @ as an infix operator that is a more convenient way to call append on two lists. However, this function is also not iterative. It scans its first argument, sets up a string of cons operations (::) and finally does them.It uses space and time, where is the length of its first argument. Its costs are independent of its second argument.An accumulating argument could make it iterative, but with considerable complication. The iterative version would still require space and time because concatenation requires copying all the elements of the first list. Therefore, we cannot hope for asymptotic gains; at best we can decrease the constant factor involved in , but complicating the code is likely to increase that factor. Never add an accumulator merely out of habit.Note append’s polymorphic type. It tells us that two lists can be joined if their element types agree.Anil Madhavapeddyhttps://www.cl.cam.ac.uk/teaching/2627/FoundsCS/forest/focs-predicate-applications/focs-predicate-applications/teaching/2627/FoundsCS/forest/focs-predicate-applications/Applications of the Predicate Functionals x=y) xs val member : 'a -> 'a list -> bool = # let inter xs ys = filter (fun x -> member x ys) xs val inter : 'a list -> 'a list -> 'a list = ]]>Testing whether two lists have no common elements all (fun y -> x<>y) ys) xs val disjoint : 'a list -> 'a list -> bool = ]]>The Lists lecture presented the function member, which tests whether a specified value can be found as a list element, and inter, which returns the “intersection” of two lists: the list of elements they have in common.But remember: the purpose of list functionals is not to replace the declarations of popular functions, which probably are available already. It is to eliminate the need for separate declarations of ad-hoc functions. When they are nested, like the calls to all in disjoint above, the inner functions are almost certainly one-offs, not worth declaring separately.Our primitives themselves can be seen as a programming language. Part of the task of programming is to extend our programming language with notation for solving the problem at hand. The levels of notation that we define should correspond to natural levels of abstraction in the problem domain.Historical Note: Alonzo Church’s -calculus gave a simple syntax, -notation, for expressing functions. It is the direct precursor of OCaml’s fun-notation. It was soon shown that his system was equivalent in computational power to Turing machines, and Church’s thesis states that this defines precisely the set of functions that can be computed effectively.The -calculus had a tremendous influence on the design of functional programming languages. McCarthy’s Lisp was something of a false start; it interpreted variable binding incorrectly, an error that stood for some 20 years. But in 1966, Peter Landin (of Queen Mary College, University of London) sketched out the main features of functional languages.Anil Madhavapeddyhttps://www.cl.cam.ac.uk/teaching/2627/FoundsCS/forest/focs-tree-traversal/focs-tree-traversal/teaching/2627/FoundsCS/forest/focs-tree-traversal/Aside: Traversing Trees (3 Methods) [] | Br (v, t1, t2) -> [v] @ preorder t1 @ preorder t2 val preorder : 'a tree -> 'a list = # let rec inorder = function | Lf -> [] | Br (v, t1, t2) -> inorder t1 @ [v] @ inorder t2 val inorder : 'a tree -> 'a list = # let rec postorder = function | Lf -> [] | Br (v, t1, t2) -> postorder t1 @ postorder t2 @ [v] val postorder : 'a tree -> 'a list = ]]>Tree traversal means examining each node of a tree in some order. D. E. Knuth has identified three forms of tree traversal: preorder, inorder and postorder. We can code these “visiting orders” as functions that convert trees into lists of labels. Algorithms based on these notions typically perform some action at each node; the functions above simply copy the nodes into lists. Consider the tree:preorder visits the label first (“Polish notation”), yielding ABDECFG inorder visits the label midway, yielding DBEAFCG postorder visits the label last (“Reverse Polish”), yielding DEBFGCA. You might be familiar with this concept if you own an ancient RPN calculator!What is the use of inorder? Consider applying it to a binary search tree: the result is a sorted list of pairs. We could use this, for example, to merge two binary search trees. It is not difficult to transform a sorted list of pairs into a binary search tree.Anil Madhavapeddyhttps://www.cl.cam.ac.uk/teaching/2627/FoundsCS/forest/focs-bfs-append/focs-bfs-append/teaching/2627/FoundsCS/forest/focs-bfs-append/Breadth-First Tree Traversal — Using Append [] | Lf :: ts -> nbreadth ts | Br (v, t, u) :: ts -> v :: nbreadth (ts @ [t; u]) val nbreadth : 'a tree list -> 'a list = ]]>Keeps an enormous queue of nodes of search, and is a wasteful use of append.Breadth-first search can be inefficient, this naive implementation especially so. When the search is at depth of the tree, the list contains all the remaining trees at depth , followed by the subtrees (all at depth ) of the trees that have already been visited. At depth 10, the list could already contain 1024 elements. It requires a lot of space, and aggravates this with a gross misuse of append. Evaluating ts@[t, u] copies the long list ts just to insert two elements.Anil Madhavapeddyhttps://www.cl.cam.ac.uk/teaching/2627/FoundsCS/forest/focs-zip/focs-zip/teaching/2627/FoundsCS/forest/focs-zip/Building a List of Pairs (x, y) :: zip xs ys | _ -> [] val zip : 'a list -> 'b list -> ('a * 'b) list = ]]>The wildcard pattern _ matches anything. We could have written a variable such as p instead, but the wildcard reminds us that the relevant clause ignores this argument.The patterns are also tested in order of their definitions: first (x::xs, y::ys), then _.A list of pairs of the form associates each with . Conceptually, a telephone directory could be regarded as such a list, where ranges over names and over the corresponding telephone number. Linear search in such a list can find the associated with a given , or vice versa—very slowly.In other cases, the pairs might have been generated by applying a function to the elements of another list . ([], []) | (x, y)::pairs -> let xs, ys = unzip pairs in (x::xs, y::ys) val unzip : ('a * 'b) list -> 'a list * 'b list = ]]>Given a list of pairs, unzip has to build two lists of results, which is awkward using recursion. The version shown above uses the local binding let p = in , where the value of is bound to the variable p within . The let-construct counts as an expression and can be used (perhaps wrapped within parentheses) wherever an expression is expected.Note especially the phrase let xs, ys = unzip pairs which binds xs and ys to the results of the recursive call. In general, the phrase let P = E matches the pattern against the value of expression . It binds all the variables in to the corresponding values.The functions zip and unzip build and take apart lists of pairs: zip pairs up corresponding list elements and unzip inverts this operation. Their types reflect what they do: 'b list -> ('a * 'b) list = # unzip - : ('a * 'b) list -> 'a list * 'b list = ]]>If the lists are of unequal length, zip discards surplus items at the end of the longer list. Its first pattern only matches a pair of non-empty lists. The second pattern is just a wildcard and could match anything. OCaml tries the clauses in the order given, so the first pattern is tried first. The second only gets arguments where at least one of the lists is empty.Anil Madhavapeddyhttps://www.cl.cam.ac.uk/teaching/2627/FoundsCS/forest/focs-unzip/focs-unzip/teaching/2627/FoundsCS/forest/focs-unzip/Building a Pair of ResultsHere is a version of unzip that replaces the local declaration by a function conspair for taking apart the pair of lists in the recursive call. It defines the same computation as the previous version of unzip and is possibly clearer, but not every local binding can be eliminated as easily. 'a list * 'b list = # let rec unzip = function | [] -> ([], []) | xy :: pairs -> conspair (xy, unzip pairs) val unzip : ('a * 'b) list -> 'a list * 'b list = ]]>Making the function iterative yields revUnzip below, which is very simple. Iteration can construct many results at once in different argument positions. Both output lists are built in reverse order, which can be corrected by reversing the input to revUnzip. The total costs will probably exceed those of unzip despite the advantages of iteration. (xs, ys) | ((x, y)::pairs, xs, ys) -> revUnzip (pairs, x::xs, y::ys) val revUnzip : ('a * 'b) list * 'a list * 'b list -> 'a list * 'b list = ]]>Anil Madhavapeddyhttps://www.cl.cam.ac.uk/teaching/2627/FoundsCS/forest/focs-nlength/focs-nlength/teaching/2627/FoundsCS/forest/focs-nlength/Computing the Length of a List 0 | x :: xs -> 1 + nlength xs val nlength : 'a list -> int = # nlength [] - : int = 0 # nlength [5; 6; 7] - : int = 3]]>Most list processing involves recursion. This is a simple example; patterns can be more complex. Observe the use of a vertical bar | to separate the function’s clauses. We have one function declaration that handles two cases. To understand its role, consider the following faulty code: int = # let rec nlength (x::xs) = 1 + nlength xs Line 1, characters 17-24: Warning 8 [partial-match]: this pattern-matching is not exhaustive. Here is an example of a case that is not matched: [] val nlength : 'a list -> int = ]]>These are two declarations, not one. First we declare nlength to be a function that handles only empty lists. Then we redeclare it to be a function that handles only non-empty lists; it can never deliver a result. We see that a second let declaration replaces any previous one rather than extending it to cover new cases.Now, let us return to our original declaration of nlength. The length function is polymorphic and applies to all lists regardless of element type! Most programming languages lack such flexibility.Unfortunately, this length computation is naive and wasteful. Like nsum earlier, it is not tail-recursive. It uses space, where is the length of its input. As usual, the solution is to add an accumulating argument.Anil Madhavapeddyhttps://www.cl.cam.ac.uk/teaching/2627/FoundsCS/forest/focs-seq-consume/focs-seq-consume/teaching/2627/FoundsCS/forest/focs-seq-consume/Consuming a Sequence [] | n, Nil -> [] | n, Cons (x, xf) -> x :: get (n-1) (xf ()) val get : int -> 'a seq -> 'a list = ]]>The above code gets the first elements as a list. xf () forces evaluation.The function get converts a sequence to a list. It takes the first elements; it takes all of them if , which can terminate only if the sequence is finite.In the last line of get, the expression xf() calls the tail function, demanding evaluation of the next element. This operation is called forcing the sequence.Anil Madhavapeddyhttps://www.cl.cam.ac.uk/teaching/2627/FoundsCS/forest/focs-dictionary-adt/focs-dictionary-adt/teaching/2627/FoundsCS/forest/focs-dictionary-adt/Dictionarieslookup: find an item in the dictionary update (insert): replace (store) an item in the dictionary delete: remove an item from the dictionary empty: the null dictionary Missing: exception for errors in lookup and deleteIdeally, an abstract type should provide these operations and hide the internal data structures.A dictionary attaches values to identifiers, called “keys”. Before choosing the internal representation for a data structure, you need to specify the full set of operations. In fact, here we only consider update (associating a value with an identifier) and lookup (retrieving such a value). Deletion is more difficult and would limit our choices. Some applications may need additional operations, such as merge (combining two dictionaries). We shall see that update can be done efficiently in a functional style, without excessive copying.An abstract type provides specified operations while hiding low-level details, such as the data structure used to represent dictionaries. Abstract types can be declared in any modern programming language. Java’s objects serve this role, as do OCaml’s modules. This course does not cover modules, and we simply declare the dictionary operations individually.An association list (a list of pairs) is the simplest dictionary representation. Lookup is by linear search, and therefore slow: . Association lists are only usable if there are few keys in use. However, they are general in that the keys do not need a concept of ordering, only equality. raise Missing | (x, y) :: pairs -> if a = x then y else lookup a pairs val lookup : 'a -> ('a * 'b) list -> 'b = # let update (l, b, y) = (b, y) :: l val update : ('a * 'b) list * 'a * 'b -> ('a * 'b) list = ]]>To enter a new (key, value) pair, simply “cons” it to the list with update. This takes constant time, which is the best we could hope for. But the space requirement is huge: linear in the number of updates, not in the number of distinct keys. Obsolete entries are never deleted: that would require first finding them, increasing the update time from to .Anil Madhavapeddyhttps://www.cl.cam.ac.uk/teaching/2627/FoundsCS/forest/focs-functional-queues/focs-functional-queues/teaching/2627/FoundsCS/forest/focs-functional-queues/Efficient Functional Queues: Code Q (List.rev tls, []) | q -> q val norm : 'a queue -> 'a queue = # let qnull = function | Q ([], []) -> true | _ -> false val qnull : 'a queue -> bool = # let enq (Q (hds, tls)) x = norm (Q (hds, x::tls)) val enq : 'a queue -> 'a -> 'a queue = # exception Empty exception Empty # let deq = function | Q (x::hds, tls) -> norm (Q (hds, tls)) | _ -> raise Empty val deq : 'a queue -> 'a queue = # let qempty = Q ([], []) val qempty : 'a queue = Q ([], []) # let qhd = function | Q (x::_, _) -> x | _ -> raise Empty val qhd : 'a queue -> 'a = ]]>The datatype of queues prevents confusion with other pairs of lists. The empty queue has both parts empty.The function norm puts a queue into normal form, ensuring that the front part is never empty unless the entire queue is empty. Functions deq and enq call norm to normalise their result.Because queues are in normal form, their head is certain to be in their front part, so qhd looks there.Let us analyse the cost of an execution comprising (in any possible order) enq operations and deq operations, starting with an empty queue. Each enq operation will perform one cons, adding an element to the rear part. Since the final queue must be empty, each element of the rear part gets transferred to the front part. The corresponding reversals perform one cons per element. Thus, the total cost of the series of queue operations is cons operations, an average of 2 per operation. The amortized time is .There is a catch. The conses need not be distributed evenly; reversing a long list could take up to of them. Unpredictable delays make the approach unsuitable for real-time programming where deadlines must be met.Anil Madhavapeddyhttps://www.cl.cam.ac.uk/teaching/2627/FoundsCS/forest/focs-functional-queues-idea/focs-functional-queues-idea/teaching/2627/FoundsCS/forest/focs-functional-queues-idea/Efficient Functional Queues: IdeaRepresent the queue by any pair of lists Add new items to the rear list Remove items from front list and if empty move rear to front Amortized time per operation is Queues require efficient access at both ends: at the front, for removal, and at the back, for insertion. Ideally, access should take constant time, . It may appear that lists cannot provide such access. If enq(q, x) performs q@[x], then this operation will be . We could represent queues by reversed lists, implementing enq(q, x) by x::q, but then the deq and qhd operations would be . Linear time is intolerable: a series of queue operations could then require time.The solution is to represent a queue by a pair of lists, whererepresents the queue .The front part of the queue is stored in order, and the rear part is stored in reverse order. The enq operation adds elements to the rear part using cons, since this list is reversed; thus, enq takes constant time. The deq and qhd operations look at the front part, which normally takes constant time, since this list is stored in order. But sometimes deq removes the last element from the front part; when this happens, it reverses the rear part, which becomes the new front part.Amortized time refers to the cost per operation averaged over the lifetime of any complete execution. Even for the worst possible execution, the average cost per operation turns out to be constant; see the analysis below.Anil Madhavapeddyhttps://www.cl.cam.ac.uk/teaching/2627/FoundsCS/forest/focs-length/focs-length/teaching/2627/FoundsCS/forest/focs-length/Efficiently Computing the Length of a List n | x::xs -> addlen (n + 1) xs val addlen : int -> 'a list -> int = # addlen 0 [5; 6; 7] - : int = 3]]>Recall that the use of function introduces an extra (unnamed) argument that is pattern matched in the subsequent clauses; in this case, to break open the list.Function addlen is again polymorphic. Its type mentions the integer accumulator.Now we may declare an efficient length function. It is simply a wrapper for addlen, supplying zero as the initial value of . int = # length [5; 6; 7; 8] - : int = 4]]>The recursive calls do not nest: this version is iterative. It takes space. Obviously its time requirement is because it takes at least steps to find the length of an -element list.Anil Madhavapeddyhttps://www.cl.cam.ac.uk/teaching/2627/FoundsCS/forest/focs-equality/focs-equality/teaching/2627/FoundsCS/forest/focs-equality/Equality Tests false | y::l -> if x = y then true else member x l val member : 'a -> 'a list -> bool = ]]>All the list functions we have encountered up to now have been “polymorphic”, working for lists of any type. Function member uses linear search to report whether or not x occurs in l.To do this generically, it uses a special feature of OCaml known as “polymorphic equality”, which manifests itself via the =, >=, <=, > and < operators. These operators inspect the structure of the values using a consistent order. Types you can legitimately compare this way include integers, strings, booleans, and tuples or lists of primitive types.More complex types can be compared this way within careful limits: recursive structures or function values will not work (we will cover function values in the Currying lecture later). For now, it is sufficient to use these magic polymorphic equality operators. As you get more familiar with OCaml and the use of higher order functions (also covered in a later lecture), you will encounter the use of explicit compare functions that are used to provide more complex equality tests.The presence of polymorphic equality is a contentious feature in OCaml. While it provides a great ease of use in smaller codebases, it starts to become more dangerous when building larger OCaml-based systems. Most large-scale users of OCaml tend towards not using it in important code, but it is just fine for our purposes while learning the beginning steps of computer science.Anil Madhavapeddyhttps://www.cl.cam.ac.uk/teaching/2627/FoundsCS/forest/focs-transpose/focs-transpose/teaching/2627/FoundsCS/forest/focs-transpose/Example: Matrix Transpose [] | rows -> (map List.hd rows) :: (transp (map List.tl rows)) val transp : 'a list list -> 'a list list = ]]>A matrix can be viewed as a list of rows, each row a list of matrix elements. This representation is not especially efficient compared with the conventional one (using arrays). Lists of lists turn up often, though, and we can see how to deal with them by taking familiar matrix operations as examples. ML for the Working Programmer goes as far as Gaussian elimination, which presents surprisingly few difficulties.The transpose of the matrix is , which in OCaml corresponds to the following transformation on lists of lists:The workings of function transp are simple. If rows is the matrix to be transposed, then map hd extracts its first column and map tl extracts its second column:A recursive call transposes the latter matrix, which is then given the column [a; d] as its first row. The two functions expressed using map would otherwise have to be declared separately.Anil Madhavapeddyhttps://www.cl.cam.ac.uk/teaching/2627/FoundsCS/forest/focs-head-tail/focs-head-tail/teaching/2627/FoundsCS/forest/focs-head-tail/Getting at the Head and Tail true | x :: l -> false val null : 'a list -> bool = # null [] - : bool = true # null [1; 2; 3] - : bool = false # let hd (x::l) = x Line 1, characters 8-14: Warning 8 [partial-match]: this pattern-matching is not exhaustive. Here is an example of a case that is not matched: [] val hd : 'a list -> 'a = # hd [1; 2; 3] - : int = 1 # let tl (x::l) = l Line 1, characters 8-14: Warning 8 [partial-match]: this pattern-matching is not exhaustive. Here is an example of a case that is not matched: [] val tl : 'a list -> 'a list = # tl [7; 6; 5] - : int list = [6; 5]]]>The empty list has neither head nor tail. Applying hd or tl to [] is an error—strictly speaking, an “exception”. The function null can be used to check for the empty list beforehand. Taking a list apart using combinations of hd and tl is hard to get right. Fortunately, it is seldom necessary because of pattern-matching.The declaration of null introduces a new concept known as "pattern matching", which we will explore more in subsequent lectures. For now, it is sufficient to observe that let null = function allows for matching on the two possible values that might be passed in as argument to null here: one for the empty list (for which it returns true) and one for non-empty lists (for which it returns false).The declaration of hd above has only one clause, for non-empty lists. They have the form x::l and the function returns x, which is the head. If you compile this program, OCaml also prints a warning to tell us that calling the function could raise an exception because not all possible inputs are handled, including a counter-example (in this case, the empty list []). The declaration of tl is similar to hd.These three primitive functions are polymorphic and allow flexibility in the types of their arguments and results. Note their types! bool = # hd - : 'a list -> 'a = # tl - : 'a list -> 'a list = ]]>Symbols 'a and 'b are called type variables and stand for any types. Code written using these functions is checked for type correctness at compile time. And this guarantees strong properties at run time, for example that the elements of any list all have the same type. They are usually read as their corresponding greek characters; 'a is "alpha", 'b is "beta", and so on.Anil Madhavapeddyhttps://www.cl.cam.ac.uk/teaching/2627/FoundsCS/forest/focs-insertion-sort/focs-insertion-sort/teaching/2627/FoundsCS/forest/focs-insertion-sort/Insertion SortAn insert operation does comparisons on average. [x] | y::ys -> if x <= y then x :: y :: ys else y :: ins x ys val ins : 'a -> 'a list -> 'a list = ]]>Insertion sort takes comparisons on average: [] | x::xs -> ins x (insort xs)]]>Items from the input are copied one at a time to the output. Each new item is inserted into the right place so that the output is always in order.We could easily write iterative versions of these functions, but to no purpose. Insertion sort is slow because it does comparisons (and a lot of list copying), not because it is recursive. Its quadratic runtime makes it nearly useless: it takes 174 seconds for our example while the next-worst figure is 1.4 seconds.Insertion sort is worth considering because it is easy to code and illustrates the concepts. Two efficient sorting algorithms, mergesort and heapsort, can be regarded as refinements of insertion sort.Anil Madhavapeddyhttps://www.cl.cam.ac.uk/teaching/2627/FoundsCS/forest/focs-while/focs-while/teaching/2627/FoundsCS/forest/focs-while/Iteration: the while command None | _::xs -> Some xs val tlopt : 'a list -> 'a list option = # let length xs = let lp = ref xs in (* list of uncounted elements *) let np = ref 0 in (* accumulated count *) let fin = ref false in while not !fin do match tlopt !lp with | None -> fin := true | Some xs -> lp := xs; np := 1 + !np done; !np (* the final count is returned *) val length : 'a list -> int = ]]>Once we can change the state, we need to do so repeatedly. Recursion can serve this purpose, but having to declare a procedure for every loop is clumsy, and compilers for conventional languages seldom exploit tail-recursion.Early programming languages provided little support for repetition. The programmer had to set up loops using goto commands, exiting the loop using another goto controlled by an if. Modern languages provide a confusing jumble of looping constructs, the most fundamental of which is while B do C. The boolean expression is evaluated, and if true, command is executed and the command repeats. If evaluates to false then the while command terminates, perhaps without executing even once.OCaml’s main looping construct is while, which returns the value (). The function length declares references to hold the list under examination (lp) and number of elements counted so far (np) as well as whether the end of the list has been reached (the boolean reference fin). While the list is non-empty, we skip over one more element (by setting it to its tail) and count that element.The body of the while loop first checks to see if the end of the list has been reached, in which case it sets the fin variable to true. If there is a tail value, then two assignments are executed in sequence. The lp reference is set to the tail of the list, and the np reference integer is incremented by one. When the while loop terminates due to the fin variable being set to true, the expression !np returns the computed length as the function’s result.Anil Madhavapeddyhttps://www.cl.cam.ac.uk/teaching/2627/FoundsCS/forest/focs-seq-append/focs-seq-append/teaching/2627/FoundsCS/forest/focs-seq-append/Joining Two Sequences yq | Cons (x, xf) -> Cons(x, fun () -> appendq (xf ()) yq) val appendq : 'a seq -> 'a seq -> 'a seq = ]]>A more fair alternative: yq | Cons (x, xf) -> Cons (x, fun () -> interleave yq (xf ())) val interleave : 'a seq -> 'a seq -> 'a seq = ]]>Most list functions and functionals have analogues on sequences, but strange things can happen. Can an infinite list be reversed?Function appendq is precisely the same idea as append from the Lists lecture; it concatenates two sequences. If the first argument is infinite, then appendq never gets to its second argument, which is lost. Concatenation of infinite sequences is not terribly interesting.The function interleave avoids this problem by exchanging the two arguments in each recursive call. It combines the two lazy lists, losing no elements. Interleaving is the right way to combine two potentially infinite information sources into one.In both function declarations, observe that each xf () is enclosed within a . Each force is enclosed within a delay. This practice makes the functions lazy. A force not enclosed in a delay, as in get above, runs the risk of evaluating the sequence in full.Anil Madhavapeddyhttps://www.cl.cam.ac.uk/teaching/2627/FoundsCS/forest/focs-lazy-lists/focs-lazy-lists/teaching/2627/FoundsCS/forest/focs-lazy-lists/Lazy Lists (or Streams)Lists of possibly infinite length Elements computed upon demand Avoids waste if there are many solutions Infinite values are a useful abstractionIn OCaml, we can implement laziness by delaying evaluation of the tail of the list.Lazy lists have practical uses. Some algorithms, like making change, can yield many solutions when only a few are required. Sometimes the original problem concerns infinite series: with lazy lists, we can pretend they really exist!We are now dealing with infinite (or at least unbounded) computations. A potentially infinite source of data is processed one element at a time, upon demand. Such programs are harder to understand than terminating ones and have more ways of going wrong.Some purely functional languages, such as Haskell, use lazy evaluation everywhere. Even the if-then-else construct can be a function, and all lists are lazy. In OCaml, we can declare a type of lists such that evaluation of the tail does not occur until demanded. Delayed evaluation is weaker than lazy evaluation, but it is good enough for our purposes and often the best compromise for performance and memory usage.The traditional word “stream” is reserved in OCaml parlance for input/output channels. Let us call lazy lists sequences instead.Anil Madhavapeddyhttps://www.cl.cam.ac.uk/teaching/2627/FoundsCS/forest/focs-def-lazy-list/focs-def-lazy-list/teaching/2627/FoundsCS/forest/focs-def-lazy-list/Lazy listDefinitionA lazy list is a list of possibly infinite length whose elements are computed upon demand; in OCaml we implement laziness by delaying evaluation of the tail of the list. The traditional word “stream” is reserved in OCaml parlance for input/output channels, so we call lazy lists sequences instead. Introduced in Lazy Lists.Part of the course glossary.Anil Madhavapeddyhttps://www.cl.cam.ac.uk/teaching/2627/FoundsCS/forest/focs-linear-search/focs-linear-search/teaching/2627/FoundsCS/forest/focs-linear-search/Linear Searchfind in list by comparing with each element obviously time simple & general ordered searching needs only indexed lookup needs only Linear search is the obvious way to find a desired item in a collection: simply look through all the items, one at a time. If is in the list, then it will be found in steps on average, and even the worst case is obviously .Large collections of data are usually ordered or indexed so that items can be found in time, which is exponentially better than . Even is achievable (using a hash table), though subject to the usual proviso that machine limits are not exceeded.Efficient indexing methods are of prime importance: consider Web search engines. Nevertheless, linear search is often used to search small collections because it is so simple and general, and it is the starting point for better algorithms.Anil Madhavapeddyhttps://www.cl.cam.ac.uk/teaching/2627/FoundsCS/forest/focs-predicate-functionals/focs-predicate-functionals/teaching/2627/FoundsCS/forest/focs-predicate-functionals/List Functionals for Predicates false | x::xs -> (p x) || (exists p xs) val exists : ('a -> bool) -> 'a list -> bool = # let rec filter p = function | [] -> [] | x::xs -> if p x then x :: filter p xs else filter p xs val filter : ('a -> bool) -> 'a list -> 'a list = ]]>A predicate is a boolean-valued function.The functional exists transforms a predicate into a predicate over lists. Given a list, exists p tests whether or not some list element satisfies p (making it return true). If it finds one, it stops searching immediately, thanks to the behaviour of the lazy || operator.Dually, we have a functional to test whether all list elements satisfy the predicate. If it finds a counterexample then it, too, stops searching. true | x::xs -> (p x) && all p xs val all : ('a -> bool) -> 'a list -> bool = ]]>The filter functional, like map, transforms lists. It applies a predicate to all the list elements, but instead of returning the resulting values (which could only be true or false), it returns the list of elements satisfying the predicate.Anil Madhavapeddyhttps://www.cl.cam.ac.uk/teaching/2627/FoundsCS/forest/focs-take-drop/focs-take-drop/teaching/2627/FoundsCS/forest/focs-take-drop/List Utilities: take and dropThis lecture examines more list utilities, illustrating more patterns of recursion, and concludes with a small program for making change.The functions take and drop divide a list into parts, returning or discarding the first elements.They can be implemented in OCaml as follows: [] | x::xs -> if i > 0 then x :: take (i - 1) xs else [] val take : int -> 'a list -> 'a list = # let rec drop i = function | [] -> [] | x::xs -> if i > 0 then drop (i-1) xs else x::xs val drop : int -> 'a list -> 'a list = ]]>Applications of take and drop will appear in future lectures. Typically, they divide a collection of items into equal parts for recursive processing.The take function is not iterative, but making it so would not improve its efficiency. The task requires copying up to list elements, which must take space and time.Function drop simply skips over list elements. This requires time but only constant space. It is iterative and much faster than take. Both functions use time, but skipping elements is faster than copying them: drop’s constant factor is smaller.Both functions take an integer and a list, returning a list of the same type. So their type is int -> 'a list -> 'a list.Anil Madhavapeddyhttps://www.cl.cam.ac.uk/teaching/2627/FoundsCS/forest/focs-strings-chars/focs-strings-chars/teaching/2627/FoundsCS/forest/focs-strings-chars/Lists, Strings and CharactersStrings are provided in most programming languages to allow text processing. Strings are essential for communication with users. Even a purely numerical program formats its results ultimately as strings.In a few programming languages, strings simply are lists of characters. In OCaml they are a separate type, unrelated to lists, reflecting the fact that strings are an abstract concept in themselves.Similarly, characters are not strings of size one, but are a primitive concept. Character constants in OCaml have the form 'c', where is any character. For example, the comma character is ','.Special characters are coded in strings using escape sequences involving the backslash character; among many others, a double quote is written and the newline character is written . For example, the string represents three text lines.In addition to the operators described above, the relations <, <=, >, and >= work for strings and yield alphabetic order (more precisely, lexicographic order with respect to ASCII character codes).Anil Madhavapeddyhttps://www.cl.cam.ac.uk/teaching/2627/FoundsCS/forest/focs-matmul-ocaml/focs-matmul-ocaml/teaching/2627/FoundsCS/forest/focs-matmul-ocaml/Matrix Multiplication in OCamlDot product of two vectors—a curried function 0.0 | x::xs, y::ys -> (x *. y) +. (dotprod xs ys) Lines 2-4, characters 5-51: Warning 8 [partial-match]: this pattern-matching is not exhaustive. Here is an example of a case that is not matched: ([], _::_) val dotprod : float list -> float list -> float = ]]>Matrix product map (dotprod row) cols) arows val matprod : float list list -> float list list -> float list list = ]]>The transp brows converts into a list of columns, yielding a list whose elements are the columns of . Each row of is obtained by multiplying a row of by the columns of .Because dotprod is curried, it can be applied to a row of . The resulting function is applied to all the columns of . We have another example of currying and partial application.The outer map applies dotprod to each row of . The inner map, using fun-notation, applies dotprod row to each column of . Compare with the version in ML for the Working Programmer (page 89) which does not use map and requires two additional function declarations.In the dot product function, the two vectors must have the same length. Otherwise, exception Match_failure is raised.Anil Madhavapeddyhttps://www.cl.cam.ac.uk/teaching/2627/FoundsCS/forest/focs-merge/focs-merge/teaching/2627/FoundsCS/forest/focs-merge/Merging Two ListsMerge joins two sorted lists. ys | xs, [] -> xs | x::xs, y::ys -> if x <= y then x :: merge (xs, y::ys) else y :: merge (x::xs, ys) val merge : 'a list * 'a list -> 'a list = ]]>Generalises insert to two lists, and does at most comparisons.Merging means combining two sorted lists to form a larger sorted list. It does at most comparisons, where and are the lengths of the input lists. If and are roughly equal then we have a fast way of constructing sorted lists; if then merging degenerates to insertion, doing much work for little gain.Merging is the basis of several sorting algorithms; we look at a divide-and-conquer one. Mergesort is seldom found in conventional programming because it is hard to code for arrays; it works nicely with lists. It divides the input (if non-trivial) into two roughly equal parts, sorts them recursively, then merges them.Function merge is not iterative; the recursion is deep. An iterative version is of little benefit for the same reasons that apply to append in the earlier lecture on Lists.Anil Madhavapeddyhttps://www.cl.cam.ac.uk/teaching/2627/FoundsCS/forest/focs-def-polymorphism/focs-def-polymorphism/teaching/2627/FoundsCS/forest/focs-def-polymorphism/PolymorphismDefinitionA polymorphic function allows flexibility in the types of its arguments and results: the list primitives null, hd and tl work whatever the element type is. Symbols 'a and 'b are called type variables and stand for any types, and code written using them is still checked for type correctness at compile time. Introduced in Getting at the Head and Tail.Part of the course glossary.Anil Madhavapeddyhttps://www.cl.cam.ac.uk/teaching/2627/FoundsCS/forest/focs-quicksort/focs-quicksort/teaching/2627/FoundsCS/forest/focs-quicksort/Quicksort: The Code [] | [x] -> [x] | a::bs -> let rec part l r = function | [] -> (quick l) @ (a :: quick r) | x::xs -> if (x <= a) then part (x::l) r xs else part l (x::r) xs in part [] [] bs val quick : 'a list -> 'a list = ]]>Our OCaml quicksort copies the items. It is still pretty fast, and it is much easier to understand. It takes roughly 0.74 seconds to sort our list of random numbers.The function declaration consists of three clauses. The first handles the empty list; the second handles singleton lists (those of the form [x]); the third handles lists of two or more elements. Often, lists of length up to five or so are treated as special cases to boost speed.The locally declared function part partitions the input using a as the pivot. The arguments l and r accumulate items for the left () and right (a]]>) parts of the input, respectively.It is not hard to prove that quicksort does comparisons, in the average case (see page 94 of Aho). With random data, the pivot usually has an average value that divides the input in two approximately equal parts. We have the recurrence and , which is . In our example, it is about 235 times faster than insertion sort.In the worst case, quicksort's running time is quadratic! An example is when its input is almost sorted or reverse sorted. Nearly all of the items end up in one partition; work is not divided evenly. We have the recurrence and , which is . Randomising the input makes the worst case highly unlikely.Anil Madhavapeddyhttps://www.cl.cam.ac.uk/teaching/2627/FoundsCS/forest/focs-rev/focs-rev/teaching/2627/FoundsCS/forest/focs-rev/Reversing a List in ys | x::xs, ys -> rev_app xs (x::ys) val rev_app : 'a list -> 'a list -> 'a list = ]]>Calling rev_app xs ys reverses the elements of xs and prepends them to ys. Now we may declare 'a list = # rev [1; 2; 3] - : int list = [3; 2; 1]]]>It is easy to see that this reverse function performs just conses, given an -element list. For both reverse functions, we could count the number of conses precisely—not just up to a constant factor. notation is still useful to describe the overall running time: the time taken by a cons varies from one system to another.The accumulator makes the function iterative. But the gain in complexity arises from the removal of append. Replacing an expensive operation (append) by a series of cheap operations (cons) is called reduction in strength and is a common technique in computer science. It originated when many computers did not have a hardware multiply instruction; the series of products for , could more efficiently be computed by repeated addition. Reduction in strength can be done in various ways; we shall see many instances of removing append.Consing to an accumulator produces the result in reverse. If that forces the use of an extra list reversal then the iterative function may be much slower than the recursive one.Anil Madhavapeddyhttps://www.cl.cam.ac.uk/teaching/2627/FoundsCS/forest/focs-nrev/focs-nrev/teaching/2627/FoundsCS/forest/focs-nrev/Reversing a List in Let us consider one way to reverse a list. [] | x::xs -> (nrev xs) @ [x] val nrev : 'a list -> 'a list = # nrev [1; 2; 3] - : int list = [3; 2; 1]]]>This reverse function is grossly inefficient due to poor usage of append, which copies its first argument. If nrev is given a list of length 0]]>, then append makes conses to copy the reversed tail. Constructing the list [x] calls cons again, for a total of calls. Reversing the tail requires more conses, and so forth. The total number of conses is:The time complexity is therefore . Space complexity is only because the copies don’t all exist at the same time.Anil Madhavapeddyhttps://www.cl.cam.ac.uk/teaching/2627/FoundsCS/forest/focs-curry-shorthand/focs-curry-shorthand/teaching/2627/FoundsCS/forest/focs-curry-shorthand/Shorthand for Curried FunctionsA function-returning function is just a function of two arguments.This curried function syntax is nicer than nested fun binders: string -> string = # let dub = prefix "Sir " val dub : string -> string = ]]>Curried functions allows partial application (to the first argument).In OCaml, an -argument curried function f can be declared using the syntax:and applied using the syntax:If f is not recursive, then it is equivalent to the function expressed via nesting as follows:We now have two ways of expressing functions of multiple arguments: either by passing a pair of arguments or by currying. Currying allows partial application which is useful when fixing the first argument yields a function that is interesting in its own right. An example from mathematics is the function , where fixing yields a function in alone, namely squaring. Similarly, yields cubing, while yields the identity function.Though the function hd (which returns the head of a list) is not curried, it may be used with the curried application syntax in some expressions:Here List.hd is applied to a list of functions, and the resulting function dub is then applied to the string "Hamilton". The idea of executing code stored in data structures reaches its full development in object-oriented programming, like in Java.Anil Madhavapeddyhttps://www.cl.cam.ac.uk/teaching/2627/FoundsCS/forest/focs-list-primitives/focs-list-primitives/teaching/2627/FoundsCS/forest/focs-list-primitives/The List PrimitivesThere are two kinds of lists:[] represents the empty list x :: l is the list with head and tail The operator :: (also called List.cons for “construct”), puts a new element on to the head of an existing list. While we should not be too preoccupied with implementation details, it is essential to know that :: is an operation. It uses constant time and space, regardless of the length of the resulting list. Lists are represented internally with a linked structure; adding a new element to a list merely hooks the new element to the front of the existing structure. Moreover, that structure continues to denote the same list as it did before; to see the new list, one must look at the new :: node (or “cons cell”) just created. We will explain the 'a notation in the next section.Here we see the element 1 being consed to the front of the list [3; 5; 9]:Given a list, taking its first element (its “head”) or its list of remaining elements (its “tail”) also takes constant time. Each operation just follows a link. In the diagram above, the first down arrow leads to the head and the leftmost right arrow leads to the tail. Once we have the tail, its head is the second element of the original list, etc.The tail is not the last element; it is the list of all elements other than the head!Anil Madhavapeddyhttps://www.cl.cam.ac.uk/teaching/2627/FoundsCS/forest/focs-merge-sort/focs-merge-sort/teaching/2627/FoundsCS/forest/focs-merge-sort/Top-down Merge sort [] | [x] -> [x] | xs -> let k = List.length xs / 2 in let l = tmergesort (take k xs) in let r = tmergesort (drop k xs) in merge (l, r) val tmergesort : 'a list -> 'a list = ]]> comparisons in worst caseMergesort’s divide stage divides the input not by choosing a pivot (as in quicksort) but by simply counting out half of the elements. The conquer stage again involves recursive calls, and the combine stage involves merging. Function tmergesort takes roughly 1.4 seconds to sort the list rs.In the worst case, mergesort does comparisons, with the same recurrence equation as in quicksort’s average case. Because take and drop divide the input in two equal parts (they differ at most by one element), we always have .Quicksort is nearly 3 times as fast in the example. But it risks a quadratic worst case! Merge sort is safe but slow. So which algorithm is best?We have seen a top-down mergesort. Bottom-up algorithms also exist. They start with a list of one-element lists and repeatedly merge adjacent lists until only one is left. A refinement, which exploits any initial order among the input, is to start with a list of increasing or decreasing runs of input items.Anil Madhavapeddyhttps://www.cl.cam.ac.uk/teaching/2627/FoundsCS/forest/focs-map/focs-map/teaching/2627/FoundsCS/forest/focs-map/map: the “Apply to All” Function [] | x::xs -> (f x) :: map f xs val map : ('a -> 'b) -> 'a list -> 'b list = # map (fun s -> s ^ "ppy") ["Hi"; "Ho"] - : string list = ["Hippy"; "Hoppy"] # map (map double) [[1]; [2; 3]] - : int list list = [[2]; [4; 6]]]]>The functional map applies a function to every element of a list, returning a list of the function’s results. “Apply to all” is a fundamental operation and we shall see several applications of it below. We again see the advantages of fun-notation, currying and map. If we did not have them, the first use of map in the above code block would require a preliminary function declaration: [] | s::ss -> (s ^ "ppy") :: sillylist ss val sillylist : string list -> string list = ]]>An expression containing several applications of functionals—such as our second example—can abbreviate a long series of declarations. Sometimes this coding style is cryptic, but it can be clear as crystal. Treating functions as values lets us capture common program structures once and for all.In the second example, double is the obvious integer doubling function we defined earlier. Note that map is a built-in OCaml function in the form of List.map. OCaml’s standard library includes, among much else, many list functions.Anil MadhavapeddyAnil Madhavapeddyhttps://www.cl.cam.ac.uk/teaching/2627/FoundsCS/forest/focs-ex-4-3/focs-ex-4-3/teaching/2627/FoundsCS/forest/focs-ex-4-3/Exercises › A non-exhaustive zipExerciseHow does this version of zip differ from the one above? (x, y) :: zip xs ys | ([], []) -> []]]>Anil MadhavapeddyAnil Madhavapeddyhttps://www.cl.cam.ac.uk/teaching/2627/FoundsCS/forest/focs-ex-9-5/focs-ex-9-5/teaching/2627/FoundsCS/forest/focs-ex-9-5/Exercises › All lists of zeroes and onesExerciseCode the lazy list whose elements are all ordinary lists of zeroes and ones, namely []; [0]; [1]; [0; 0]; [0; 1]; [1; 0]; [1; 1]; [0; 0; 0]; …. (Taken from the exam question 2003 Paper 1 Question 5.)Anil MadhavapeddyAnil Madhavapeddyhttps://www.cl.cam.ac.uk/teaching/2627/FoundsCS/forest/focs-ex-4-4/focs-ex-4-4/teaching/2627/FoundsCS/forest/focs-ex-4-4/Exercises › Assumptions in making changeExerciseWhat assumptions do the ‘making change’ functions make about the variables till and amt? Describe what could happen if these assumptions were violated.Anil MadhavapeddyAnil Madhavapeddyhttps://www.cl.cam.ac.uk/teaching/2627/FoundsCS/forest/focs-ex-9-2/focs-ex-9-2/teaching/2627/FoundsCS/forest/focs-ex-9-2/Exercises › Concatenating infinite sequencesExerciseConsider the list function concat, which concatenates a list of lists to form a single list. Can it be generalised to concatenate a sequence of sequences? What can go wrong? [] | l::ls -> l @ concat ls]]>Anil MadhavapeddyAnil Madhavapeddyhttps://www.cl.cam.ac.uk/teaching/2627/FoundsCS/forest/focs-ex-9-6/focs-ex-9-6/teaching/2627/FoundsCS/forest/focs-ex-9-6/Exercises › Enumerating palindromesExercise(Continuing the previous exercise.) A palindrome is a list that equals its own reverse. Code the lazy list whose elements are all palindromes of 0s and 1s, namely []; [0]; [1]; [0; 0]; [0; 0; 0]; [0; 1; 0]; [1; 1]; [1; 0; 1]; [1; 1; 1]; [0; 0; 0; 0]; , …. You may take the reversal function List.rev as given.Anil MadhavapeddyAnil Madhavapeddyhttps://www.cl.cam.ac.uk/teaching/2627/FoundsCS/forest/focs-ex-3-3/focs-ex-3-3/teaching/2627/FoundsCS/forest/focs-ex-3-3/Exercises › Even-numbered elementsExerciseCode a function to return the list consisting of the even-numbered elements of the list given as its argument. For example, given [a; b; c; d] it should return [b; d].Anil MadhavapeddyAnil Madhavapeddyhttps://www.cl.cam.ac.uk/teaching/2627/FoundsCS/forest/focs-ex-5-4/focs-ex-5-4/teaching/2627/FoundsCS/forest/focs-ex-5-4/Exercises › Implementing bubble sortExerciseImplement bubble sort (see previous exercise) using OCaml.Anil MadhavapeddyAnil Madhavapeddyhttps://www.cl.cam.ac.uk/teaching/2627/FoundsCS/forest/focs-ex-5-2/focs-ex-5-2/teaching/2627/FoundsCS/forest/focs-ex-5-2/Exercises › Implementing selection sortExerciseImplement selection sort (see previous exercise) using OCaml.Anil MadhavapeddyAnil Madhavapeddyhttps://www.cl.cam.ac.uk/teaching/2627/FoundsCS/forest/focs-ex-3-2/focs-ex-3-2/teaching/2627/FoundsCS/forest/focs-ex-3-2/Exercises › Last element of a listExerciseCode a function to return the last element of a non-empty list. How efficiently can this be done?Anil MadhavapeddyAnil Madhavapeddyhttps://www.cl.cam.ac.uk/teaching/2627/FoundsCS/forest/focs-ex-11-1/focs-ex-11-1/teaching/2627/FoundsCS/forest/focs-ex-11-1/Exercises › Lists of referencesExerciseComment, with examples, on the differences between an int ref list and an int list ref.Anil MadhavapeddyAnil Madhavapeddyhttps://www.cl.cam.ac.uk/teaching/2627/FoundsCS/forest/focs-ex-9-3/focs-ex-9-3/teaching/2627/FoundsCS/forest/focs-ex-9-3/Exercises › Making change lazilyExerciseCode a function to make change using lazy lists, delivering the sequence of all possible ways of making change. Using sequences allows us to compute solutions one at a time when there exists an astronomical number. Represent lists of coins using ordinary lists. (Hint: to benefit from laziness you may need to pass around the sequence of alternative solutions as a function of type unit -> (int list) seq.)Anil MadhavapeddyAnil Madhavapeddyhttps://www.cl.cam.ac.uk/teaching/2627/FoundsCS/forest/focs-ex-8-5/focs-ex-8-5/teaching/2627/FoundsCS/forest/focs-ex-8-5/Exercises › Making change with mapExerciseRecall the making change function of More on Lists: [ [] ] | [] , _ -> [] | c::till , amt -> if amt < c then change till amt else let rec allc = function | [] -> [] | cs :: css -> (c::cs) :: allc css in allc (change (c::till) (amt - c)) @ change till amt val change : int list -> int -> int list list = ]]>Function allc applies the function ‘cons a c’ to every element of a list. Eliminate it by declaring a curried cons function and applying map.Anil MadhavapeddyAnil Madhavapeddyhttps://www.cl.cam.ac.uk/teaching/2627/FoundsCS/forest/focs-ex-8-3/focs-ex-8-3/teaching/2627/FoundsCS/forest/focs-ex-8-3/Exercises › Mapping over lists of listsExerciseWithout using map write a function map2 such that map2 f is equivalent to map (map f). The obvious solution requires declaring two recursive functions. Try to get away with only one by exploiting nested pattern-matching.Anil MadhavapeddyAnil Madhavapeddyhttps://www.cl.cam.ac.uk/teaching/2627/FoundsCS/forest/focs-ex-4-2/focs-ex-4-2/teaching/2627/FoundsCS/forest/focs-ex-4-2/Exercises › Partitioning integers by signExerciseCode a function that takes a list of integers and returns two lists, the first consisting of all non-negative numbers found in the input and the second consisting of all the negative numbers.Anil MadhavapeddyAnil Madhavapeddyhttps://www.cl.cam.ac.uk/teaching/2627/FoundsCS/forest/focs-ex-4-6/focs-ex-4-6/teaching/2627/FoundsCS/forest/focs-ex-4-6/Exercises › Reasoning from polymorphic typesExerciseWe know nothing about the functions f and g other than their polymorphic types: val f : 'a * 'b -> 'b * 'a and val g : 'a -> 'a list. Suppose that f (1, true) and g 0 are evaluated and return their results. State, with reasons, what you think the resulting values will be.Anil MadhavapeddyAnil Madhavapeddyhttps://www.cl.cam.ac.uk/teaching/2627/FoundsCS/forest/focs-ex-4-1/focs-ex-4-1/teaching/2627/FoundsCS/forest/focs-ex-4-1/Exercises › Set union without repetitionsExerciseSets can be represented in OCaml using lists containing no duplicated items (i.e. where no item is equal to another using polymorphic comparison).Using the member function defined above, code a function to implement set union. It should avoid introducing repetitions, for example the union of the lists [4; 7; 1] and [6; 4; 7] should be [1; 6; 4; 7] (though the order does not matter).Anil MadhavapeddyAnil Madhavapeddyhttps://www.cl.cam.ac.uk/teaching/2627/FoundsCS/forest/focs-ex-3-1/focs-ex-3-1/teaching/2627/FoundsCS/forest/focs-ex-3-1/Exercises › Summing a listExerciseCode a recursive function to compute the sum of a list’s elements. Then code an iterative version and comment on the improvement in efficiency.Anil MadhavapeddyAnil Madhavapeddyhttps://www.cl.cam.ac.uk/teaching/2627/FoundsCS/forest/focs-ex-3-5/focs-ex-3-5/teaching/2627/FoundsCS/forest/focs-ex-3-5/Exercises › Tails of a listExerciseCode a function tails to return the list of the tails of its argument. For example, given [1; 2; 3] it should return [[1; 2; 3]; [2; 3]; [3]; []].Anil Madhavapeddyhttps://www.cl.cam.ac.uk/teaching/2627/FoundsCS/forest/focs-topic-sorting/focs-topic-sorting/teaching/2627/FoundsCS/forest/focs-topic-sorting/SortingAnil Madhavapeddyhttps://www.cl.cam.ac.uk/teaching/2627/FoundsCS/forest/focs-quicksort-append-free/focs-quicksort-append-free/teaching/2627/FoundsCS/forest/focs-quicksort-append-free/Append-Free Quicksort sorted | ([x], sorted) -> x::sorted | a::bs, sorted -> let rec part = function | l, r, [] -> quik (l, a :: quik (r, sorted)) | l, r, x::xs -> if x <= a then part (x::l, r, xs) else part (l, x::r, xs) in part ([], [], bs) val quik : 'a list * 'a list -> 'a list = ]]>The list sorted accumulates the result in the combine stage of the quicksort algorithm. We have again used the standard technique for eliminating append. Calling quik(xs, sorted) reverses the elements of xs and prepends them to the list sorted.Looking closely at part, observe that quik(r, sorted) is performed first. Then a is consed to this sorted list. Finally, quik is called again to sort the elements of l.The speedup is significant. An imperative quicksort coded in Pascal (taken from Sedgewick) is just slightly faster than function quik. The near-agreement is surprising because the computational overheads of lists exceed those of arrays. In realistic applications, comparisons are the dominant cost and the overheads matter even less.Anil Madhavapeddyhttps://www.cl.cam.ac.uk/teaching/2627/FoundsCS/forest/focs-sorting-bounds/focs-sorting-bounds/teaching/2627/FoundsCS/forest/focs-sorting-bounds/How Fast Can We Sort?typically count comparisons there are permutations of elements each comparison eliminates half of the permutations therefore The usual measure of efficiency for sorting algorithms is the number of comparison operations required. Mergesort requires only comparisons to sort an input of items. It is straightforward to prove that this complexity is the best possible. There are permutations of elements and each comparison distinguishes two permutations. The lower bound on the number of comparisons, , is obtained by solving ; therefore .In order to compare the sorting algorithms, we use the following source of pseudo-random numbers. Never mind how this works: generating statistically good random numbers is hard. Much effort has gone into those few lines of code. float = # let rec randlist (seed, seeds) = function | 0 -> (seed, seeds) | n -> randlist (nextrandom seed, seed::seeds) (n-1) val randlist : float * float list -> int -> float * float list = ]]>We can now bind the identifier rs to a list of 10,000 random numbers.Anil Madhavapeddyhttps://www.cl.cam.ac.uk/teaching/2627/FoundsCS/forest/focs-insertion-sort/focs-insertion-sort/teaching/2627/FoundsCS/forest/focs-insertion-sort/Insertion SortAn insert operation does comparisons on average. [x] | y::ys -> if x <= y then x :: y :: ys else y :: ins x ys val ins : 'a -> 'a list -> 'a list = ]]>Insertion sort takes comparisons on average: [] | x::xs -> ins x (insort xs)]]>Items from the input are copied one at a time to the output. Each new item is inserted into the right place so that the output is always in order.We could easily write iterative versions of these functions, but to no purpose. Insertion sort is slow because it does comparisons (and a lot of list copying), not because it is recursive. Its quadratic runtime makes it nearly useless: it takes 174 seconds for our example while the next-worst figure is 1.4 seconds.Insertion sort is worth considering because it is easy to code and illustrates the concepts. Two efficient sorting algorithms, mergesort and heapsort, can be regarded as refinements of insertion sort.Anil Madhavapeddyhttps://www.cl.cam.ac.uk/teaching/2627/FoundsCS/forest/focs-merge/focs-merge/teaching/2627/FoundsCS/forest/focs-merge/Merging Two ListsMerge joins two sorted lists. ys | xs, [] -> xs | x::xs, y::ys -> if x <= y then x :: merge (xs, y::ys) else y :: merge (x::xs, ys) val merge : 'a list * 'a list -> 'a list = ]]>Generalises insert to two lists, and does at most comparisons.Merging means combining two sorted lists to form a larger sorted list. It does at most comparisons, where and are the lengths of the input lists. If and are roughly equal then we have a fast way of constructing sorted lists; if then merging degenerates to insertion, doing much work for little gain.Merging is the basis of several sorting algorithms; we look at a divide-and-conquer one. Mergesort is seldom found in conventional programming because it is hard to code for arrays; it works nicely with lists. It divides the input (if non-trivial) into two roughly equal parts, sorts them recursively, then merges them.Function merge is not iterative; the recursion is deep. An iterative version is of little benefit for the same reasons that apply to append in the earlier lecture on Lists.Anil Madhavapeddyhttps://www.cl.cam.ac.uk/teaching/2627/FoundsCS/forest/focs-partial-application/focs-partial-application/teaching/2627/FoundsCS/forest/focs-partial-application/Partial Application: A Curried Insertion Sort [x] | y::ys -> if lessequal x y then x :: y :: ys else y :: ins x ys in let rec sort = function | [] -> [] | x::xs -> ins x (sort xs) in sort val insort : ('a -> 'a -> bool) -> 'a list -> 'a list = ]]>The sorting functions we discussed in earlier lectures are coded to sort floating-point numbers. They can be generalised to an arbitrary ordered type by passing the ordering predicate lessequal as an argument.Functions ins and sort are declared locally, referring to lessequal. Though it may not be obvious, insort is a curried function. Given its first argument, a predicate for comparing some particular type of items, it returns the function sort for sorting lists of that type of items.Some examples of its use:=) [5; 3; 9; 8] - : int list = [9; 8; 5; 3]]]>An obscure point: the syntax (<=) denotes the comparison operator as a function, which is then given to insort. Passing the relation for lessequal gives a decreasing sort. This is no coding trick; it is justified in mathematics, since if is a partial ordering then so is .Anil Madhavapeddyhttps://www.cl.cam.ac.uk/teaching/2627/FoundsCS/forest/focs-quicksort/focs-quicksort/teaching/2627/FoundsCS/forest/focs-quicksort/Quicksort: The Code [] | [x] -> [x] | a::bs -> let rec part l r = function | [] -> (quick l) @ (a :: quick r) | x::xs -> if (x <= a) then part (x::l) r xs else part l (x::r) xs in part [] [] bs val quick : 'a list -> 'a list = ]]>Our OCaml quicksort copies the items. It is still pretty fast, and it is much easier to understand. It takes roughly 0.74 seconds to sort our list of random numbers.The function declaration consists of three clauses. The first handles the empty list; the second handles singleton lists (those of the form [x]); the third handles lists of two or more elements. Often, lists of length up to five or so are treated as special cases to boost speed.The locally declared function part partitions the input using a as the pivot. The arguments l and r accumulate items for the left () and right (a]]>) parts of the input, respectively.It is not hard to prove that quicksort does comparisons, in the average case (see page 94 of Aho). With random data, the pivot usually has an average value that divides the input in two approximately equal parts. We have the recurrence and , which is . In our example, it is about 235 times faster than insertion sort.In the worst case, quicksort's running time is quadratic! An example is when its input is almost sorted or reverse sorted. Nearly all of the items end up in one partition; work is not divided evenly. We have the recurrence and , which is . Randomising the input makes the worst case highly unlikely.Anil Madhavapeddyhttps://www.cl.cam.ac.uk/teaching/2627/FoundsCS/forest/focs-quicksort-idea/focs-quicksort-idea/teaching/2627/FoundsCS/forest/focs-quicksort-idea/Quicksort: The IdeaThe Quicksort algorithm has the following flow:Choose a pivot element, Divide to partition the input into two sublists: those at most in value those exceeding Conquer using recursive calls to sort the sublists Combine the sorted lists by appending one to the otherQuicksort was invented by Sir Anthony Hoare (1934–2026), who in later life worked at Microsoft Research, Cambridge. Quicksort works by divide and conquer, a basic algorithm design principle. Quicksort chooses from the input some value , called the pivot. It partitions the remaining items into two parts: those , and those a]]>. It sorts each part recursively, then puts the smaller part before the greater.The cleverest feature of Hoare's algorithm was that the partition could be done in place by exchanging array elements. Quicksort was invented before recursion was well known, and people found it extremely hard to understand. As usual, we shall consider a list version based on functional programming.Anil Madhavapeddyhttps://www.cl.cam.ac.uk/teaching/2627/FoundsCS/forest/focs-sorting-summary/focs-sorting-summary/teaching/2627/FoundsCS/forest/focs-sorting-summary/Summary of Sorting AlgorithmsOptimal is comparisons Insertion sort: simple to code; too slow (quadratic) [174 secs] Quicksort: fast on average; quadratic in worst case [0.53 secs] Mergesort: optimal in theory; often slower than quicksort [1.4 secs] Match the algorithm to the applicationQuicksort’s worst case cannot be ignored. For large , a complexity of is catastrophic. Mergesort has an worst case running time, which is optimal, but it is typically slower than quicksort for random data.Non-comparison sorting deserves mentioning. We can sort a large number of small integers using their radix representation in time. This result does not contradict the comparison-counting argument because comparisons are not used at all. Linear time is achievable only if the greatest integer is fixed in advance; as goes to infinity, increasingly many of the items are the same. It is a simple special case.Many other sorting algorithms exist. A few are outlined in the exercises.Anil Madhavapeddyhttps://www.cl.cam.ac.uk/teaching/2627/FoundsCS/forest/focs-merge-sort/focs-merge-sort/teaching/2627/FoundsCS/forest/focs-merge-sort/Top-down Merge sort [] | [x] -> [x] | xs -> let k = List.length xs / 2 in let l = tmergesort (take k xs) in let r = tmergesort (drop k xs) in merge (l, r) val tmergesort : 'a list -> 'a list = ]]> comparisons in worst caseMergesort’s divide stage divides the input not by choosing a pivot (as in quicksort) but by simply counting out half of the elements. The conquer stage again involves recursive calls, and the combine stage involves merging. Function tmergesort takes roughly 1.4 seconds to sort the list rs.In the worst case, mergesort does comparisons, with the same recurrence equation as in quicksort’s average case. Because take and drop divide the input in two equal parts (they differ at most by one element), we always have .Quicksort is nearly 3 times as fast in the example. But it risks a quadratic worst case! Merge sort is safe but slow. So which algorithm is best?We have seen a top-down mergesort. Bottom-up algorithms also exist. They start with a list of one-element lists and repeatedly merge adjacent lists until only one is left. A refinement, which exploits any initial order among the input, is to start with a list of increasing or decreasing runs of input items.Anil MadhavapeddyAnil Madhavapeddyhttps://www.cl.cam.ac.uk/teaching/2627/FoundsCS/forest/focs-ex-5-3/focs-ex-5-3/teaching/2627/FoundsCS/forest/focs-ex-5-3/Exercises › Complexity of bubble sortExerciseAnother sorting algorithm (bubble sort) consists of looking at adjacent pairs of elements, exchanging them if they are out of order and repeating this process until no more exchanges are possible. State, with justification, the time complexity of this approach.Anil MadhavapeddyAnil Madhavapeddyhttps://www.cl.cam.ac.uk/teaching/2627/FoundsCS/forest/focs-ex-5-1/focs-ex-5-1/teaching/2627/FoundsCS/forest/focs-ex-5-1/Exercises › Complexity of selection sortExerciseAnother sorting algorithm (selection sort) consists of looking at the elements to be sorted, identifying and removing a minimal element, which is placed at the head of the result. The tail is obtained by recursively sorting the remaining elements. State, with justification, the time complexity of this approach.Anil MadhavapeddyAnil Madhavapeddyhttps://www.cl.cam.ac.uk/teaching/2627/FoundsCS/forest/focs-ex-5-4/focs-ex-5-4/teaching/2627/FoundsCS/forest/focs-ex-5-4/Exercises › Implementing bubble sortExerciseImplement bubble sort (see previous exercise) using OCaml.Anil MadhavapeddyAnil Madhavapeddyhttps://www.cl.cam.ac.uk/teaching/2627/FoundsCS/forest/focs-ex-5-2/focs-ex-5-2/teaching/2627/FoundsCS/forest/focs-ex-5-2/Exercises › Implementing selection sortExerciseImplement selection sort (see previous exercise) using OCaml.Anil MadhavapeddyAnil Madhavapeddyhttps://www.cl.cam.ac.uk/teaching/2627/FoundsCS/forest/focs-ex-8-2/focs-ex-8-2/teaching/2627/FoundsCS/forest/focs-ex-8-2/Exercises › Lexicographic orderingsExerciseThere are many ways of combining orderings. The lexicographic ordering uses two keys for comparisons. It is specified byWrite an OCaml function to lexicographically combine two orderings, supplied as functions. Explain how it allows function insort to sort a list of pairs.Anil Madhavapeddyhttps://www.cl.cam.ac.uk/teaching/2627/FoundsCS/forest/focs-topic-datatypes/focs-topic-datatypes/teaching/2627/FoundsCS/forest/focs-topic-datatypes/DatatypesAnil Madhavapeddyhttps://www.cl.cam.ac.uk/teaching/2627/FoundsCS/forest/focs-constructors-args/focs-constructors-args/teaching/2627/FoundsCS/forest/focs-constructors-args/A Datatype whose Constructors have ArgumentsConstructors with arguments (like Lorry) are distinct values. (So Car true is distinct from Car false). Different kinds of vehicle can belong to one list: [Bike; Car true; Motorbike 450]OCaml generalises the notion of enumeration type to allow data to be associated with each constructor. The constructor Bike is a vehicle all by itself, but the other three constructors create vehicles from arguments.Since we might find it hard to remember what the various int and bool components are for, it is wise to include comments in complex declarations. In OCaml, comments are enclosed in the brackets and . Programmers should comment their code to explain design decisions and key features of the algorithms (sometimes by citing a reference work).The list [Bike; Car true; Motorbike 450] represents a bicycle, a Reliant Robin and a large motorbike. It can be almost seen as a mixed-type list containing integers and booleans. It is actually a list of vehicles; datatypes lessen the impact of the restriction that all list elements must have the same type.Anil Madhavapeddyhttps://www.cl.cam.ac.uk/teaching/2627/FoundsCS/forest/focs-wheels/focs-wheels/teaching/2627/FoundsCS/forest/focs-wheels/A Finer Wheel ComputationWe now define a wheels function to calculate the number of wheels in any vehicle. This requires pattern matching to retrieve the constructors and their associated arguments, around which we build the logic: 2 | Motorbike _ -> 2 | Car robin -> if robin then 3 else 4 | Lorry w -> w val wheels : vehicle -> int = ]]>This function consists of four clauses:A Bike has two wheels. This is a constant result. A Motorbike has two wheels. The _ signifies a "wildcard" pattern match that we discard, since the engine size of the bike is not relevant to the number of wheels. A Reliant Robin has three wheels; all other cars have four. We bind robin to the bool argument and then use it in the right hand side of the pattern match, much like a let binding in normal code. A Lorry has the number of wheels stored with its constructor, and we simply return that.There is no overlap between the Motorbike and Lorry cases. Although Motorbike and Lorry both hold an integer, OCaml takes the constructor into account and keeps any Motorbike distinct from any Lorry.Vehicles are one example of a concept consisting of several varieties with distinct features. Most programming languages can represent such concepts using something analogous to datatypes. (They are sometimes called union types or variant records whose tag fields play the role of the constructors.)A pattern may be built from the constructors of several datatypes, including lists. A pattern may also contain integer and string constants. There is no limit to the size of patterns or the number of clauses in a function declaration. OCaml performs pattern-matching efficiently (you do not need to understand the details of how it optimises pattern-matching at this stage).Anil Madhavapeddyhttps://www.cl.cam.ac.uk/teaching/2627/FoundsCS/forest/focs-def-abstract-data-type/focs-def-abstract-data-type/teaching/2627/FoundsCS/forest/focs-def-abstract-data-type/Abstract data typeDefinitionAn abstract type provides specified operations while hiding low-level details, such as the data structure used to represent them. Before choosing the internal representation for a data structure, you need to specify the full set of operations. Abstract types can be declared in any modern programming language: Java’s objects serve this role, as do OCaml’s modules. Introduced in Dictionaries, and applied to queues and stacks later on.Part of the course glossary.Anil Madhavapeddyhttps://www.cl.cam.ac.uk/teaching/2627/FoundsCS/forest/focs-def-algebraic-datatype/focs-def-algebraic-datatype/teaching/2627/FoundsCS/forest/focs-def-algebraic-datatype/Algebraic datatypeDefinitionOCaml generalises the notion of enumeration type to allow data to be associated with each constructor: some constructors are values all by themselves, while others create values from arguments. Constructors with arguments are distinct values, and values built from different constructors of one datatype can belong to a single list. Introduced in A Datatype whose Constructors have Arguments.Part of the course glossary.Anil Madhavapeddyhttps://www.cl.cam.ac.uk/teaching/2627/FoundsCS/forest/focs-queue-adt/focs-queue-adt/teaching/2627/FoundsCS/forest/focs-queue-adt/An Abstract Data Type: Queuesqempty is the empty queue qnull tests whether a queue is empty qhd returns the element at the head of a queue deq discards the element at the head of a queue enq adds an element at the end of a queueBreadth-first search becomes much faster if we replace the lists by queues. A queue represents a sequence, allowing elements to be taken from the head and added to the tail. This is a First-In-First-Out (FIFO) discipline: the item next to be removed is the one that has been in the queue for the longest time. Lists can implement queues, but append is a poor means of adding elements to the tail.Our functional arrays are suitable, provided we augment them with a function to delete the first array element. (See ML for the Working Programmer page 156.) Each operation would take time for a queue of length .We shall describe a representation of queues that is purely functional, based upon lists, and efficient. Operations take time when “amortized”: averaged over the lifetime of a queue.A conventional programming technique is to represent a queue by an array. Two indices point to the front and back of the queue, which may wrap around the end of the array. The coding is somewhat tricky. Worse, the length of the queue must be given a fixed upper bound.Anil Madhavapeddyhttps://www.cl.cam.ac.uk/teaching/2627/FoundsCS/forest/focs-enumeration/focs-enumeration/teaching/2627/FoundsCS/forest/focs-enumeration/An Enumeration TypeWe will now learn how to define more expressive types than the basic ones supplied with the core OCaml language.We have declared a new type named vehicle. along with four new constants. They are the constructors of the datatype.The type declaration adds a new type to our OCaml session. Type vehicle is as good as any built-in type and even admits pattern-matching (as we used with the built-in list types earlier). The four new identifiers of type vehicle are called constructors.We could represent the various vehicles by the numbers 0–3. However, the code would be hard to read and even harder to maintain. Consider adding Tricycle as a new vehicle. If we wanted to add it before Bike, then all the numbers would have to be changed. Using type, such additions are trivial and the compiler can (at least sometimes) warn us when it encounters a function declaration that doesn’t yet have a case for Tricycle.Representing vehicles by strings like "Bike", "Car", etc., is also bad. Comparing string values is slow and the compiler can’t warn us of misspellings like "MOtorbike": they will make our code fail.Most programming languages allow the declaration of types like vehicle. Because they consist of a series of identifiers, they are called enumeration types. Other common examples are days of the week or colours. The compiler chooses the integers for us; type-checking prevents us from confusing Bike with Red or Sunday.Anil Madhavapeddyhttps://www.cl.cam.ac.uk/teaching/2627/FoundsCS/forest/focs-binary-trees/focs-binary-trees/teaching/2627/FoundsCS/forest/focs-binary-trees/Binary Trees, a Recursive DatatypeA data structure with multiple branching is called a “tree”. Trees can represent mathematical expressions, logical formulae, computer programs, the phrase structure of English sentences, etc.Binary trees are nearly as fundamental as lists. They can provide efficient storage and retrieval of information. In a binary tree, each node is empty (), or is a branch () with a label and two subtrees.OCaml lists are a datatype and could be declared as follows:We could even declare :: as an infix constructor. The only thing we could not define is the [...] notation, which is part of the OCaml grammar (although there does exist a mechanism to use a similar syntax for custom indexed datatypes).A recursive type does not have to be polymorphic. For example, here is a simple datatype of tree shapes with no attached data that is recursive but not polymorphic.The datatype 'a option (mentioned above) is the opposite – it is polymorphic, but not recursive.Anil Madhavapeddyhttps://www.cl.cam.ac.uk/teaching/2627/FoundsCS/forest/focs-vehicles/focs-vehicles/teaching/2627/FoundsCS/forest/focs-vehicles/Declaring a Function on Vehicles 2 | Motorbike -> 2 | Car -> 4 | Lorry -> 18 val wheels : vehicle -> int = ]]>Datatype constructors can be used in patterns. Pattern-matching is fast, even complicated nested patterns. Notice the type of the argument is vehicle, which we defined earlier.The beauty of datatype declarations is that the new types behave as if they were built into OCaml. Type-checking catches common errors, such as mixing up different datatypes in a function like wheels, as well as missing and redundant patterns.Anil Madhavapeddyhttps://www.cl.cam.ac.uk/teaching/2627/FoundsCS/forest/focs-dates/focs-dates/teaching/2627/FoundsCS/forest/focs-dates/Example 1: DatesAbstract level: dates over a certain interval Concrete level: could be 6 characters: YYMMDD (where each character is represented by 8 bits) Date crises caused by inadequate internal formats: Digital’s PDP-10: using 12-bit dates (good for at most 11 years) 2000 crisis: 48 bits could be good for lifetime of universe!Digital Equipment Corporation’s date crisis occurred in 1975. The PDP-10 was a 36-bit mainframe computer. It represented dates using a 12-bit format designed for the tiny PDP-8. With 12 bits, one can distinguish days or 11 years.Last century, the most common industry format for dates used six characters: two for the year, two for the month and two for the day. The most common “solution” to the year 2000 crisis is to add two further characters, thereby altering file sizes. Others have noticed that the existing six characters consist of 48 bits, already sufficient to represent all dates over the projected lifetime of the universe: = days = years!Mathematicians think in terms of unbounded ranges, but the representation we choose for the computer usually imposes hard limits. A good programming language like OCaml lets one easily change the representation used in the program. But if files in the old representation exist all over the place, there will still be conversion problems. The need for compatibility with older systems causes problems across the computer industry.Anil Madhavapeddyhttps://www.cl.cam.ac.uk/teaching/2627/FoundsCS/forest/focs-ocaml-exceptions/focs-ocaml-exceptions/teaching/2627/FoundsCS/forest/focs-ocaml-exceptions/Exceptions in OCamlEach exception declaration introduces a distinct sort of exception, which can be handled separately from others. If raises an exception, then its evaluation has failed; handling an exception means evaluating another expression and returning its value instead. One exception handler can specify separate expressions for different sorts of exceptions.Exception names are constructors of the special datatype exn. This is a peculiarity of OCaml that lets exception-handlers use pattern-matching. Note that exception Failure is just an error indication, while NoChange n carries further information: the integer . print_endline "handled a NoChange exception" Line 3, characters 5-23: Warning 21 [nonreturning-statement]: this statement never returns (or has an unsound type.) pre exception handled a NoChange exception - : unit = ()]]>The effect of raise <expr> is to jump to the most recently-encountered handler that matches <expr>. The matching handler can only be found dynamically (during execution); contrast with how OCaml associates occurrences of identifiers with their matching declarations, which does not require running the program. A handler is introduced via the try keyword, which executes the subexpression and dispatches any exceptions encountered to the corresponding pattern match for exceptions defined in the with block.This is also the first time that we have encountered the unit type. This represents a type that has no values, and is used to indicate that a block has no meaningful return value. We will come back to this when learning more about imperative programming later on. For now, it is sufficient to understand that print_endline will print out the argument to the console output, and return a unit type. The semicolon at the end of the expression is a convenient way to execute sequential statements that return the unit type.One criticism of OCaml’s exceptions is that—unlike the Java language—nothing in a function declaration indicates which exceptions it might raise. One alternative to exceptions is to instead return a value of datatype option.None signifies an error, while Some x returns the solution . This approach looks clean, but the drawback is that many places in the code would have to check for None. Despite this, there is a builtin option type in OCaml as it is so useful. We will see in later lectures how to define our own version of option using polymorphic datatype definitions.Anil Madhavapeddyhttps://www.cl.cam.ac.uk/teaching/2627/FoundsCS/forest/focs-lazy-ocaml/focs-lazy-ocaml/teaching/2627/FoundsCS/forest/focs-lazy-ocaml/Lazy Lists in OCamlThe empty tuple () and its type unit Delayed version of is fun () -> E 'a seq) type 'a seq = Nil | Cons of 'a * (unit -> 'a seq) # let head (Cons (x, _)) = x Line 1, characters 10-23: Warning 8 [partial-match]: this pattern-matching is not exhaustive. Here is an example of a case that is not matched: Nil val head : 'a seq -> 'a = # let tail (Cons (_, xf)) = xf () Line 1, characters 10-24: Warning 8 [partial-match]: this pattern-matching is not exhaustive. Here is an example of a case that is not matched: Nil val tail : 'a seq -> 'a seq = ]]> has head and tail function The primitive OCaml type unit has one element, which is written (). This element may be regarded as a 0-tuple, and unit as the nullary Cartesian product. (Think of the connection between multiplication and the number 1.)The empty tuple serves as a placeholder in situations where no information is required. It may:appear in a data structure. For example, a unit-valued dictionary represents a set of keys. be the argument of a function, where its effect is to delay evaluation. be the argument or result of a procedure. (see the Procedural Programming section)The empty tuple, like all tuples, is a constructor and is allowed in patterns; for example: let f () = ...In particular is the function that takes an argument of type unit and returns the value of as its result. Expression is not evaluated until the function is called, even though the only possible argument is (). The function simply delays the evaluation of .Anil Madhavapeddyhttps://www.cl.cam.ac.uk/teaching/2627/FoundsCS/forest/focs-def-pattern-matching/focs-def-pattern-matching/teaching/2627/FoundsCS/forest/focs-def-pattern-matching/Pattern matchingDefinitionPattern matching takes a value apart by matching it against the forms that value can have, letting one function declaration handle several cases, separated by a vertical bar |. Taking a list apart using combinations of hd and tl is hard to get right; fortunately, it is seldom necessary because of pattern-matching. Introduced in Getting at the Head and Tail.Part of the course glossary.Anil Madhavapeddyhttps://www.cl.cam.ac.uk/teaching/2627/FoundsCS/forest/focs-references-comparison/focs-references-comparison/teaching/2627/FoundsCS/forest/focs-references-comparison/References: OCaml vs conventional languagesWe must write !p to get the contents of p We write just p for the address of p We can store private reference cells (like balance) in functions—analogous to elements of object-oriented programming OCaml’s assignment syntax is instead of = OCaml has few control structures: while, match, if and for (the latter is not covered in this course) OCaml has syntax for updating an array via the a.(i) <- v syntax which is the same as Array.set a i v.Conventional syntax for variables and assignments has hardly changed since Fortran, the first high-level language. In conventional languages, virtually all variables can be updated. We declare something like p: int, mentioning no reference type even if the language provides them. If we do not specify an initial value, we may get whatever bits were previously at that address. Illegal values arising from uninitialised variables can cause errors that are almost impossible to diagnose.Dereferencing operators (like OCaml’s !) are especially unpopular, because they clutter the program text. Virtually all programming languages make dereferencing implicit (that is, automatic).It is generally accepted these days that a two-dimensional array is nothing but an array of arrays. An assignment to such an array is typically written something like ; in C, the syntax is A[i][j] = x. Higher dimensions are treated analogously. The corresponding OCaml code can either declare an array of arrays, or use the A.(i) syntax to calculate the linear offset into a single array.You can use the constructs we have learnt to easily create linked (mutable) lists as an alternative to arrays.It is worth mentioning that OCaml’s references fully suffice for coding the sort of linked data structures taught in algorithms courses, as the mlist type above illustrates. The programming style is a little different from the usual, but the principles are the same. OCaml also provides comprehensive input/output primitives for various types of file and operating system.OCaml’s system of modules include structures, which can be seen as encapsulated groups of declarations, and signatures, which are specifications of structures listing the name and type of each component. Finally, there are functors, which are analogous to functions that combine a number of argument structures, and which can be used to plug program components together. These primitives are useful for managing large programming projects.Anil Madhavapeddyhttps://www.cl.cam.ac.uk/teaching/2627/FoundsCS/forest/focs-list-primitives/focs-list-primitives/teaching/2627/FoundsCS/forest/focs-list-primitives/The List PrimitivesThere are two kinds of lists:[] represents the empty list x :: l is the list with head and tail The operator :: (also called List.cons for “construct”), puts a new element on to the head of an existing list. While we should not be too preoccupied with implementation details, it is essential to know that :: is an operation. It uses constant time and space, regardless of the length of the resulting list. Lists are represented internally with a linked structure; adding a new element to a list merely hooks the new element to the front of the existing structure. Moreover, that structure continues to denote the same list as it did before; to see the new list, one must look at the new :: node (or “cons cell”) just created. We will explain the 'a notation in the next section.Here we see the element 1 being consed to the front of the list [3; 5; 9]:Given a list, taking its first element (its “head”) or its list of remaining elements (its “tail”) also takes constant time. Each operation just follows a link. In the diagram above, the first down arrow leads to the head and the leftmost right arrow leads to the tail. Once we have the tail, its head is the second element of the original list, etc.The tail is not the last element; it is the list of all elements other than the head!Anil MadhavapeddyAnil Madhavapeddyhttps://www.cl.cam.ac.uk/teaching/2627/FoundsCS/forest/focs-ex-8-4/focs-ex-8-4/teaching/2627/FoundsCS/forest/focs-ex-8-4/Exercises › A map for optionsExerciseThe type 'a option, declared below, can be viewed as a type of lists having at most one element. (It is typically used as an alternative to exceptions.) Declare an analogue of the function map for type 'a option.Anil MadhavapeddyAnil Madhavapeddyhttps://www.cl.cam.ac.uk/teaching/2627/FoundsCS/forest/focs-ex-6-4/focs-ex-6-4/teaching/2627/FoundsCS/forest/focs-ex-6-4/Exercises › A type for arithmetic expressionsExerciseGive the declaration of an OCaml type for arithmetic expressions that have the following possibilities: floating-point numbers, variables (represented by strings), or expressions of the form , , .Anil MadhavapeddyAnil Madhavapeddyhttps://www.cl.cam.ac.uk/teaching/2627/FoundsCS/forest/focs-ex-6-1/focs-ex-6-1/teaching/2627/FoundsCS/forest/focs-ex-6-1/Exercises › Days of the week typeExerciseGive the declaration of an OCaml type for the days of the week. Comment on the practicality of such a type in a calendar application.Anil MadhavapeddyAnil Madhavapeddyhttps://www.cl.cam.ac.uk/teaching/2627/FoundsCS/forest/focs-ex-6-5/focs-ex-6-5/teaching/2627/FoundsCS/forest/focs-ex-6-5/Exercises › Evaluating expressions with exceptionsExerciseContinuing the previous exercise, write a function that evaluates an expression. If the expression contains any variables, your function should raise an exception indicating the variable name.Anil MadhavapeddyAnil Madhavapeddyhttps://www.cl.cam.ac.uk/teaching/2627/FoundsCS/forest/focs-ex-9-4/focs-ex-9-4/teaching/2627/FoundsCS/forest/focs-ex-9-4/Exercises › Lazy binary treesExerciseA lazy binary tree is either empty or is a branch containing a label and two lazy binary trees, possibly to infinite depth. Present an OCaml datatype to represent lazy binary trees, along with a function that accepts a lazy binary tree and produces a lazy list that contains all of the tree’s labels. (Taken from the exam question 2008 Paper 1 Question 5.)Anil Madhavapeddyhttps://www.cl.cam.ac.uk/teaching/2627/FoundsCS/forest/focs-topic-trees/focs-topic-trees/teaching/2627/FoundsCS/forest/focs-topic-trees/TreesAnil Madhavapeddyhttps://www.cl.cam.ac.uk/teaching/2627/FoundsCS/forest/focs-search-methods/focs-search-methods/teaching/2627/FoundsCS/forest/focs-search-methods/A Survey of Search MethodsDepth-first: use a stack (efficient but incomplete) Breadth-first: use a queue (uses too much space!) Iterative deepening: use depth-first to get benefits of breadth-first (trades time for space) Best-first: use a priority queue (heuristic search)The data structure determines the search!Search procedures can be classified by the data structure used to store pending subtrees. Depth-first search stores them on a stack, which is implicit in functions like inorder, but can be made explicit. Breadth-first search stores such nodes in a queue.An important variation is to store the nodes in a priority queue, which is an ordered sequence. The priority queue applies some sort of ranking function to the nodes, placing higher-ranked nodes before lower-ranked ones. The ranking function typically estimates the distance from the node to a solution. If the estimate is good, the solution is located swiftly. This method is called best-first search.The priority queue can be kept as a sorted list, although this is slow. Binary search trees would be much better on average, and fancier data structures improve matters further.Anil Madhavapeddyhttps://www.cl.cam.ac.uk/teaching/2627/FoundsCS/forest/focs-tree-traversal/focs-tree-traversal/teaching/2627/FoundsCS/forest/focs-tree-traversal/Aside: Traversing Trees (3 Methods) [] | Br (v, t1, t2) -> [v] @ preorder t1 @ preorder t2 val preorder : 'a tree -> 'a list = # let rec inorder = function | Lf -> [] | Br (v, t1, t2) -> inorder t1 @ [v] @ inorder t2 val inorder : 'a tree -> 'a list = # let rec postorder = function | Lf -> [] | Br (v, t1, t2) -> postorder t1 @ postorder t2 @ [v] val postorder : 'a tree -> 'a list = ]]>Tree traversal means examining each node of a tree in some order. D. E. Knuth has identified three forms of tree traversal: preorder, inorder and postorder. We can code these “visiting orders” as functions that convert trees into lists of labels. Algorithms based on these notions typically perform some action at each node; the functions above simply copy the nodes into lists. Consider the tree:preorder visits the label first (“Polish notation”), yielding ABDECFG inorder visits the label midway, yielding DBEAFCG postorder visits the label last (“Reverse Polish”), yielding DEBFGCA. You might be familiar with this concept if you own an ancient RPN calculator!What is the use of inorder? Consider applying it to a binary search tree: the result is a sorted list of pairs. We could use this, for example, to merge two binary search trees. It is not difficult to transform a sorted list of pairs into a binary search tree.Anil Madhavapeddyhttps://www.cl.cam.ac.uk/teaching/2627/FoundsCS/forest/focs-tree-properties/focs-tree-properties/teaching/2627/FoundsCS/forest/focs-tree-properties/Basic Properties of Binary Trees 0 (* number of branch nodes *) | Br (v, t1, t2) -> 1 + count t1 + count t2 val count : 'a tree -> int = # let rec depth = function | Lf -> 0 (* length of longest path *) | Br (v, t1, t2) -> 1 + max (depth t1) (depth t2) val depth : 'a tree -> int = ]]>The invariant holds in the functions above.Functions on trees are expressed recursively using pattern-matching. Both functions above are analogous to on lists. Here is a third measure of a tree’s size: 1 | Br (v, t1, t2) -> leaves t1 + leaves t2 val leaves : 'a tree -> int = ]]>This function is redundant because of a basic fact about trees, which can be proved by induction: for every tree , we have . The invariant stated above also has an elementary proof by induction.A tree of depth 20 can store or approximately one million elements. The access paths to these elements are short, particularly when compared with a million-element list!Anil Madhavapeddyhttps://www.cl.cam.ac.uk/teaching/2627/FoundsCS/forest/focs-bst/focs-bst/teaching/2627/FoundsCS/forest/focs-bst/Binary Search TreesA dictionary associates values (here, numbers) with keys.Binary search trees are an important application of binary trees. They work for keys that have a total ordering, such as strings. Each branch of the tree carries a pair; its left subtree holds smaller keys; the right subtree holds greater keys. If the tree remains reasonably balanced, then update and lookup both take for a tree of size . These times hold in the average case; given random data, the tree is likely to remain balanced.At a given node, all keys in the left subtree are smaller (or equal) while all trees in the right subtree are greater.An unbalanced tree has a linear access time in the worst case. Examples include building a tree by repeated insertions of elements in increasing or decreasing order; there is a close resemblance to quicksort. Building a binary search tree, then converting it to inorder, yields a sorting algorithm called treesort.Self-balancing trees, such as Red-Black trees, attain in the worst case. They are complicated to implement.Anil Madhavapeddyhttps://www.cl.cam.ac.uk/teaching/2627/FoundsCS/forest/focs-binary-trees/focs-binary-trees/teaching/2627/FoundsCS/forest/focs-binary-trees/Binary Trees, a Recursive DatatypeA data structure with multiple branching is called a “tree”. Trees can represent mathematical expressions, logical formulae, computer programs, the phrase structure of English sentences, etc.Binary trees are nearly as fundamental as lists. They can provide efficient storage and retrieval of information. In a binary tree, each node is empty (), or is a branch () with a label and two subtrees.OCaml lists are a datatype and could be declared as follows:We could even declare :: as an infix constructor. The only thing we could not define is the [...] notation, which is part of the OCaml grammar (although there does exist a mechanism to use a similar syntax for custom indexed datatypes).A recursive type does not have to be polymorphic. For example, here is a simple datatype of tree shapes with no attached data that is recursive but not polymorphic.The datatype 'a option (mentioned above) is the opposite – it is polymorphic, but not recursive.Anil Madhavapeddyhttps://www.cl.cam.ac.uk/teaching/2627/FoundsCS/forest/focs-def-binary-search-tree/focs-def-binary-search-tree/teaching/2627/FoundsCS/forest/focs-def-binary-search-tree/Binary search treeDefinitionA binary search tree works for keys that have a total ordering, such as strings. Each branch of the tree carries a pair; its left subtree holds smaller keys and its right subtree holds greater keys. If the tree remains reasonably balanced, then update and lookup both take for a tree of size . Introduced in Binary Search Trees.Part of the course glossary.Anil Madhavapeddyhttps://www.cl.cam.ac.uk/teaching/2627/FoundsCS/forest/focs-bfs-append/focs-bfs-append/teaching/2627/FoundsCS/forest/focs-bfs-append/Breadth-First Tree Traversal — Using Append [] | Lf :: ts -> nbreadth ts | Br (v, t, u) :: ts -> v :: nbreadth (ts @ [t; u]) val nbreadth : 'a tree list -> 'a list = ]]>Keeps an enormous queue of nodes of search, and is a wasteful use of append.Breadth-first search can be inefficient, this naive implementation especially so. When the search is at depth of the tree, the list contains all the remaining trees at depth , followed by the subtrees (all at depth ) of the trees that have already been visited. At depth 10, the list could already contain 1024 elements. It requires a lot of space, and aggravates this with a gross misuse of append. Evaluating ts@[t, u] copies the long list ts just to insert two elements.Anil Madhavapeddyhttps://www.cl.cam.ac.uk/teaching/2627/FoundsCS/forest/focs-bfs-queues/focs-bfs-queues/teaching/2627/FoundsCS/forest/focs-bfs-queues/Breadth-First Tree Traversal — Using Queues breadth (deq q) | Br (v, t, u) -> v :: breadth (enq (enq (deq q) t) u) val breadth : 'a tree queue -> 'a list = ]]>This function implements the same algorithm as nbreadth but uses a different data structure. It represents queues using type queue instead of type list.To compare their efficiency, I applied both functions to the full binary tree of depth 12, which contains 4095 labels. The function nbreadth took 30 seconds while breadth took only 0.15 seconds: faster by a factor of 200.For larger trees, the speedup would be greater. Choosing the right data structure pays handsomely.Anil Madhavapeddyhttps://www.cl.cam.ac.uk/teaching/2627/FoundsCS/forest/focs-bfs-dfs/focs-bfs-dfs/teaching/2627/FoundsCS/forest/focs-bfs-dfs/Breadth-First v Depth-First Tree Traversalbinary trees as decision trees look for solution nodes Depth-first: search one subtree in full before moving on Breadth-first: search all nodes at level before moving to finds all solutions — nearest first!Preorder, inorder and postorder tree traversals all have something in common: they are depth-first. At each node, the left subtree is entirely traversed before the right subtree. Depth-first traversals are easy to code and can be efficient, but they are ill-suited for some problems.Suppose the tree represents the possible moves in a puzzle, and the purpose of the traversal is to search for a node containing a solution. Then a depth-first traversal may find one solution node deep in the left subtree, when another solution is at the very top of the right subtree. Often we want the shortest path to a solution.Suppose the tree is infinite or simply extremely large. Depth-first search is almost useless with such trees, for if the left subtree is infinite then the search will never reach the right subtree. OCaml can represent infinite trees by the means discussed in the lecture on laziness. Another tree representation (suitable for solving solitaire, for example) is by a function next : pos -> pos list, which maps a board position to a list of the positions possible after the next move. For simplicity, the examples below use the OCaml datatype tree, which has only finite trees.A breadth-first traversal explores the nodes horizontally rather than vertically. When visiting a node, it does not traverse the subtrees until it has visited all other nodes at the current depth. This is easily implemented by keeping a list of trees to visit. Initially, this list consists of one element: the entire tree. Each iteration removes a tree from the head of the list and adds its subtrees after the end of the list.Anil Madhavapeddyhttps://www.cl.cam.ac.uk/teaching/2627/FoundsCS/forest/focs-tree-traversal-efficient/focs-tree-traversal-efficient/teaching/2627/FoundsCS/forest/focs-tree-traversal-efficient/Efficiently Traversing TreesUnfortunately, the traversal functions are quadratic in the worst case: the appends in the recursive calls are inefficient. To correct that problem, we (as usual) add an accumulating argument. Observe how each function constructs its result list and compare with how appends were eliminated from quicksort in the Sorting lecture. vs | Br (v, t1, t2), vs -> v :: preord (t1, preord (t2, vs)) val preord : 'a tree * 'a list -> 'a list = # let rec inord = function | Lf, vs -> vs | Br (v, t1, t2), vs -> inord (t1, v::inord (t2, vs)) val inord : 'a tree * 'a list -> 'a list = # let rec postord = function | Lf, vs -> vs | Br (v, t1, t2), vs -> postord (t1, postord (t2, v::vs)) val postord : 'a tree * 'a list -> 'a list = ]]>One can prove equations relating each of these functions to its counterpart on the previous section. For example:These three types of tree traversal are related in that all are depth-first. They each traverse the left subtree in full before traversing the right subtree. Breadth-first search (from the Queues lecture) is another possibility. That involves going through the levels of a tree one at a time.Anil Madhavapeddyhttps://www.cl.cam.ac.uk/teaching/2627/FoundsCS/forest/focs-functional-arrays/focs-functional-arrays/teaching/2627/FoundsCS/forest/focs-functional-arrays/Functional Arrays as Binary TreesThe path to element follows the binary code for (its “subscript”).This simple representation (credited to W. Braun) ensures that the tree is balanced. Complexity of access is always , which is optimal. For actual running time, access to conventional arrays is much faster: it requires only a few hardware instructions. Array access is often taken to be , which (as always) presumes that hardware limits are never exceeded.The lower bound for array subscripts (or “indices”) is one. The upper bound starts at zero (which signifies the empty array) and can grow without limit. Inspection of the diagram above should make it clear that these trees are always balanced: the left subtree can have at most one node more than the right subtree, recursively all the way down. (This assumes that the array is defined for subscripts with no gaps; an array defined only for odd numbers, for example, would obviously be unbalanced.)The numbers in the diagram above are not the labels of branch nodes, but indicate the positions of array elements. For example, the label corresponding to is at the position shown. The nodes of a functional array are labelled with the data we want to store, not with these integers.Anil Madhavapeddyhttps://www.cl.cam.ac.uk/teaching/2627/FoundsCS/forest/focs-iterative-deepening/focs-iterative-deepening/teaching/2627/FoundsCS/forest/focs-iterative-deepening/Iterative deepening: Another Exhaustive SearchBreadth-first search examines nodes: Recompute nodes at depth instead of storing them Time factor is if 1]]>; complexity is still Space required at depth drops from to Breadth-first search is not practical for infinite trees: it uses too much space. Large parts of the tree have to be stored. Consider the slightly more general problem of searching trees whose branching factor is (for binary trees, ). Then breadth-first search to depth examines nodes, which is , ignoring the constant factor of . Since all nodes that are examined are also stored, the space and time requirements are both .Depth-first iterative deepening combines the space efficiency of depth-first with the “nearest-first” property of breadth-first search. It performs repeated depth-first searches with increasing depth bounds, each time discarding the result of the previous search. Thus it searches to depth 1, then to depth 2, and so on until it finds a solution. We can afford to discard previous results because the number of nodes is growing exponentially. There are nodes at level ; if , this number actually exceeds the total number of nodes of all previous levels put together, namely .Korf shows that the time needed for iterative deepening to reach depth is only times that for breadth-first search, if 1]]>. This is a constant factor; both algorithms have the same time complexity, . In typical applications where the extra factor of is quite tolerable. The reduction in the space requirement is exponential, from for breadth-first to for iterative deepening. Of course, this assumes that the tree itself is not stored in memory.Anil Madhavapeddyhttps://www.cl.cam.ac.uk/teaching/2627/FoundsCS/forest/focs-bst-lookup/focs-bst-lookup/teaching/2627/FoundsCS/forest/focs-bst-lookup/Lookup: Seeks Left or Rightfocs-binary-trees if b < a then lookup b t1 else if a < b then lookup b t2 else x | Lf -> raise (Missing b) val lookup : string -> (string * 'a) tree -> 'a = ]]>This has guaranteed access time if the tree is balanced!Lookup in the binary search tree goes to the left subtree if the desired key is smaller than the current one and to the right if it is greater. It raises Missing if it encounters an empty tree.Since an ordering is involved, we have to declare the functions for a specific type, here string. Now exception Missing mentions that type: if lookup fails, the exception returns the missing key. The exception could be eliminated using type option of our earlier Datatypes lecture, using the constructor None for failure.Anil Madhavapeddyhttps://www.cl.cam.ac.uk/teaching/2627/FoundsCS/forest/focs-funarray-lookup/focs-funarray-lookup/teaching/2627/FoundsCS/forest/focs-funarray-lookup/The Lookup Function raise Subscript (* Not found *) | Br (v, t1, t2), k -> if k = 1 then v else if k mod 2 = 0 then sub (t1, k / 2) else sub (t2, k / 2) exception Subscript val sub : 'a tree * int -> 'a = # let rec sub = function (* Alternative implementation *) | Lf, _ -> raise Subscript | Br (v, t1, t2), 1 -> v | Br (v, t1, t2), k when k mod 2 = 0 -> sub (t1, k / 2) | Br (v, t1, t2), k -> sub (t2, k / 2) val sub : 'a tree * int -> 'a = ]]>Notice that we have used a new keyword when above, which changes pattern clauses to be only matched if the expression evalutes to true. This can be equivalently expressed by moving the corresponding checks into an if clause on the right hand side of the pattern match, but is often more readable using when (as above).The lookup function sub, divides the subscript by 2 until 1 is reached. If the remainder is 0 then the function follows the left subtree, otherwise the right. If it reaches a leaf, it signals error by raising exception Subscript.Array access can also be understood in terms of the subscript’s binary code. Because the subscript must be a positive integer, in binary it has a leading one. Discard this one and reverse the remaining bits. Interpreting zero as left and one as right yields the path from the root to the subscript.Popular literature often explains the importance of binary as being led by hardware: because a circuit is either on or off. The truth is almost the opposite. Designers of digital electronics go to a lot of trouble to suppress the continuous behaviour that would naturally arise. The real reason why binary is important is its role in algorithms: an if-then-else decision leads to binary branching.Data structures, such as trees, and algorithms, such as mergesort, use binary branching in order to reduce a cost from to . Two is the smallest integer divisor that achieves this reduction. (Larger divisors are only occasionally helpful, as in the case of B-trees, where they reduce the constant factor.) The simplicity of binary arithmetic compared with decimal arithmetic is just another instance of the simplicity of algorithms based on binary choices.Anil Madhavapeddyhttps://www.cl.cam.ac.uk/teaching/2627/FoundsCS/forest/focs-funarray-update/focs-funarray-update/teaching/2627/FoundsCS/forest/focs-funarray-update/The Update Function if k = 1 then Br (w, Lf, Lf) else raise Subscript (* Gap in tree *) | Br (v, t1, t2), k, w -> if k = 1 then Br (w, t1, t2) else if k mod 2 = 0 then Br (v, update (t1, k / 2, w), t2) else Br (v, t1, update (t2, k / 2, w)) val update : 'a tree * int * 'a -> 'a tree = ]]>The update function also divides the subscript repeatedly by two. When it reaches a value of one, it has identified the element position. Then it replaces the branch node by another branch with the new label.A leaf may be replaced by a branch, extending the array, provided no intervening nodes have to be generated. This suffices for arrays without gaps in their subscripting. (The data structure can be modified to allow sparse arrays, where most subscript positions are undefined.) Exception Subscript indicates that the subscript position does not exist and cannot be created. This use of exceptions is not easily replaced by None and Some.Note that there are two tests involving . If we have reached a leaf, it returns a branch, extending the array by one. If we are still at a branch node, then the effect is to update an existing array element.A similar function can shrink an array by one.Anil Madhavapeddyhttps://www.cl.cam.ac.uk/teaching/2627/FoundsCS/forest/focs-bst-update/focs-bst-update/teaching/2627/FoundsCS/forest/focs-bst-update/Update Br ((k, v), Lf, Lf) | Br ((a, x), t1, t2) -> if k < a then Br ((a, x), update k v t1, t2) else if a < k then Br ((a, x), t1, update k v t2) else (* a = k *) Br ((a, v), t1, t2) val update : 'a -> 'b -> ('a * 'b) tree -> ('a * 'b) tree = ]]>This is also as it copies the path only, and not whole subtrees!If you are familiar with the usual update operation for this sort of tree, you may wonder whether it can be implemented in OCaml, where there is no direct way to replace part of a data structure by something else.The update operation is a nice piece of functional programming. It searches in the same manner as lookup, but the recursive calls reconstruct a new tree around the result of the update. One subtree is updated and the other left unchanged. The internal representation of trees ensures that unchanged parts of the tree are not copied, but shared. Therefore, update copies only the path from the root to the new node. Its time and space requirements, for a reasonably balanced tree, are both .The comparison between and allows three cases:smaller: update the left subtree; share the right greater: update the right subtree; share the left equal: update the label and share both subtreesNote: in the function definition, (* a = k *) is a comment. Comments in OCaml are enclosed in the brackets and .Anil MadhavapeddyAnil Madhavapeddyhttps://www.cl.cam.ac.uk/teaching/2627/FoundsCS/forest/focs-ex-10-5/focs-ex-10-5/teaching/2627/FoundsCS/forest/focs-ex-10-5/Exercises › An implicit infinite treeExerciseConsider the following OCaml function.If we regard it as representing a tree, where the subtrees are computed from the current label, what tree does next 1 represent?Anil MadhavapeddyAnil Madhavapeddyhttps://www.cl.cam.ac.uk/teaching/2627/FoundsCS/forest/focs-ex-7-4/focs-ex-7-4/teaching/2627/FoundsCS/forest/focs-ex-7-4/Exercises › Deleting from a search treeExerciseDescribe an algorithm for deleting an entry from a binary search tree. Comment on the suitability of your approach.Anil MadhavapeddyAnil Madhavapeddyhttps://www.cl.cam.ac.uk/teaching/2627/FoundsCS/forest/focs-ex-7-5/focs-ex-7-5/teaching/2627/FoundsCS/forest/focs-ex-7-5/Exercises › Implementing deletionExerciseCode the delete function outlined in the previous exercise.Anil MadhavapeddyAnil Madhavapeddyhttps://www.cl.cam.ac.uk/teaching/2627/FoundsCS/forest/focs-ex-7-1/focs-ex-7-1/teaching/2627/FoundsCS/forest/focs-ex-7-1/Exercises › Insertion order in search treesExerciseDraw the binary search tree that arises from successively inserting the following pairs into the empty tree: ("Alice", 6), ("Tobias", 2), ("Gerald", 8), ("Lucy", 9). Then repeat this task using the order ("Gerald", 8), ("Alice", 6), ("Lucy", 9), ("Tobias", 2). Why are results different?Anil MadhavapeddyAnil Madhavapeddyhttps://www.cl.cam.ac.uk/teaching/2627/FoundsCS/forest/focs-ex-7-2/focs-ex-7-2/teaching/2627/FoundsCS/forest/focs-ex-7-2/Exercises › Insertion with collision detectionExerciseCode an insertion function for binary search trees. It should resemble the existing update function except that it should raise the exception Collision if the item to be inserted is already present.Anil MadhavapeddyAnil Madhavapeddyhttps://www.cl.cam.ac.uk/teaching/2627/FoundsCS/forest/focs-ex-9-4/focs-ex-9-4/teaching/2627/FoundsCS/forest/focs-ex-9-4/Exercises › Lazy binary treesExerciseA lazy binary tree is either empty or is a branch containing a label and two lazy binary trees, possibly to infinite depth. Present an OCaml datatype to represent lazy binary trees, along with a function that accepts a lazy binary tree and produces a lazy list that contains all of the tree’s labels. (Taken from the exam question 2008 Paper 1 Question 5.)Anil MadhavapeddyAnil Madhavapeddyhttps://www.cl.cam.ac.uk/teaching/2627/FoundsCS/forest/focs-ex-7-7/focs-ex-7-7/teaching/2627/FoundsCS/forest/focs-ex-7-7/Exercises › Linear traversalsExerciseShow that the functions preord, inord and postord all take linear time in the size of the tree.Anil MadhavapeddyAnil Madhavapeddyhttps://www.cl.cam.ac.uk/teaching/2627/FoundsCS/forest/focs-ex-7-6/focs-ex-7-6/teaching/2627/FoundsCS/forest/focs-ex-7-6/Exercises › Quadratic traversalsExerciseShow that the functions preorder, inorder and postorder all require time in the worst case, where is the size of the tree.Anil MadhavapeddyAnil Madhavapeddyhttps://www.cl.cam.ac.uk/teaching/2627/FoundsCS/forest/focs-ex-10-3/focs-ex-10-3/teaching/2627/FoundsCS/forest/focs-ex-10-3/Exercises › Rewriting breadth with letExerciseWrite a version of the function breadth using a nested let construction rather than match.Anil MadhavapeddyAnil Madhavapeddyhttps://www.cl.cam.ac.uk/teaching/2627/FoundsCS/forest/focs-ex-7-8/focs-ex-7-8/teaching/2627/FoundsCS/forest/focs-ex-7-8/Exercises › Shifting a functional arrayExerciseWrite a function to remove the first element from a functional array. All the other elements are to have their subscripts reduced by one. The cost of this operation should be linear in the size of the array.Anil MadhavapeddyAnil Madhavapeddyhttps://www.cl.cam.ac.uk/teaching/2627/FoundsCS/forest/focs-ex-6-2/focs-ex-6-2/teaching/2627/FoundsCS/forest/focs-ex-6-2/Exercises › Summing tree labelsExerciseWrite an OCaml function taking a binary tree labelled with integers and returning their sum.Anil MadhavapeddyAnil Madhavapeddyhttps://www.cl.cam.ac.uk/teaching/2627/FoundsCS/forest/focs-ex-10-1/focs-ex-10-1/teaching/2627/FoundsCS/forest/focs-ex-10-1/Exercises › Tree-based queuesExerciseSuppose that we have an implementation of queues, based on binary trees, such that each operation takes logarithmic time in the worst case. Outline the advantages and drawbacks of such an implementation compared with one presented above.Anil MadhavapeddyAnil Madhavapeddyhttps://www.cl.cam.ac.uk/teaching/2627/FoundsCS/forest/focs-ex-6-3/focs-ex-6-3/teaching/2627/FoundsCS/forest/focs-ex-6-3/Exercises › What ftree buildsExerciseUsing the definition of 'a tree from before:Examine the following function declaration. What does ftree (1, n) accomplish?Anil Madhavapeddyhttps://www.cl.cam.ac.uk/teaching/2627/FoundsCS/forest/focs-topic-exceptions/focs-topic-exceptions/teaching/2627/FoundsCS/forest/focs-topic-exceptions/ExceptionsAnil Madhavapeddyhttps://www.cl.cam.ac.uk/teaching/2627/FoundsCS/forest/focs-exceptions/focs-exceptions/teaching/2627/FoundsCS/forest/focs-exceptions/Error Handling: ExceptionsDuring a computation, what happens if something goes wrong?Division by zero Pattern matching failureException-handling lets us recover gracefully.Raising an exception abandons the current computation. Handling the exception attempts an alternative computation. The raising and handling can be far apart in the code. Errors of different sorts can be handled separately.Exceptions are necessary because it is not always possible to tell in advance whether or not a search will lead to a dead end or whether a numerical calculation will encounter errors such as overflow or divide by zero. Rather than just crashing, programs should check whether things have gone wrong, and perhaps attempt an alternative computation (perhaps using a different algorithm or higher precision). A number of modern languages provide exception handling.Anil Madhavapeddyhttps://www.cl.cam.ac.uk/teaching/2627/FoundsCS/forest/focs-def-exception/focs-def-exception/teaching/2627/FoundsCS/forest/focs-def-exception/ExceptionDefinitionAn exception signals that something has gone wrong during a computation, such as division by zero or pattern matching failure. Raising an exception abandons the current computation; handling the exception attempts an alternative computation. The raising and the handling can be far apart in the code, and errors of different sorts can be handled separately. Introduced in Error Handling: Exceptions.Part of the course glossary.Anil Madhavapeddyhttps://www.cl.cam.ac.uk/teaching/2627/FoundsCS/forest/focs-ocaml-exceptions/focs-ocaml-exceptions/teaching/2627/FoundsCS/forest/focs-ocaml-exceptions/Exceptions in OCamlEach exception declaration introduces a distinct sort of exception, which can be handled separately from others. If raises an exception, then its evaluation has failed; handling an exception means evaluating another expression and returning its value instead. One exception handler can specify separate expressions for different sorts of exceptions.Exception names are constructors of the special datatype exn. This is a peculiarity of OCaml that lets exception-handlers use pattern-matching. Note that exception Failure is just an error indication, while NoChange n carries further information: the integer . print_endline "handled a NoChange exception" Line 3, characters 5-23: Warning 21 [nonreturning-statement]: this statement never returns (or has an unsound type.) pre exception handled a NoChange exception - : unit = ()]]>The effect of raise <expr> is to jump to the most recently-encountered handler that matches <expr>. The matching handler can only be found dynamically (during execution); contrast with how OCaml associates occurrences of identifiers with their matching declarations, which does not require running the program. A handler is introduced via the try keyword, which executes the subexpression and dispatches any exceptions encountered to the corresponding pattern match for exceptions defined in the with block.This is also the first time that we have encountered the unit type. This represents a type that has no values, and is used to indicate that a block has no meaningful return value. We will come back to this when learning more about imperative programming later on. For now, it is sufficient to understand that print_endline will print out the argument to the console output, and return a unit type. The semicolon at the end of the expression is a convenient way to execute sequential statements that return the unit type.One criticism of OCaml’s exceptions is that—unlike the Java language—nothing in a function declaration indicates which exceptions it might raise. One alternative to exceptions is to instead return a value of datatype option.None signifies an error, while Some x returns the solution . This approach looks clean, but the drawback is that many places in the code would have to check for None. Despite this, there is a builtin option type in OCaml as it is so useful. We will see in later lectures how to define our own version of option using polymorphic datatype definitions.Anil Madhavapeddyhttps://www.cl.cam.ac.uk/teaching/2627/FoundsCS/forest/focs-bst-lookup/focs-bst-lookup/teaching/2627/FoundsCS/forest/focs-bst-lookup/Lookup: Seeks Left or Rightfocs-binary-trees if b < a then lookup b t1 else if a < b then lookup b t2 else x | Lf -> raise (Missing b) val lookup : string -> (string * 'a) tree -> 'a = ]]>This has guaranteed access time if the tree is balanced!Lookup in the binary search tree goes to the left subtree if the desired key is smaller than the current one and to the right if it is greater. It raises Missing if it encounters an empty tree.Since an ordering is involved, we have to declare the functions for a specific type, here string. Now exception Missing mentions that type: if lookup fails, the exception returns the missing key. The exception could be eliminated using type option of our earlier Datatypes lecture, using the constructor None for failure.Anil Madhavapeddyhttps://www.cl.cam.ac.uk/teaching/2627/FoundsCS/forest/focs-change-exceptions/focs-change-exceptions/teaching/2627/FoundsCS/forest/focs-change-exceptions/Making Change with Exceptions [] | [], _ -> raise Change | c::till, amt -> if amt < 0 then raise Change else try c :: change (c::till) (amt - c) with Change -> change till amt exception Change val change : int list -> int -> int list = ]]>In the Lists lectures, we considered the problem of making change. The greedy algorithm presented there could not express “6 using 5 and 2” because it always took the largest coin. Returning the list of all possible solutions avoids that problem rather expensively: we only need one solution.Using exceptions, we can code a backtracking algorithm: one that can undo past decisions if it comes to a dead end. The exception Change is raised if we run out of coins (with a non-zero amount) or if the amount goes negative. We always try the largest coin, but enclose the recursive call in an exception handler, which undoes the choice if it goes wrong.Carefully observe how exceptions interact with recursion. The exception handler always undoes the most recent choice, leaving others possibly to be undone later. If making change really is impossible, then eventually exception Change will be raised with no handler to catch it, and it will be reported at top level.Anil Madhavapeddyhttps://www.cl.cam.ac.uk/teaching/2627/FoundsCS/forest/focs-change-trace/focs-change-trace/teaching/2627/FoundsCS/forest/focs-change-trace/Making Change: A TraceHere is the full execution. Observe how the exception handlers nest and how they drop away once the given expression has returned a value. change [2] 6}\\ \Rightarrow &\; \text{try 5::(try 5::change [5; 2] (-4)}\\ &\; \text{with Change -> change [2] 1)}\\ &\; \text{with Change -> change [2] 6}\\ \Rightarrow &\; \text{5::(change [2] 1)}\\ &\; \text{with Change -> change [2] 6}\\ \Rightarrow &\; \text{try 5::(try 2::change [2] (-1)}\\ &\; \text{with Change -> change [] 1)}\\ &\; \text{with Change -> change [2] 6} \\ \Rightarrow &\; \text{try 5::(change [] 1)}\\ &\; \text{with Change -> change [2] 6} \\ \Rightarrow &\; \text{change [2] 6} \\ \Rightarrow &\; \text{try 2::change [2] 4}\\ &\; \text{with Change -> change [] 6} \\ \Rightarrow &\; \text{try 2::(try 2::change [2] 2}\\ &\; \text{with Change -> change [] 4)}\\ &\; \text{with Change -> change [] 6} \\ \Rightarrow &\; \text{try 2::(try 2::(try 2::change [2] 0 }\\ &\; \text{with Change -> change [] 2)}\\ &\; \text{with Change -> change [] 4)}\\ &\; \text{with Change -> change [] 6} \\ \Rightarrow &\; \text{try 2::(try 2::[2]}\\ &\; \text{with Change -> change [] 4)}\\ &\; \text{with Change -> change [] 6} \\ \Rightarrow &\; \text{try 2::[2; 2]}\\ &\; \text{with Change -> change [] 6} \\ \Rightarrow &\; \text{[2; 2; 2]} \end{aligned}]]>Anil Madhavapeddyhttps://www.cl.cam.ac.uk/teaching/2627/FoundsCS/forest/focs-private-references/focs-private-references/teaching/2627/FoundsCS/forest/focs-private-references/Private, Persistent References !balance then raise (TooMuch (amt - !balance)) else begin balance := !balance - amt; !balance end in withdraw val makeAccount : int -> int -> int = ]]>As you may have noticed, OCaml’s programming style looks clumsy compared with that of languages like C. OCaml omits the defaults and abbreviations they provide to shorten programs. However, OCaml’s explicitness makes it ideal for teaching the fine points of references and arrays. OCaml’s references are more flexible than those found in other languages.The function makeAccount models a bank. Calling the function with a specified initial balance creates a new reference (balance) to maintain the account balance and returns a function (withdraw) having sole access to that reference. Calling withdraw reduces the balance by the specified amount and returns the new balance. You can pay money in by withdrawing a negative amount. The if-construct prevents the account from going overdrawn, raising an exception.Look at the sequence inside begin … end in the else part above. The first expression updates the account balance and returns the trivial value (). The second expression, !balance, returns the current balance but does not return the reference itself: that would allow unauthorised updates.This example is based on one by Dr A C Norman.Anil Madhavapeddyhttps://www.cl.cam.ac.uk/teaching/2627/FoundsCS/forest/focs-funarray-lookup/focs-funarray-lookup/teaching/2627/FoundsCS/forest/focs-funarray-lookup/The Lookup Function raise Subscript (* Not found *) | Br (v, t1, t2), k -> if k = 1 then v else if k mod 2 = 0 then sub (t1, k / 2) else sub (t2, k / 2) exception Subscript val sub : 'a tree * int -> 'a = # let rec sub = function (* Alternative implementation *) | Lf, _ -> raise Subscript | Br (v, t1, t2), 1 -> v | Br (v, t1, t2), k when k mod 2 = 0 -> sub (t1, k / 2) | Br (v, t1, t2), k -> sub (t2, k / 2) val sub : 'a tree * int -> 'a = ]]>Notice that we have used a new keyword when above, which changes pattern clauses to be only matched if the expression evalutes to true. This can be equivalently expressed by moving the corresponding checks into an if clause on the right hand side of the pattern match, but is often more readable using when (as above).The lookup function sub, divides the subscript by 2 until 1 is reached. If the remainder is 0 then the function follows the left subtree, otherwise the right. If it reaches a leaf, it signals error by raising exception Subscript.Array access can also be understood in terms of the subscript’s binary code. Because the subscript must be a positive integer, in binary it has a leading one. Discard this one and reverse the remaining bits. Interpreting zero as left and one as right yields the path from the root to the subscript.Popular literature often explains the importance of binary as being led by hardware: because a circuit is either on or off. The truth is almost the opposite. Designers of digital electronics go to a lot of trouble to suppress the continuous behaviour that would naturally arise. The real reason why binary is important is its role in algorithms: an if-then-else decision leads to binary branching.Data structures, such as trees, and algorithms, such as mergesort, use binary branching in order to reduce a cost from to . Two is the smallest integer divisor that achieves this reduction. (Larger divisors are only occasionally helpful, as in the case of B-trees, where they reduce the constant factor.) The simplicity of binary arithmetic compared with decimal arithmetic is just another instance of the simplicity of algorithms based on binary choices.Anil Madhavapeddyhttps://www.cl.cam.ac.uk/teaching/2627/FoundsCS/forest/focs-funarray-update/focs-funarray-update/teaching/2627/FoundsCS/forest/focs-funarray-update/The Update Function if k = 1 then Br (w, Lf, Lf) else raise Subscript (* Gap in tree *) | Br (v, t1, t2), k, w -> if k = 1 then Br (w, t1, t2) else if k mod 2 = 0 then Br (v, update (t1, k / 2, w), t2) else Br (v, t1, update (t2, k / 2, w)) val update : 'a tree * int * 'a -> 'a tree = ]]>The update function also divides the subscript repeatedly by two. When it reaches a value of one, it has identified the element position. Then it replaces the branch node by another branch with the new label.A leaf may be replaced by a branch, extending the array, provided no intervening nodes have to be generated. This suffices for arrays without gaps in their subscripting. (The data structure can be modified to allow sparse arrays, where most subscript positions are undefined.) Exception Subscript indicates that the subscript position does not exist and cannot be created. This use of exceptions is not easily replaced by None and Some.Note that there are two tests involving . If we have reached a leaf, it returns a branch, extending the array by one. If we are still at a branch node, then the effect is to update an existing array element.A similar function can shrink an array by one.Anil MadhavapeddyAnil Madhavapeddyhttps://www.cl.cam.ac.uk/teaching/2627/FoundsCS/forest/focs-ex-6-5/focs-ex-6-5/teaching/2627/FoundsCS/forest/focs-ex-6-5/Exercises › Evaluating expressions with exceptionsExerciseContinuing the previous exercise, write a function that evaluates an expression. If the expression contains any variables, your function should raise an exception indicating the variable name.Anil MadhavapeddyAnil Madhavapeddyhttps://www.cl.cam.ac.uk/teaching/2627/FoundsCS/forest/focs-ex-7-2/focs-ex-7-2/teaching/2627/FoundsCS/forest/focs-ex-7-2/Exercises › Insertion with collision detectionExerciseCode an insertion function for binary search trees. It should resemble the existing update function except that it should raise the exception Collision if the item to be inserted is already present.Anil MadhavapeddyAnil Madhavapeddyhttps://www.cl.cam.ac.uk/teaching/2627/FoundsCS/forest/focs-ex-7-3/focs-ex-7-3/teaching/2627/FoundsCS/forest/focs-ex-7-3/Exercises › Returning the old value on collisionExerciseContinuing the previous exercise, it would be natural for exceptional Collision to return the value previously stored in the dictionary. Why is that goal difficult to achieve?Anil Madhavapeddyhttps://www.cl.cam.ac.uk/teaching/2627/FoundsCS/forest/focs-topic-dictionaries/focs-topic-dictionaries/teaching/2627/FoundsCS/forest/focs-topic-dictionaries/DictionariesAnil Madhavapeddyhttps://www.cl.cam.ac.uk/teaching/2627/FoundsCS/forest/focs-def-abstract-data-type/focs-def-abstract-data-type/teaching/2627/FoundsCS/forest/focs-def-abstract-data-type/Abstract data typeDefinitionAn abstract type provides specified operations while hiding low-level details, such as the data structure used to represent them. Before choosing the internal representation for a data structure, you need to specify the full set of operations. Abstract types can be declared in any modern programming language: Java’s objects serve this role, as do OCaml’s modules. Introduced in Dictionaries, and applied to queues and stacks later on.Part of the course glossary.Anil Madhavapeddyhttps://www.cl.cam.ac.uk/teaching/2627/FoundsCS/forest/focs-arrays/focs-arrays/teaching/2627/FoundsCS/forest/focs-arrays/ArraysA conventional array is an indexed storage area. It is updated in place by the command a.(k) <- x The concept is inherently imperative. A functional array is a finite map from integers to data. Updating implies copying to return update(A, k, x) The new array equals A except that A.(k) = x. Can we do updates efficiently?The elements of a list can only be reached by counting from the front. Elements of a tree are reached by following a path from the root. An array hides such structural matters; its elements are uniformly designated by number. Immediate access to arbitrary parts of a data structure is called random access.Arrays are the dominant data structure in conventional programming languages. The ingenious use of arrays is the key to many of the great classical algorithms, such as Hoare’s original quicksort (the partition step) and Warshall’s transitive-closure algorithm.The drawback is that subscripting is a chief cause of programmer error. That is why arrays play little role in this introductory course.Functional arrays are described below in order to illustrate another way of using trees to organise data. Here is a summary of basic dictionary data structures in order of decreasing generality and increasing efficiency:Linear search: Most general, needing only equality on keys, but inefficient: linear time. Binary search: Needs an ordering on keys. Logarithmic access time in the average case, but our binary search trees are linear in the worst case. Array subscripting: Least general, requiring keys to be integers, but even worst-case time is logarithmic.Anil Madhavapeddyhttps://www.cl.cam.ac.uk/teaching/2627/FoundsCS/forest/focs-bst/focs-bst/teaching/2627/FoundsCS/forest/focs-bst/Binary Search TreesA dictionary associates values (here, numbers) with keys.Binary search trees are an important application of binary trees. They work for keys that have a total ordering, such as strings. Each branch of the tree carries a pair; its left subtree holds smaller keys; the right subtree holds greater keys. If the tree remains reasonably balanced, then update and lookup both take for a tree of size . These times hold in the average case; given random data, the tree is likely to remain balanced.At a given node, all keys in the left subtree are smaller (or equal) while all trees in the right subtree are greater.An unbalanced tree has a linear access time in the worst case. Examples include building a tree by repeated insertions of elements in increasing or decreasing order; there is a close resemblance to quicksort. Building a binary search tree, then converting it to inorder, yields a sorting algorithm called treesort.Self-balancing trees, such as Red-Black trees, attain in the worst case. They are complicated to implement.Anil Madhavapeddyhttps://www.cl.cam.ac.uk/teaching/2627/FoundsCS/forest/focs-def-binary-search-tree/focs-def-binary-search-tree/teaching/2627/FoundsCS/forest/focs-def-binary-search-tree/Binary search treeDefinitionA binary search tree works for keys that have a total ordering, such as strings. Each branch of the tree carries a pair; its left subtree holds smaller keys and its right subtree holds greater keys. If the tree remains reasonably balanced, then update and lookup both take for a tree of size . Introduced in Binary Search Trees.Part of the course glossary.Anil Madhavapeddyhttps://www.cl.cam.ac.uk/teaching/2627/FoundsCS/forest/focs-dictionary-adt/focs-dictionary-adt/teaching/2627/FoundsCS/forest/focs-dictionary-adt/Dictionarieslookup: find an item in the dictionary update (insert): replace (store) an item in the dictionary delete: remove an item from the dictionary empty: the null dictionary Missing: exception for errors in lookup and deleteIdeally, an abstract type should provide these operations and hide the internal data structures.A dictionary attaches values to identifiers, called “keys”. Before choosing the internal representation for a data structure, you need to specify the full set of operations. In fact, here we only consider update (associating a value with an identifier) and lookup (retrieving such a value). Deletion is more difficult and would limit our choices. Some applications may need additional operations, such as merge (combining two dictionaries). We shall see that update can be done efficiently in a functional style, without excessive copying.An abstract type provides specified operations while hiding low-level details, such as the data structure used to represent dictionaries. Abstract types can be declared in any modern programming language. Java’s objects serve this role, as do OCaml’s modules. This course does not cover modules, and we simply declare the dictionary operations individually.An association list (a list of pairs) is the simplest dictionary representation. Lookup is by linear search, and therefore slow: . Association lists are only usable if there are few keys in use. However, they are general in that the keys do not need a concept of ordering, only equality. raise Missing | (x, y) :: pairs -> if a = x then y else lookup a pairs val lookup : 'a -> ('a * 'b) list -> 'b = # let update (l, b, y) = (b, y) :: l val update : ('a * 'b) list * 'a * 'b -> ('a * 'b) list = ]]>To enter a new (key, value) pair, simply “cons” it to the list with update. This takes constant time, which is the best we could hope for. But the space requirement is huge: linear in the number of updates, not in the number of distinct keys. Obsolete entries are never deleted: that would require first finding them, increasing the update time from to .Anil Madhavapeddyhttps://www.cl.cam.ac.uk/teaching/2627/FoundsCS/forest/focs-bst-lookup/focs-bst-lookup/teaching/2627/FoundsCS/forest/focs-bst-lookup/Lookup: Seeks Left or Rightfocs-binary-trees if b < a then lookup b t1 else if a < b then lookup b t2 else x | Lf -> raise (Missing b) val lookup : string -> (string * 'a) tree -> 'a = ]]>This has guaranteed access time if the tree is balanced!Lookup in the binary search tree goes to the left subtree if the desired key is smaller than the current one and to the right if it is greater. It raises Missing if it encounters an empty tree.Since an ordering is involved, we have to declare the functions for a specific type, here string. Now exception Missing mentions that type: if lookup fails, the exception returns the missing key. The exception could be eliminated using type option of our earlier Datatypes lecture, using the constructor None for failure.Anil Madhavapeddyhttps://www.cl.cam.ac.uk/teaching/2627/FoundsCS/forest/focs-bst-update/focs-bst-update/teaching/2627/FoundsCS/forest/focs-bst-update/Update Br ((k, v), Lf, Lf) | Br ((a, x), t1, t2) -> if k < a then Br ((a, x), update k v t1, t2) else if a < k then Br ((a, x), t1, update k v t2) else (* a = k *) Br ((a, v), t1, t2) val update : 'a -> 'b -> ('a * 'b) tree -> ('a * 'b) tree = ]]>This is also as it copies the path only, and not whole subtrees!If you are familiar with the usual update operation for this sort of tree, you may wonder whether it can be implemented in OCaml, where there is no direct way to replace part of a data structure by something else.The update operation is a nice piece of functional programming. It searches in the same manner as lookup, but the recursive calls reconstruct a new tree around the result of the update. One subtree is updated and the other left unchanged. The internal representation of trees ensures that unchanged parts of the tree are not copied, but shared. Therefore, update copies only the path from the root to the new node. Its time and space requirements, for a reasonably balanced tree, are both .The comparison between and allows three cases:smaller: update the left subtree; share the right greater: update the right subtree; share the left equal: update the label and share both subtreesNote: in the function definition, (* a = k *) is a comment. Comments in OCaml are enclosed in the brackets and .Anil MadhavapeddyAnil Madhavapeddyhttps://www.cl.cam.ac.uk/teaching/2627/FoundsCS/forest/focs-ex-7-4/focs-ex-7-4/teaching/2627/FoundsCS/forest/focs-ex-7-4/Exercises › Deleting from a search treeExerciseDescribe an algorithm for deleting an entry from a binary search tree. Comment on the suitability of your approach.Anil MadhavapeddyAnil Madhavapeddyhttps://www.cl.cam.ac.uk/teaching/2627/FoundsCS/forest/focs-ex-7-5/focs-ex-7-5/teaching/2627/FoundsCS/forest/focs-ex-7-5/Exercises › Implementing deletionExerciseCode the delete function outlined in the previous exercise.Anil MadhavapeddyAnil Madhavapeddyhttps://www.cl.cam.ac.uk/teaching/2627/FoundsCS/forest/focs-ex-7-1/focs-ex-7-1/teaching/2627/FoundsCS/forest/focs-ex-7-1/Exercises › Insertion order in search treesExerciseDraw the binary search tree that arises from successively inserting the following pairs into the empty tree: ("Alice", 6), ("Tobias", 2), ("Gerald", 8), ("Lucy", 9). Then repeat this task using the order ("Gerald", 8), ("Alice", 6), ("Lucy", 9), ("Tobias", 2). Why are results different?Anil MadhavapeddyAnil Madhavapeddyhttps://www.cl.cam.ac.uk/teaching/2627/FoundsCS/forest/focs-ex-7-2/focs-ex-7-2/teaching/2627/FoundsCS/forest/focs-ex-7-2/Exercises › Insertion with collision detectionExerciseCode an insertion function for binary search trees. It should resemble the existing update function except that it should raise the exception Collision if the item to be inserted is already present.Anil MadhavapeddyAnil Madhavapeddyhttps://www.cl.cam.ac.uk/teaching/2627/FoundsCS/forest/focs-ex-7-3/focs-ex-7-3/teaching/2627/FoundsCS/forest/focs-ex-7-3/Exercises › Returning the old value on collisionExerciseContinuing the previous exercise, it would be natural for exceptional Collision to return the value previously stored in the dictionary. Why is that goal difficult to achieve?Anil Madhavapeddyhttps://www.cl.cam.ac.uk/teaching/2627/FoundsCS/forest/focs-topic-arrays/focs-topic-arrays/teaching/2627/FoundsCS/forest/focs-topic-arrays/ArraysAnil Madhavapeddyhttps://www.cl.cam.ac.uk/teaching/2627/FoundsCS/forest/focs-array-examples/focs-array-examples/teaching/2627/FoundsCS/forest/focs-array-examples/Array ExamplesIn the following session, the identifier ar is bound to an array of 20 elements, which are initially set to the squares of their subscripts. The array’s third element (which actually has subscript 2) is inspected and found to be four. The second call to Array.get supplies a subscript that is out of range, so OCaml rejects it. i * i) val ar : int array = [|0; 1; 4; 9; 16; 25; 36; 49; 64; 81; 100; 121; 144; 169; 196; 225; 256; 289; 324; 361|] # Array.get ar 2 - : int = 4 # Array.get ar 20 Exception: Invalid_argument "index out of bounds". # Array.set ar 2 33; ar - : int array = [|0; 1; 33; 9; 16; 25; 36; 49; 64; 81; 100; 121; 144; 169; 196; 225; 256; 289; 324; 361|]]]>By calling Array.set, we then modify the element with subscript 2. Note however that we cannot modify the array’s length. If we outgrow the array, we have to create a new one, copy the data into it, and then forget the old array. Typically the new array would be double the size of the old one, so that the cost of copying is insignificant.OCaml provides numerous operators for modifying, computing over and searching in arrays. Many are analogous to functions on lists. For example, Array.exists takes a boolean-valued function and returns true if an array element satisfies it. i > 200) ar - : bool = true # Array.exists (fun i -> i < 0) ar - : bool = false]]>Anil Madhavapeddyhttps://www.cl.cam.ac.uk/teaching/2627/FoundsCS/forest/focs-arrays/focs-arrays/teaching/2627/FoundsCS/forest/focs-arrays/ArraysA conventional array is an indexed storage area. It is updated in place by the command a.(k) <- x The concept is inherently imperative. A functional array is a finite map from integers to data. Updating implies copying to return update(A, k, x) The new array equals A except that A.(k) = x. Can we do updates efficiently?The elements of a list can only be reached by counting from the front. Elements of a tree are reached by following a path from the root. An array hides such structural matters; its elements are uniformly designated by number. Immediate access to arbitrary parts of a data structure is called random access.Arrays are the dominant data structure in conventional programming languages. The ingenious use of arrays is the key to many of the great classical algorithms, such as Hoare’s original quicksort (the partition step) and Warshall’s transitive-closure algorithm.The drawback is that subscripting is a chief cause of programmer error. That is why arrays play little role in this introductory course.Functional arrays are described below in order to illustrate another way of using trees to organise data. Here is a summary of basic dictionary data structures in order of decreasing generality and increasing efficiency:Linear search: Most general, needing only equality on keys, but inefficient: linear time. Binary search: Needs an ordering on keys. Logarithmic access time in the average case, but our binary search trees are linear in the worst case. Array subscripting: Least general, requiring keys to be integers, but even worst-case time is logarithmic.Anil Madhavapeddyhttps://www.cl.cam.ac.uk/teaching/2627/FoundsCS/forest/focs-functional-arrays/focs-functional-arrays/teaching/2627/FoundsCS/forest/focs-functional-arrays/Functional Arrays as Binary TreesThe path to element follows the binary code for (its “subscript”).This simple representation (credited to W. Braun) ensures that the tree is balanced. Complexity of access is always , which is optimal. For actual running time, access to conventional arrays is much faster: it requires only a few hardware instructions. Array access is often taken to be , which (as always) presumes that hardware limits are never exceeded.The lower bound for array subscripts (or “indices”) is one. The upper bound starts at zero (which signifies the empty array) and can grow without limit. Inspection of the diagram above should make it clear that these trees are always balanced: the left subtree can have at most one node more than the right subtree, recursively all the way down. (This assumes that the array is defined for subscripts with no gaps; an array defined only for odd numbers, for example, would obviously be unbalanced.)The numbers in the diagram above are not the labels of branch nodes, but indicate the positions of array elements. For example, the label corresponding to is at the position shown. The nodes of a functional array are labelled with the data we want to store, not with these integers.Anil Madhavapeddyhttps://www.cl.cam.ac.uk/teaching/2627/FoundsCS/forest/focs-array-primitives/focs-array-primitives/teaching/2627/FoundsCS/forest/focs-array-primitives/OCaml Primitives for Arrays i * 10) (* array[5] initialised to (fun i) *) val aa : int array = [|0; 10; 20; 30; 40|] # Array.get aa 3 (* retrieve the 4th cell in the array *) - : int = 30 # Array.set aa 3 42 (* set the 4th cell's value to 42 *) - : unit = ()]]>There are many other array operations in the Array module in the OCaml standard library. 'a -> 'a array = # Array.init - : int -> (int -> 'a) -> 'a array = # Array.get - : 'a array -> int -> 'a = # Array.set - : 'a array -> int -> 'a -> unit = ]]>OCaml arrays are like references that hold several elements instead of one. The elements of an -element array are designated by the integers from 0 to . The th array element is usually written . If is a type then array is the type of arrays (of any size) with elements from .Calling Array.init n f creates an array of the size specified in by function . Initially, element holds the value of for , …, . Like ref, it allocates mutable storage to hold the specified values.Calling Array.get A i returns the contents of .Calling Array.set A i E modifies the array by storing the value of as the new contents of ; it returns () as its value.OCaml’s arrays are much safer than C’s. In C, an array is nothing more than an address indicating the start of a storage area. Nothing indicates the size of the area. Therefore C programs are vulnerable to buffer overrun attacks: an attacker sends more data than the receiving program expects, overrunning the area of storage set aside to hold it. The attack eventually overwrites the program itself, replacing it with code controlled by the attacker.Anil Madhavapeddyhttps://www.cl.cam.ac.uk/teaching/2627/FoundsCS/forest/focs-references-comparison/focs-references-comparison/teaching/2627/FoundsCS/forest/focs-references-comparison/References: OCaml vs conventional languagesWe must write !p to get the contents of p We write just p for the address of p We can store private reference cells (like balance) in functions—analogous to elements of object-oriented programming OCaml’s assignment syntax is instead of = OCaml has few control structures: while, match, if and for (the latter is not covered in this course) OCaml has syntax for updating an array via the a.(i) <- v syntax which is the same as Array.set a i v.Conventional syntax for variables and assignments has hardly changed since Fortran, the first high-level language. In conventional languages, virtually all variables can be updated. We declare something like p: int, mentioning no reference type even if the language provides them. If we do not specify an initial value, we may get whatever bits were previously at that address. Illegal values arising from uninitialised variables can cause errors that are almost impossible to diagnose.Dereferencing operators (like OCaml’s !) are especially unpopular, because they clutter the program text. Virtually all programming languages make dereferencing implicit (that is, automatic).It is generally accepted these days that a two-dimensional array is nothing but an array of arrays. An assignment to such an array is typically written something like ; in C, the syntax is A[i][j] = x. Higher dimensions are treated analogously. The corresponding OCaml code can either declare an array of arrays, or use the A.(i) syntax to calculate the linear offset into a single array.You can use the constructs we have learnt to easily create linked (mutable) lists as an alternative to arrays.It is worth mentioning that OCaml’s references fully suffice for coding the sort of linked data structures taught in algorithms courses, as the mlist type above illustrates. The programming style is a little different from the usual, but the principles are the same. OCaml also provides comprehensive input/output primitives for various types of file and operating system.OCaml’s system of modules include structures, which can be seen as encapsulated groups of declarations, and signatures, which are specifications of structures listing the name and type of each component. Finally, there are functors, which are analogous to functions that combine a number of argument structures, and which can be used to plug program components together. These primitives are useful for managing large programming projects.Anil Madhavapeddyhttps://www.cl.cam.ac.uk/teaching/2627/FoundsCS/forest/focs-funarray-lookup/focs-funarray-lookup/teaching/2627/FoundsCS/forest/focs-funarray-lookup/The Lookup Function raise Subscript (* Not found *) | Br (v, t1, t2), k -> if k = 1 then v else if k mod 2 = 0 then sub (t1, k / 2) else sub (t2, k / 2) exception Subscript val sub : 'a tree * int -> 'a = # let rec sub = function (* Alternative implementation *) | Lf, _ -> raise Subscript | Br (v, t1, t2), 1 -> v | Br (v, t1, t2), k when k mod 2 = 0 -> sub (t1, k / 2) | Br (v, t1, t2), k -> sub (t2, k / 2) val sub : 'a tree * int -> 'a = ]]>Notice that we have used a new keyword when above, which changes pattern clauses to be only matched if the expression evalutes to true. This can be equivalently expressed by moving the corresponding checks into an if clause on the right hand side of the pattern match, but is often more readable using when (as above).The lookup function sub, divides the subscript by 2 until 1 is reached. If the remainder is 0 then the function follows the left subtree, otherwise the right. If it reaches a leaf, it signals error by raising exception Subscript.Array access can also be understood in terms of the subscript’s binary code. Because the subscript must be a positive integer, in binary it has a leading one. Discard this one and reverse the remaining bits. Interpreting zero as left and one as right yields the path from the root to the subscript.Popular literature often explains the importance of binary as being led by hardware: because a circuit is either on or off. The truth is almost the opposite. Designers of digital electronics go to a lot of trouble to suppress the continuous behaviour that would naturally arise. The real reason why binary is important is its role in algorithms: an if-then-else decision leads to binary branching.Data structures, such as trees, and algorithms, such as mergesort, use binary branching in order to reduce a cost from to . Two is the smallest integer divisor that achieves this reduction. (Larger divisors are only occasionally helpful, as in the case of B-trees, where they reduce the constant factor.) The simplicity of binary arithmetic compared with decimal arithmetic is just another instance of the simplicity of algorithms based on binary choices.Anil Madhavapeddyhttps://www.cl.cam.ac.uk/teaching/2627/FoundsCS/forest/focs-funarray-update/focs-funarray-update/teaching/2627/FoundsCS/forest/focs-funarray-update/The Update Function if k = 1 then Br (w, Lf, Lf) else raise Subscript (* Gap in tree *) | Br (v, t1, t2), k, w -> if k = 1 then Br (w, t1, t2) else if k mod 2 = 0 then Br (v, update (t1, k / 2, w), t2) else Br (v, t1, update (t2, k / 2, w)) val update : 'a tree * int * 'a -> 'a tree = ]]>The update function also divides the subscript repeatedly by two. When it reaches a value of one, it has identified the element position. Then it replaces the branch node by another branch with the new label.A leaf may be replaced by a branch, extending the array, provided no intervening nodes have to be generated. This suffices for arrays without gaps in their subscripting. (The data structure can be modified to allow sparse arrays, where most subscript positions are undefined.) Exception Subscript indicates that the subscript position does not exist and cannot be created. This use of exceptions is not easily replaced by None and Some.Note that there are two tests involving . If we have reached a leaf, it returns a branch, extending the array by one. If we are still at a branch node, then the effect is to update an existing array element.A similar function can shrink an array by one.Anil MadhavapeddyAnil Madhavapeddyhttps://www.cl.cam.ac.uk/teaching/2627/FoundsCS/forest/focs-ex-10-2/focs-ex-10-2/teaching/2627/FoundsCS/forest/focs-ex-10-2/Exercises › Array-based circular queuesExerciseThe traditional way to implement queues uses a fixed-length array. Two indices into the array indicate the start and end of the queue, which wraps around from the end of the array to the start. How appropriate is such a data structure for implementing breadth-first search?Anil MadhavapeddyAnil Madhavapeddyhttps://www.cl.cam.ac.uk/teaching/2627/FoundsCS/forest/focs-ex-11-5/focs-ex-11-5/teaching/2627/FoundsCS/forest/focs-ex-11-5/Exercises › Matrix operations on arraysExerciseArrays of multiple dimensions are represented in OCaml by arrays of arrays. Write functions to (a) create an identity matrix, given , and (b) to transpose an matrix. Identity matrices have the following form:Anil MadhavapeddyAnil Madhavapeddyhttps://www.cl.cam.ac.uk/teaching/2627/FoundsCS/forest/focs-ex-7-8/focs-ex-7-8/teaching/2627/FoundsCS/forest/focs-ex-7-8/Exercises › Shifting a functional arrayExerciseWrite a function to remove the first element from a functional array. All the other elements are to have their subscripts reduced by one. The cost of this operation should be linear in the size of the array.Anil Madhavapeddyhttps://www.cl.cam.ac.uk/teaching/2627/FoundsCS/forest/focs-topic-higher-order-functions/focs-topic-higher-order-functions/teaching/2627/FoundsCS/forest/focs-topic-higher-order-functions/Higher order functionsAnil Madhavapeddyhttps://www.cl.cam.ac.uk/teaching/2627/FoundsCS/forest/focs-predicate-applications/focs-predicate-applications/teaching/2627/FoundsCS/forest/focs-predicate-applications/Applications of the Predicate Functionals x=y) xs val member : 'a -> 'a list -> bool = # let inter xs ys = filter (fun x -> member x ys) xs val inter : 'a list -> 'a list -> 'a list = ]]>Testing whether two lists have no common elements all (fun y -> x<>y) ys) xs val disjoint : 'a list -> 'a list -> bool = ]]>The Lists lecture presented the function member, which tests whether a specified value can be found as a list element, and inter, which returns the “intersection” of two lists: the list of elements they have in common.But remember: the purpose of list functionals is not to replace the declarations of popular functions, which probably are available already. It is to eliminate the need for separate declarations of ad-hoc functions. When they are nested, like the calls to all in disjoint above, the inner functions are almost certainly one-offs, not worth declaring separately.Our primitives themselves can be seen as a programming language. Part of the task of programming is to extend our programming language with notation for solving the problem at hand. The levels of notation that we define should correspond to natural levels of abstraction in the problem domain.Historical Note: Alonzo Church’s -calculus gave a simple syntax, -notation, for expressing functions. It is the direct precursor of OCaml’s fun-notation. It was soon shown that his system was equivalent in computational power to Turing machines, and Church’s thesis states that this defines precisely the set of functions that can be computed effectively.The -calculus had a tremendous influence on the design of functional programming languages. McCarthy’s Lisp was something of a false start; it interpreted variable binding incorrectly, an error that stood for some 20 years. But in 1966, Peter Landin (of Queen Mary College, University of London) sketched out the main features of functional languages.Anil Madhavapeddyhttps://www.cl.cam.ac.uk/teaching/2627/FoundsCS/forest/focs-currying/focs-currying/teaching/2627/FoundsCS/forest/focs-currying/Curried FunctionsA curried function returns another function as its result. We use the string concatenation operator (^) to illustrate how this works. string -> string = # let prefix = fun a -> fun b -> a ^ b val prefix : string -> string -> string = # let promote = prefix "Senior " val promote : string -> string = # prefix "Junior " "Professor" - : string = "Junior Professor" # promote "Professor" - : string = "Senior Professor"]]>A short form for the definition of prefix is simply to pass multiple arguments to the function definition. The following two definitions are equivalent in OCaml: fun b -> a ^ b val prefix : string -> string -> string = # let prefix a b = a ^ b val prefix : string -> string -> string = ]]>Currying is the technique of expressing a function taking multiple arguments as nested functions, each taking a single argument. The fun-notation lets us package n*2 as the function fun n -> n * 2, but what if there are several variables, as in fun n -> n * 2 + k? A function of two arguments could be coded using pattern-matching on pairs, writing fun (n, k) -> n * 2 + k.Currying is an alternative, where we nest the fun-notation: fun n -> n * 2 + k - : int -> int -> int = ]]>Applying this curried function to the argument 1 yields another function, in which k has been replaced by 1: fun n -> n * 2 + k val fn : int -> int -> int = # let fn' = fn 1 (* n * 2 + 1 *) val fn' : int -> int = # fn' 3 (* 3 * 2 + 1 *) - : int = 7]]>And this function, when applied to 3, yields the result 7. The two arguments are supplied one after another.The earlier prefix example is similar and uses the expression a ^ b, where ^ is the infix operator for string concatenation. Function promote binds the first argument of prefix to "Senior "; the resulting function prefixes that title to any string to which it is applied.Anil Madhavapeddyhttps://www.cl.cam.ac.uk/teaching/2627/FoundsCS/forest/focs-def-currying/focs-def-currying/teaching/2627/FoundsCS/forest/focs-def-currying/CurryingDefinitionCurrying is the technique of expressing a function taking multiple arguments as nested functions, each taking a single argument. A curried function returns another function as its result, so the arguments are supplied one after another. Introduced in Curried Functions.Part of the course glossary.Anil Madhavapeddyhttps://www.cl.cam.ac.uk/teaching/2627/FoundsCS/forest/focs-transpose/focs-transpose/teaching/2627/FoundsCS/forest/focs-transpose/Example: Matrix Transpose [] | rows -> (map List.hd rows) :: (transp (map List.tl rows)) val transp : 'a list list -> 'a list list = ]]>A matrix can be viewed as a list of rows, each row a list of matrix elements. This representation is not especially efficient compared with the conventional one (using arrays). Lists of lists turn up often, though, and we can see how to deal with them by taking familiar matrix operations as examples. ML for the Working Programmer goes as far as Gaussian elimination, which presents surprisingly few difficulties.The transpose of the matrix is , which in OCaml corresponds to the following transformation on lists of lists:The workings of function transp are simple. If rows is the matrix to be transposed, then map hd extracts its first column and map tl extracts its second column:A recursive call transposes the latter matrix, which is then given the column [a; d] as its first row. The two functions expressed using map would otherwise have to be declared separately.Anil Madhavapeddyhttps://www.cl.cam.ac.uk/teaching/2627/FoundsCS/forest/focs-seq-functionals/focs-seq-functionals/teaching/2627/FoundsCS/forest/focs-seq-functionals/Functionals for Lazy ListsFiltering lazy lists: Nil | Cons (x, xf) -> if p x then Cons (x, fun () -> filterq p (xf ())) else filterq p (xf ()) val filterq : ('a -> bool) -> 'a seq -> 'a seq = ]]>The infinite sequence , , , … iterates f (f x)) val iterates : ('a -> 'a) -> 'a -> 'a seq = ]]>The functional filterq demands elements of xq until it finds one satisfying p. (Recall filter, the analogous operation for ordinary lists.) It contains a force not protected by a delay. If xq is infinite and contains no satisfactory element, then filterq runs forever.The functional iterates generalises from. It creates the next element not by adding one but by calling the function f.Anil Madhavapeddyhttps://www.cl.cam.ac.uk/teaching/2627/FoundsCS/forest/focs-anonymous-functions/focs-anonymous-functions/teaching/2627/FoundsCS/forest/focs-anonymous-functions/Functions Without NamesIf functions are to be regarded as computational values, then we need a notation for them. The fun notation expresses a non-recursive function value without giving the function a name. is the function such that . The function fun n -> n*2 is a doubling function. n * 2 - : int -> int = # (fun n -> n * 2) 17 - : int = 34]]>The main purpose of fun-notation is to package up small expressions that are to be applied repeatedly using some other function. The expression fun n -> n*2 has the same value as the identifier double, declared as follows: int = ]]>The fun notation can also do pattern matching, and the function keyword adds an anonymous variable name to pattern match against. The following functions are all equivalent, with the latter definitions bound to the is_zero value and the earlier ones anonymous: match x with 0 -> true | _ -> false - : int -> bool = # function 0 -> true | _ -> false - : int -> bool = # let is_zero = fun x -> match x with 0 -> true | _ -> false val is_zero : int -> bool = # let is_zero = function 0 -> true | _ -> false val is_zero : int -> bool = ]]>Anil Madhavapeddyhttps://www.cl.cam.ac.uk/teaching/2627/FoundsCS/forest/focs-def-higher-order-function/focs-def-higher-order-function/teaching/2627/FoundsCS/forest/focs-def-higher-order-function/Higher-order functionDefinitionIn mathematics, a functional or higher-order function is a function that operates on other functions. In OCaml, functions can be passed as arguments to other functions, returned as results, and put into lists, trees and so on—but not tested for equality. Introduced in Functions as Values.Part of the course glossary.Anil Madhavapeddyhttps://www.cl.cam.ac.uk/teaching/2627/FoundsCS/forest/focs-predicate-functionals/focs-predicate-functionals/teaching/2627/FoundsCS/forest/focs-predicate-functionals/List Functionals for Predicates false | x::xs -> (p x) || (exists p xs) val exists : ('a -> bool) -> 'a list -> bool = # let rec filter p = function | [] -> [] | x::xs -> if p x then x :: filter p xs else filter p xs val filter : ('a -> bool) -> 'a list -> 'a list = ]]>A predicate is a boolean-valued function.The functional exists transforms a predicate into a predicate over lists. Given a list, exists p tests whether or not some list element satisfies p (making it return true). If it finds one, it stops searching immediately, thanks to the behaviour of the lazy || operator.Dually, we have a functional to test whether all list elements satisfy the predicate. If it finds a counterexample then it, too, stops searching. true | x::xs -> (p x) && all p xs val all : ('a -> bool) -> 'a list -> bool = ]]>The filter functional, like map, transforms lists. It applies a predicate to all the list elements, but instead of returning the resulting values (which could only be true or false), it returns the list of elements satisfying the predicate.Anil Madhavapeddyhttps://www.cl.cam.ac.uk/teaching/2627/FoundsCS/forest/focs-matmul-ocaml/focs-matmul-ocaml/teaching/2627/FoundsCS/forest/focs-matmul-ocaml/Matrix Multiplication in OCamlDot product of two vectors—a curried function 0.0 | x::xs, y::ys -> (x *. y) +. (dotprod xs ys) Lines 2-4, characters 5-51: Warning 8 [partial-match]: this pattern-matching is not exhaustive. Here is an example of a case that is not matched: ([], _::_) val dotprod : float list -> float list -> float = ]]>Matrix product map (dotprod row) cols) arows val matprod : float list list -> float list list -> float list list = ]]>The transp brows converts into a list of columns, yielding a list whose elements are the columns of . Each row of is obtained by multiplying a row of by the columns of .Because dotprod is curried, it can be applied to a row of . The resulting function is applied to all the columns of . We have another example of currying and partial application.The outer map applies dotprod to each row of . The inner map, using fun-notation, applies dotprod row to each column of . Compare with the version in ML for the Working Programmer (page 89) which does not use map and requires two additional function declarations.In the dot product function, the two vectors must have the same length. Otherwise, exception Match_failure is raised.Anil Madhavapeddyhttps://www.cl.cam.ac.uk/teaching/2627/FoundsCS/forest/focs-seq-numerical/focs-seq-numerical/teaching/2627/FoundsCS/forest/focs-seq-numerical/Numerical Computations on Infinite Sequences float -> float = ]]>Close enough? match xf () with | Cons (y, yf) -> if abs_float (x -. y) <= eps then y else within eps (Cons (y, yf)) Lines 3-6, characters 7-41: Warning 8 [partial-match]: this pattern-matching is not exhaustive. Here is an example of a case that is not matched: Nil Lines 1-6, characters 22-41: Warning 8 [partial-match]: this pattern-matching is not exhaustive. Here is an example of a case that is not matched: Nil val within : float -> float seq -> float = ]]>Square Roots: float = ]]>The Newton-Raphson method is widely used for computing square roots. The infinite series converges rapidly to . The initial approximation, , is typically retrieved from a table, and is accurate enough that only a few iterations of the method are necessary. Calling iterates (next a) x0 generates the infinite series of approximations to the square root of using the Newton-Raphson method. To compute , the resulting series begins 1, 1.5, 1.41667, 1.4142157, 1.414213562, …, and this last figure is already accurate to 10 significant digits!Function within searches down the lazy list for two points whose difference is less than eps. It tests their absolute difference. Relative difference and other “close enough” tests can be coded. Such components can be used to implement other numerical functions directly as functions over sequences. The point is to build programs from small, interchangeable parts.Function root uses within, iterates and next to to apply Newton-Raphson with a tolerance of and a (poor) initial approximation of 1.0.This treatment of numerical computation has received some attention in the research literature; a recurring example is Richardson extrapolation.Anil Madhavapeddyhttps://www.cl.cam.ac.uk/teaching/2627/FoundsCS/forest/focs-partial-application/focs-partial-application/teaching/2627/FoundsCS/forest/focs-partial-application/Partial Application: A Curried Insertion Sort [x] | y::ys -> if lessequal x y then x :: y :: ys else y :: ins x ys in let rec sort = function | [] -> [] | x::xs -> ins x (sort xs) in sort val insort : ('a -> 'a -> bool) -> 'a list -> 'a list = ]]>The sorting functions we discussed in earlier lectures are coded to sort floating-point numbers. They can be generalised to an arbitrary ordered type by passing the ordering predicate lessequal as an argument.Functions ins and sort are declared locally, referring to lessequal. Though it may not be obvious, insort is a curried function. Given its first argument, a predicate for comparing some particular type of items, it returns the function sort for sorting lists of that type of items.Some examples of its use:=) [5; 3; 9; 8] - : int list = [9; 8; 5; 3]]]>An obscure point: the syntax (<=) denotes the comparison operator as a function, which is then given to insort. Passing the relation for lessequal gives a decreasing sort. This is no coding trick; it is justified in mathematics, since if is a partial ordering then so is .Anil Madhavapeddyhttps://www.cl.cam.ac.uk/teaching/2627/FoundsCS/forest/focs-private-references/focs-private-references/teaching/2627/FoundsCS/forest/focs-private-references/Private, Persistent References !balance then raise (TooMuch (amt - !balance)) else begin balance := !balance - amt; !balance end in withdraw val makeAccount : int -> int -> int = ]]>As you may have noticed, OCaml’s programming style looks clumsy compared with that of languages like C. OCaml omits the defaults and abbreviations they provide to shorten programs. However, OCaml’s explicitness makes it ideal for teaching the fine points of references and arrays. OCaml’s references are more flexible than those found in other languages.The function makeAccount models a bank. Calling the function with a specified initial balance creates a new reference (balance) to maintain the account balance and returns a function (withdraw) having sole access to that reference. Calling withdraw reduces the balance by the specified amount and returns the new balance. You can pay money in by withdrawing a negative amount. The if-construct prevents the account from going overdrawn, raising an exception.Look at the sequence inside begin … end in the else part above. The first expression updates the account balance and returns the trivial value (). The second expression, !balance, returns the current balance but does not return the reference itself: that would allow unauthorised updates.This example is based on one by Dr A C Norman.Anil Madhavapeddyhttps://www.cl.cam.ac.uk/teaching/2627/FoundsCS/forest/focs-curry-shorthand/focs-curry-shorthand/teaching/2627/FoundsCS/forest/focs-curry-shorthand/Shorthand for Curried FunctionsA function-returning function is just a function of two arguments.This curried function syntax is nicer than nested fun binders: string -> string = # let dub = prefix "Sir " val dub : string -> string = ]]>Curried functions allows partial application (to the first argument).In OCaml, an -argument curried function f can be declared using the syntax:and applied using the syntax:If f is not recursive, then it is equivalent to the function expressed via nesting as follows:We now have two ways of expressing functions of multiple arguments: either by passing a pair of arguments or by currying. Currying allows partial application which is useful when fixing the first argument yields a function that is interesting in its own right. An example from mathematics is the function , where fixing yields a function in alone, namely squaring. Similarly, yields cubing, while yields the identity function.Though the function hd (which returns the head of a list) is not curried, it may be used with the curried application syntax in some expressions:Here List.hd is applied to a list of functions, and the resulting function dub is then applied to the string "Hamilton". The idea of executing code stored in data structures reaches its full development in object-oriented programming, like in Java.Anil Madhavapeddyhttps://www.cl.cam.ac.uk/teaching/2627/FoundsCS/forest/focs-bank-accounts/focs-bank-accounts/teaching/2627/FoundsCS/forest/focs-bank-accounts/Two Bank Accounts int = # let director = makeAccount 4000000 val director : int -> int = # student 5 (* coach fare *) - : int = 495 # director 150000 (* Tesla *) - : int = 3850000 # student 500 (* oh oh *) Exception: TooMuch 5.]]>Each call to makeAccount returns a copy of withdraw holding a fresh instance of the reference balance. As with a real bank pass-book, there is no access to the account balance except via the corresponding withdraw function. If that function is discarded, the reference cell becomes unreachable; the computer will eventually reclaim it, just as banks close down dormant accounts.Here we see two people managing their accounts. For better or worse, neither can take money from the other.We could generalise makeAccount to return several functions that jointly manage information held in shared references. The functions might be packaged using OCaml records, which are not discussed in this course. Most procedural languages do not properly support the concept of private references, although object-oriented languages take them as a basic theme.Anil Madhavapeddyhttps://www.cl.cam.ac.uk/teaching/2627/FoundsCS/forest/focs-map/focs-map/teaching/2627/FoundsCS/forest/focs-map/map: the “Apply to All” Function [] | x::xs -> (f x) :: map f xs val map : ('a -> 'b) -> 'a list -> 'b list = # map (fun s -> s ^ "ppy") ["Hi"; "Ho"] - : string list = ["Hippy"; "Hoppy"] # map (map double) [[1]; [2; 3]] - : int list list = [[2]; [4; 6]]]]>The functional map applies a function to every element of a list, returning a list of the function’s results. “Apply to all” is a fundamental operation and we shall see several applications of it below. We again see the advantages of fun-notation, currying and map. If we did not have them, the first use of map in the above code block would require a preliminary function declaration: [] | s::ss -> (s ^ "ppy") :: sillylist ss val sillylist : string list -> string list = ]]>An expression containing several applications of functionals—such as our second example—can abbreviate a long series of declarations. Sometimes this coding style is cryptic, but it can be clear as crystal. Treating functions as values lets us capture common program structures once and for all.In the second example, double is the obvious integer doubling function we defined earlier. Note that map is a built-in OCaml function in the form of List.map. OCaml’s standard library includes, among much else, many list functions.Anil MadhavapeddyAnil Madhavapeddyhttps://www.cl.cam.ac.uk/teaching/2627/FoundsCS/forest/focs-ex-8-4/focs-ex-8-4/teaching/2627/FoundsCS/forest/focs-ex-8-4/Exercises › A map for optionsExerciseThe type 'a option, declared below, can be viewed as a type of lists having at most one element. (It is typically used as an alternative to exceptions.) Declare an analogue of the function map for type 'a option.Anil MadhavapeddyAnil Madhavapeddyhttps://www.cl.cam.ac.uk/teaching/2627/FoundsCS/forest/focs-ex-9-1/focs-ex-9-1/teaching/2627/FoundsCS/forest/focs-ex-9-1/Exercises › A map for sequencesExerciseCode an analogue of map for sequences.Anil MadhavapeddyAnil Madhavapeddyhttps://www.cl.cam.ac.uk/teaching/2627/FoundsCS/forest/focs-ex-8-2/focs-ex-8-2/teaching/2627/FoundsCS/forest/focs-ex-8-2/Exercises › Lexicographic orderingsExerciseThere are many ways of combining orderings. The lexicographic ordering uses two keys for comparisons. It is specified byWrite an OCaml function to lexicographically combine two orderings, supplied as functions. Explain how it allows function insort to sort a list of pairs.Anil MadhavapeddyAnil Madhavapeddyhttps://www.cl.cam.ac.uk/teaching/2627/FoundsCS/forest/focs-ex-8-5/focs-ex-8-5/teaching/2627/FoundsCS/forest/focs-ex-8-5/Exercises › Making change with mapExerciseRecall the making change function of More on Lists: [ [] ] | [] , _ -> [] | c::till , amt -> if amt < c then change till amt else let rec allc = function | [] -> [] | cs :: css -> (c::cs) :: allc css in allc (change (c::till) (amt - c)) @ change till amt val change : int list -> int -> int list list = ]]>Function allc applies the function ‘cons a c’ to every element of a list. Eliminate it by declaring a curried cons function and applying map.Anil MadhavapeddyAnil Madhavapeddyhttps://www.cl.cam.ac.uk/teaching/2627/FoundsCS/forest/focs-ex-8-3/focs-ex-8-3/teaching/2627/FoundsCS/forest/focs-ex-8-3/Exercises › Mapping over lists of listsExerciseWithout using map write a function map2 such that map2 f is equivalent to map (map f). The obvious solution requires declaring two recursive functions. Try to get away with only one by exploiting nested pattern-matching.Anil MadhavapeddyAnil Madhavapeddyhttps://www.cl.cam.ac.uk/teaching/2627/FoundsCS/forest/focs-ex-8-1/focs-ex-8-1/teaching/2627/FoundsCS/forest/focs-ex-8-1/Exercises › The argument-swapping functionalExerciseWhat does the following function do, and what are its uses?Anil Madhavapeddyhttps://www.cl.cam.ac.uk/teaching/2627/FoundsCS/forest/focs-topic-laziness/focs-topic-laziness/teaching/2627/FoundsCS/forest/focs-topic-laziness/LazinessAnil Madhavapeddyhttps://www.cl.cam.ac.uk/teaching/2627/FoundsCS/forest/focs-pipeline/focs-pipeline/teaching/2627/FoundsCS/forest/focs-pipeline/A PipelineProduce sequence of items Filter sequence in stages Consume results as needed Lazy lists join the stages togetherTwo types of program can be distinguished. A sequential program accepts a problem to solve, processes for a while, and finally terminates with its result. A typical example is the huge numerical simulations that are run on supercomputers. Most of our OCaml functions also fit this model.At the other extreme are reactive programs, whose job is to interact with the environment. They communicate constantly during their operation and run for as long as is necessary. A typical example is the software that controls many modern aircraft. Reactive programs often consist of concurrent processes running at the same time and communicating with one another.Concurrency is too difficult to consider in this course, but we can model simple pipelines such as that shown above. The Producer represents one or more sources of data, which it outputs as a stream. The Filter stages convert the input stream to an output stream, perhaps consuming several input items to yield a single output item. The Consumer takes as many elements as necessary.The Consumer drives the pipeline: nothing is computed except in response to its demand for an additional datum. Execution of the Filter stages is interleaved as required for the computation to go through. The programmer sets up the data dependencies but has no clear idea of what happens when. We have the illusion of concurrent computation.The Unix operating system provides similar ideas through its pipes that link processes together. In OCaml, we can model pipelines using lazy lists.Anil Madhavapeddyhttps://www.cl.cam.ac.uk/teaching/2627/FoundsCS/forest/focs-seq-consume/focs-seq-consume/teaching/2627/FoundsCS/forest/focs-seq-consume/Consuming a Sequence [] | n, Nil -> [] | n, Cons (x, xf) -> x :: get (n-1) (xf ()) val get : int -> 'a seq -> 'a list = ]]>The above code gets the first elements as a list. xf () forces evaluation.The function get converts a sequence to a list. It takes the first elements; it takes all of them if , which can terminate only if the sequence is finite.In the last line of get, the expression xf() calls the tail function, demanding evaluation of the next element. This operation is called forcing the sequence.Anil Madhavapeddyhttps://www.cl.cam.ac.uk/teaching/2627/FoundsCS/forest/focs-seq-functionals/focs-seq-functionals/teaching/2627/FoundsCS/forest/focs-seq-functionals/Functionals for Lazy ListsFiltering lazy lists: Nil | Cons (x, xf) -> if p x then Cons (x, fun () -> filterq p (xf ())) else filterq p (xf ()) val filterq : ('a -> bool) -> 'a seq -> 'a seq = ]]>The infinite sequence , , , … iterates f (f x)) val iterates : ('a -> 'a) -> 'a -> 'a seq = ]]>The functional filterq demands elements of xq until it finds one satisfying p. (Recall filter, the analogous operation for ordinary lists.) It contains a force not protected by a delay. If xq is infinite and contains no satisfactory element, then filterq runs forever.The functional iterates generalises from. It creates the next element not by adding one but by calling the function f.Anil Madhavapeddyhttps://www.cl.cam.ac.uk/teaching/2627/FoundsCS/forest/focs-seq-append/focs-seq-append/teaching/2627/FoundsCS/forest/focs-seq-append/Joining Two Sequences yq | Cons (x, xf) -> Cons(x, fun () -> appendq (xf ()) yq) val appendq : 'a seq -> 'a seq -> 'a seq = ]]>A more fair alternative: yq | Cons (x, xf) -> Cons (x, fun () -> interleave yq (xf ())) val interleave : 'a seq -> 'a seq -> 'a seq = ]]>Most list functions and functionals have analogues on sequences, but strange things can happen. Can an infinite list be reversed?Function appendq is precisely the same idea as append from the Lists lecture; it concatenates two sequences. If the first argument is infinite, then appendq never gets to its second argument, which is lost. Concatenation of infinite sequences is not terribly interesting.The function interleave avoids this problem by exchanging the two arguments in each recursive call. It combines the two lazy lists, losing no elements. Interleaving is the right way to combine two potentially infinite information sources into one.In both function declarations, observe that each xf () is enclosed within a . Each force is enclosed within a delay. This practice makes the functions lazy. A force not enclosed in a delay, as in get above, runs the risk of evaluating the sequence in full.Anil Madhavapeddyhttps://www.cl.cam.ac.uk/teaching/2627/FoundsCS/forest/focs-lazy-lists/focs-lazy-lists/teaching/2627/FoundsCS/forest/focs-lazy-lists/Lazy Lists (or Streams)Lists of possibly infinite length Elements computed upon demand Avoids waste if there are many solutions Infinite values are a useful abstractionIn OCaml, we can implement laziness by delaying evaluation of the tail of the list.Lazy lists have practical uses. Some algorithms, like making change, can yield many solutions when only a few are required. Sometimes the original problem concerns infinite series: with lazy lists, we can pretend they really exist!We are now dealing with infinite (or at least unbounded) computations. A potentially infinite source of data is processed one element at a time, upon demand. Such programs are harder to understand than terminating ones and have more ways of going wrong.Some purely functional languages, such as Haskell, use lazy evaluation everywhere. Even the if-then-else construct can be a function, and all lists are lazy. In OCaml, we can declare a type of lists such that evaluation of the tail does not occur until demanded. Delayed evaluation is weaker than lazy evaluation, but it is good enough for our purposes and often the best compromise for performance and memory usage.The traditional word “stream” is reserved in OCaml parlance for input/output channels. Let us call lazy lists sequences instead.Anil Madhavapeddyhttps://www.cl.cam.ac.uk/teaching/2627/FoundsCS/forest/focs-lazy-ocaml/focs-lazy-ocaml/teaching/2627/FoundsCS/forest/focs-lazy-ocaml/Lazy Lists in OCamlThe empty tuple () and its type unit Delayed version of is fun () -> E 'a seq) type 'a seq = Nil | Cons of 'a * (unit -> 'a seq) # let head (Cons (x, _)) = x Line 1, characters 10-23: Warning 8 [partial-match]: this pattern-matching is not exhaustive. Here is an example of a case that is not matched: Nil val head : 'a seq -> 'a = # let tail (Cons (_, xf)) = xf () Line 1, characters 10-24: Warning 8 [partial-match]: this pattern-matching is not exhaustive. Here is an example of a case that is not matched: Nil val tail : 'a seq -> 'a seq = ]]> has head and tail function The primitive OCaml type unit has one element, which is written (). This element may be regarded as a 0-tuple, and unit as the nullary Cartesian product. (Think of the connection between multiplication and the number 1.)The empty tuple serves as a placeholder in situations where no information is required. It may:appear in a data structure. For example, a unit-valued dictionary represents a set of keys. be the argument of a function, where its effect is to delay evaluation. be the argument or result of a procedure. (see the Procedural Programming section)The empty tuple, like all tuples, is a constructor and is allowed in patterns; for example: let f () = ...In particular is the function that takes an argument of type unit and returns the value of as its result. Expression is not evaluated until the function is called, even though the only possible argument is (). The function simply delays the evaluation of .Anil Madhavapeddyhttps://www.cl.cam.ac.uk/teaching/2627/FoundsCS/forest/focs-def-lazy-list/focs-def-lazy-list/teaching/2627/FoundsCS/forest/focs-def-lazy-list/Lazy listDefinitionA lazy list is a list of possibly infinite length whose elements are computed upon demand; in OCaml we implement laziness by delaying evaluation of the tail of the list. The traditional word “stream” is reserved in OCaml parlance for input/output channels, so we call lazy lists sequences instead. Introduced in Lazy Lists.Part of the course glossary.Anil Madhavapeddyhttps://www.cl.cam.ac.uk/teaching/2627/FoundsCS/forest/focs-seq-numerical/focs-seq-numerical/teaching/2627/FoundsCS/forest/focs-seq-numerical/Numerical Computations on Infinite Sequences float -> float = ]]>Close enough? match xf () with | Cons (y, yf) -> if abs_float (x -. y) <= eps then y else within eps (Cons (y, yf)) Lines 3-6, characters 7-41: Warning 8 [partial-match]: this pattern-matching is not exhaustive. Here is an example of a case that is not matched: Nil Lines 1-6, characters 22-41: Warning 8 [partial-match]: this pattern-matching is not exhaustive. Here is an example of a case that is not matched: Nil val within : float -> float seq -> float = ]]>Square Roots: float = ]]>The Newton-Raphson method is widely used for computing square roots. The infinite series converges rapidly to . The initial approximation, , is typically retrieved from a table, and is accurate enough that only a few iterations of the method are necessary. Calling iterates (next a) x0 generates the infinite series of approximations to the square root of using the Newton-Raphson method. To compute , the resulting series begins 1, 1.5, 1.41667, 1.4142157, 1.414213562, …, and this last figure is already accurate to 10 significant digits!Function within searches down the lazy list for two points whose difference is less than eps. It tests their absolute difference. Relative difference and other “close enough” tests can be coded. Such components can be used to implement other numerical functions directly as functions over sequences. The point is to build programs from small, interchangeable parts.Function root uses within, iterates and next to to apply Newton-Raphson with a tolerance of and a (poor) initial approximation of 1.0.This treatment of numerical computation has received some attention in the research literature; a recurring example is Richardson extrapolation.Anil Madhavapeddyhttps://www.cl.cam.ac.uk/teaching/2627/FoundsCS/forest/focs-seq-evaluation/focs-seq-evaluation/teaching/2627/FoundsCS/forest/focs-seq-evaluation/Sample EvaluationHere we ask for two elements of the infinite sequence. In fact, three elements are computed: 6, 7 and 8. Our implementation is slightly too eager. A more complicated type declaration could avoid this problem. Another problem is that if one repeatedly examines some particular list element using forcing, that element is repeatedly evaluated. In a lazy programming language, the result of the first evaluation would be stored for later reference. To get the same effect in OCaml requires the use of references.We should be grateful that the potentially infinite computation is kept finite. The tail of the original sequence even contains the unevaluated expression 6+1.Anil Madhavapeddyhttps://www.cl.cam.ac.uk/teaching/2627/FoundsCS/forest/focs-infinite-sequence/focs-infinite-sequence/teaching/2627/FoundsCS/forest/focs-infinite-sequence/The Infinite Sequence: , , , … from (k+1)) val from : int -> int seq = # let it = from 1 val it : int seq = Cons (1, ) # let it = tail it val it : int seq = Cons (2, ) # let it = tail it val it : int seq = Cons (3, )]]>Function from constructs the infinite sequence of integers starting from . Execution terminates because of the fun enclosing the recursive call. OCaml displays the tail of a sequence as fun, which stands for some function value. Each call to tail generates the next sequence element. We could do this forever.This example is of little practical value because the cost of computing a sequence element will be dominated by that of creating the dummy function. Lazy lists tend to have high overheads.Anil MadhavapeddyAnil Madhavapeddyhttps://www.cl.cam.ac.uk/teaching/2627/FoundsCS/forest/focs-ex-9-1/focs-ex-9-1/teaching/2627/FoundsCS/forest/focs-ex-9-1/Exercises › A map for sequencesExerciseCode an analogue of map for sequences.Anil MadhavapeddyAnil Madhavapeddyhttps://www.cl.cam.ac.uk/teaching/2627/FoundsCS/forest/focs-ex-9-5/focs-ex-9-5/teaching/2627/FoundsCS/forest/focs-ex-9-5/Exercises › All lists of zeroes and onesExerciseCode the lazy list whose elements are all ordinary lists of zeroes and ones, namely []; [0]; [1]; [0; 0]; [0; 1]; [1; 0]; [1; 1]; [0; 0; 0]; …. (Taken from the exam question 2003 Paper 1 Question 5.)Anil MadhavapeddyAnil Madhavapeddyhttps://www.cl.cam.ac.uk/teaching/2627/FoundsCS/forest/focs-ex-9-2/focs-ex-9-2/teaching/2627/FoundsCS/forest/focs-ex-9-2/Exercises › Concatenating infinite sequencesExerciseConsider the list function concat, which concatenates a list of lists to form a single list. Can it be generalised to concatenate a sequence of sequences? What can go wrong? [] | l::ls -> l @ concat ls]]>Anil MadhavapeddyAnil Madhavapeddyhttps://www.cl.cam.ac.uk/teaching/2627/FoundsCS/forest/focs-ex-9-6/focs-ex-9-6/teaching/2627/FoundsCS/forest/focs-ex-9-6/Exercises › Enumerating palindromesExercise(Continuing the previous exercise.) A palindrome is a list that equals its own reverse. Code the lazy list whose elements are all palindromes of 0s and 1s, namely []; [0]; [1]; [0; 0]; [0; 0; 0]; [0; 1; 0]; [1; 1]; [1; 0; 1]; [1; 1; 1]; [0; 0; 0; 0]; , …. You may take the reversal function List.rev as given.Anil MadhavapeddyAnil Madhavapeddyhttps://www.cl.cam.ac.uk/teaching/2627/FoundsCS/forest/focs-ex-9-4/focs-ex-9-4/teaching/2627/FoundsCS/forest/focs-ex-9-4/Exercises › Lazy binary treesExerciseA lazy binary tree is either empty or is a branch containing a label and two lazy binary trees, possibly to infinite depth. Present an OCaml datatype to represent lazy binary trees, along with a function that accepts a lazy binary tree and produces a lazy list that contains all of the tree’s labels. (Taken from the exam question 2008 Paper 1 Question 5.)Anil MadhavapeddyAnil Madhavapeddyhttps://www.cl.cam.ac.uk/teaching/2627/FoundsCS/forest/focs-ex-9-3/focs-ex-9-3/teaching/2627/FoundsCS/forest/focs-ex-9-3/Exercises › Making change lazilyExerciseCode a function to make change using lazy lists, delivering the sequence of all possible ways of making change. Using sequences allows us to compute solutions one at a time when there exists an astronomical number. Represent lists of coins using ordinary lists. (Hint: to benefit from laziness you may need to pass around the sequence of alternative solutions as a function of type unit -> (int list) seq.)Anil Madhavapeddyhttps://www.cl.cam.ac.uk/teaching/2627/FoundsCS/forest/focs-topic-queues/focs-topic-queues/teaching/2627/FoundsCS/forest/focs-topic-queues/QueuesAnil Madhavapeddyhttps://www.cl.cam.ac.uk/teaching/2627/FoundsCS/forest/focs-search-methods/focs-search-methods/teaching/2627/FoundsCS/forest/focs-search-methods/A Survey of Search MethodsDepth-first: use a stack (efficient but incomplete) Breadth-first: use a queue (uses too much space!) Iterative deepening: use depth-first to get benefits of breadth-first (trades time for space) Best-first: use a priority queue (heuristic search)The data structure determines the search!Search procedures can be classified by the data structure used to store pending subtrees. Depth-first search stores them on a stack, which is implicit in functions like inorder, but can be made explicit. Breadth-first search stores such nodes in a queue.An important variation is to store the nodes in a priority queue, which is an ordered sequence. The priority queue applies some sort of ranking function to the nodes, placing higher-ranked nodes before lower-ranked ones. The ranking function typically estimates the distance from the node to a solution. If the estimate is good, the solution is located swiftly. This method is called best-first search.The priority queue can be kept as a sorted list, although this is slow. Binary search trees would be much better on average, and fancier data structures improve matters further.Anil Madhavapeddyhttps://www.cl.cam.ac.uk/teaching/2627/FoundsCS/forest/focs-queue-adt/focs-queue-adt/teaching/2627/FoundsCS/forest/focs-queue-adt/An Abstract Data Type: Queuesqempty is the empty queue qnull tests whether a queue is empty qhd returns the element at the head of a queue deq discards the element at the head of a queue enq adds an element at the end of a queueBreadth-first search becomes much faster if we replace the lists by queues. A queue represents a sequence, allowing elements to be taken from the head and added to the tail. This is a First-In-First-Out (FIFO) discipline: the item next to be removed is the one that has been in the queue for the longest time. Lists can implement queues, but append is a poor means of adding elements to the tail.Our functional arrays are suitable, provided we augment them with a function to delete the first array element. (See ML for the Working Programmer page 156.) Each operation would take time for a queue of length .We shall describe a representation of queues that is purely functional, based upon lists, and efficient. Operations take time when “amortized”: averaged over the lifetime of a queue.A conventional programming technique is to represent a queue by an array. Two indices point to the front and back of the queue, which may wrap around the end of the array. The coding is somewhat tricky. Worse, the length of the queue must be given a fixed upper bound.Anil Madhavapeddyhttps://www.cl.cam.ac.uk/teaching/2627/FoundsCS/forest/focs-stacks/focs-stacks/teaching/2627/FoundsCS/forest/focs-stacks/Another Abstract Data Type: Stacksempty is the empty stack null tests whether a stack is empty top returns the element at the top of a stack pop discards the element at the top of a stack push adds an element at the top of a stackA stack is a sequence such that items can be added or removed from the head only. A stack obeys a Last-In-First-Out (LIFO) discipline: the item next to be removed is the one that has been on the stack for the shortest time. Lists can easily implement stacks because both cons and hd affect the head. But unlike lists, stacks are often regarded as an imperative data structure: the effect of push or pop is to change an existing stack, not return a new one.In conventional programming languages, a stack is often implemented by storing the elements in an array, using a variable (the “stack pointer”) to count them. Most language processors keep track of recursive function calls using an internal stack.Anil Madhavapeddyhttps://www.cl.cam.ac.uk/teaching/2627/FoundsCS/forest/focs-bfs-queues/focs-bfs-queues/teaching/2627/FoundsCS/forest/focs-bfs-queues/Breadth-First Tree Traversal — Using Queues breadth (deq q) | Br (v, t, u) -> v :: breadth (enq (enq (deq q) t) u) val breadth : 'a tree queue -> 'a list = ]]>This function implements the same algorithm as nbreadth but uses a different data structure. It represents queues using type queue instead of type list.To compare their efficiency, I applied both functions to the full binary tree of depth 12, which contains 4095 labels. The function nbreadth took 30 seconds while breadth took only 0.15 seconds: faster by a factor of 200.For larger trees, the speedup would be greater. Choosing the right data structure pays handsomely.Anil Madhavapeddyhttps://www.cl.cam.ac.uk/teaching/2627/FoundsCS/forest/focs-functional-queues/focs-functional-queues/teaching/2627/FoundsCS/forest/focs-functional-queues/Efficient Functional Queues: Code Q (List.rev tls, []) | q -> q val norm : 'a queue -> 'a queue = # let qnull = function | Q ([], []) -> true | _ -> false val qnull : 'a queue -> bool = # let enq (Q (hds, tls)) x = norm (Q (hds, x::tls)) val enq : 'a queue -> 'a -> 'a queue = # exception Empty exception Empty # let deq = function | Q (x::hds, tls) -> norm (Q (hds, tls)) | _ -> raise Empty val deq : 'a queue -> 'a queue = # let qempty = Q ([], []) val qempty : 'a queue = Q ([], []) # let qhd = function | Q (x::_, _) -> x | _ -> raise Empty val qhd : 'a queue -> 'a = ]]>The datatype of queues prevents confusion with other pairs of lists. The empty queue has both parts empty.The function norm puts a queue into normal form, ensuring that the front part is never empty unless the entire queue is empty. Functions deq and enq call norm to normalise their result.Because queues are in normal form, their head is certain to be in their front part, so qhd looks there.Let us analyse the cost of an execution comprising (in any possible order) enq operations and deq operations, starting with an empty queue. Each enq operation will perform one cons, adding an element to the rear part. Since the final queue must be empty, each element of the rear part gets transferred to the front part. The corresponding reversals perform one cons per element. Thus, the total cost of the series of queue operations is cons operations, an average of 2 per operation. The amortized time is .There is a catch. The conses need not be distributed evenly; reversing a long list could take up to of them. Unpredictable delays make the approach unsuitable for real-time programming where deadlines must be met.Anil Madhavapeddyhttps://www.cl.cam.ac.uk/teaching/2627/FoundsCS/forest/focs-functional-queues-idea/focs-functional-queues-idea/teaching/2627/FoundsCS/forest/focs-functional-queues-idea/Efficient Functional Queues: IdeaRepresent the queue by any pair of lists Add new items to the rear list Remove items from front list and if empty move rear to front Amortized time per operation is Queues require efficient access at both ends: at the front, for removal, and at the back, for insertion. Ideally, access should take constant time, . It may appear that lists cannot provide such access. If enq(q, x) performs q@[x], then this operation will be . We could represent queues by reversed lists, implementing enq(q, x) by x::q, but then the deq and qhd operations would be . Linear time is intolerable: a series of queue operations could then require time.The solution is to represent a queue by a pair of lists, whererepresents the queue .The front part of the queue is stored in order, and the rear part is stored in reverse order. The enq operation adds elements to the rear part using cons, since this list is reversed; thus, enq takes constant time. The deq and qhd operations look at the front part, which normally takes constant time, since this list is stored in order. But sometimes deq removes the last element from the front part; when this happens, it reverses the rear part, which becomes the new front part.Amortized time refers to the cost per operation averaged over the lifetime of any complete execution. Even for the worst possible execution, the average cost per operation turns out to be constant; see the analysis below.Anil MadhavapeddyAnil Madhavapeddyhttps://www.cl.cam.ac.uk/teaching/2627/FoundsCS/forest/focs-ex-10-2/focs-ex-10-2/teaching/2627/FoundsCS/forest/focs-ex-10-2/Exercises › Array-based circular queuesExerciseThe traditional way to implement queues uses a fixed-length array. Two indices into the array indicate the start and end of the queue, which wraps around from the end of the array to the start. How appropriate is such a data structure for implementing breadth-first search?Anil MadhavapeddyAnil Madhavapeddyhttps://www.cl.cam.ac.uk/teaching/2627/FoundsCS/forest/focs-ex-10-3/focs-ex-10-3/teaching/2627/FoundsCS/forest/focs-ex-10-3/Exercises › Rewriting breadth with letExerciseWrite a version of the function breadth using a nested let construction rather than match.Anil MadhavapeddyAnil Madhavapeddyhttps://www.cl.cam.ac.uk/teaching/2627/FoundsCS/forest/focs-ex-10-1/focs-ex-10-1/teaching/2627/FoundsCS/forest/focs-ex-10-1/Exercises › Tree-based queuesExerciseSuppose that we have an implementation of queues, based on binary trees, such that each operation takes logarithmic time in the worst case. Outline the advantages and drawbacks of such an implementation compared with one presented above.Anil Madhavapeddyhttps://www.cl.cam.ac.uk/teaching/2627/FoundsCS/forest/focs-topic-search/focs-topic-search/teaching/2627/FoundsCS/forest/focs-topic-search/SearchAnil Madhavapeddyhttps://www.cl.cam.ac.uk/teaching/2627/FoundsCS/forest/focs-search-methods/focs-search-methods/teaching/2627/FoundsCS/forest/focs-search-methods/A Survey of Search MethodsDepth-first: use a stack (efficient but incomplete) Breadth-first: use a queue (uses too much space!) Iterative deepening: use depth-first to get benefits of breadth-first (trades time for space) Best-first: use a priority queue (heuristic search)The data structure determines the search!Search procedures can be classified by the data structure used to store pending subtrees. Depth-first search stores them on a stack, which is implicit in functions like inorder, but can be made explicit. Breadth-first search stores such nodes in a queue.An important variation is to store the nodes in a priority queue, which is an ordered sequence. The priority queue applies some sort of ranking function to the nodes, placing higher-ranked nodes before lower-ranked ones. The ranking function typically estimates the distance from the node to a solution. If the estimate is good, the solution is located swiftly. This method is called best-first search.The priority queue can be kept as a sorted list, although this is slow. Binary search trees would be much better on average, and fancier data structures improve matters further.Anil Madhavapeddyhttps://www.cl.cam.ac.uk/teaching/2627/FoundsCS/forest/focs-predicate-applications/focs-predicate-applications/teaching/2627/FoundsCS/forest/focs-predicate-applications/Applications of the Predicate Functionals x=y) xs val member : 'a -> 'a list -> bool = # let inter xs ys = filter (fun x -> member x ys) xs val inter : 'a list -> 'a list -> 'a list = ]]>Testing whether two lists have no common elements all (fun y -> x<>y) ys) xs val disjoint : 'a list -> 'a list -> bool = ]]>The Lists lecture presented the function member, which tests whether a specified value can be found as a list element, and inter, which returns the “intersection” of two lists: the list of elements they have in common.But remember: the purpose of list functionals is not to replace the declarations of popular functions, which probably are available already. It is to eliminate the need for separate declarations of ad-hoc functions. When they are nested, like the calls to all in disjoint above, the inner functions are almost certainly one-offs, not worth declaring separately.Our primitives themselves can be seen as a programming language. Part of the task of programming is to extend our programming language with notation for solving the problem at hand. The levels of notation that we define should correspond to natural levels of abstraction in the problem domain.Historical Note: Alonzo Church’s -calculus gave a simple syntax, -notation, for expressing functions. It is the direct precursor of OCaml’s fun-notation. It was soon shown that his system was equivalent in computational power to Turing machines, and Church’s thesis states that this defines precisely the set of functions that can be computed effectively.The -calculus had a tremendous influence on the design of functional programming languages. McCarthy’s Lisp was something of a false start; it interpreted variable binding incorrectly, an error that stood for some 20 years. But in 1966, Peter Landin (of Queen Mary College, University of London) sketched out the main features of functional languages.Anil Madhavapeddyhttps://www.cl.cam.ac.uk/teaching/2627/FoundsCS/forest/focs-bst/focs-bst/teaching/2627/FoundsCS/forest/focs-bst/Binary Search TreesA dictionary associates values (here, numbers) with keys.Binary search trees are an important application of binary trees. They work for keys that have a total ordering, such as strings. Each branch of the tree carries a pair; its left subtree holds smaller keys; the right subtree holds greater keys. If the tree remains reasonably balanced, then update and lookup both take for a tree of size . These times hold in the average case; given random data, the tree is likely to remain balanced.At a given node, all keys in the left subtree are smaller (or equal) while all trees in the right subtree are greater.An unbalanced tree has a linear access time in the worst case. Examples include building a tree by repeated insertions of elements in increasing or decreasing order; there is a close resemblance to quicksort. Building a binary search tree, then converting it to inorder, yields a sorting algorithm called treesort.Self-balancing trees, such as Red-Black trees, attain in the worst case. They are complicated to implement.Anil Madhavapeddyhttps://www.cl.cam.ac.uk/teaching/2627/FoundsCS/forest/focs-bfs-append/focs-bfs-append/teaching/2627/FoundsCS/forest/focs-bfs-append/Breadth-First Tree Traversal — Using Append [] | Lf :: ts -> nbreadth ts | Br (v, t, u) :: ts -> v :: nbreadth (ts @ [t; u]) val nbreadth : 'a tree list -> 'a list = ]]>Keeps an enormous queue of nodes of search, and is a wasteful use of append.Breadth-first search can be inefficient, this naive implementation especially so. When the search is at depth of the tree, the list contains all the remaining trees at depth , followed by the subtrees (all at depth ) of the trees that have already been visited. At depth 10, the list could already contain 1024 elements. It requires a lot of space, and aggravates this with a gross misuse of append. Evaluating ts@[t, u] copies the long list ts just to insert two elements.Anil Madhavapeddyhttps://www.cl.cam.ac.uk/teaching/2627/FoundsCS/forest/focs-bfs-queues/focs-bfs-queues/teaching/2627/FoundsCS/forest/focs-bfs-queues/Breadth-First Tree Traversal — Using Queues breadth (deq q) | Br (v, t, u) -> v :: breadth (enq (enq (deq q) t) u) val breadth : 'a tree queue -> 'a list = ]]>This function implements the same algorithm as nbreadth but uses a different data structure. It represents queues using type queue instead of type list.To compare their efficiency, I applied both functions to the full binary tree of depth 12, which contains 4095 labels. The function nbreadth took 30 seconds while breadth took only 0.15 seconds: faster by a factor of 200.For larger trees, the speedup would be greater. Choosing the right data structure pays handsomely.Anil Madhavapeddyhttps://www.cl.cam.ac.uk/teaching/2627/FoundsCS/forest/focs-bfs-dfs/focs-bfs-dfs/teaching/2627/FoundsCS/forest/focs-bfs-dfs/Breadth-First v Depth-First Tree Traversalbinary trees as decision trees look for solution nodes Depth-first: search one subtree in full before moving on Breadth-first: search all nodes at level before moving to finds all solutions — nearest first!Preorder, inorder and postorder tree traversals all have something in common: they are depth-first. At each node, the left subtree is entirely traversed before the right subtree. Depth-first traversals are easy to code and can be efficient, but they are ill-suited for some problems.Suppose the tree represents the possible moves in a puzzle, and the purpose of the traversal is to search for a node containing a solution. Then a depth-first traversal may find one solution node deep in the left subtree, when another solution is at the very top of the right subtree. Often we want the shortest path to a solution.Suppose the tree is infinite or simply extremely large. Depth-first search is almost useless with such trees, for if the left subtree is infinite then the search will never reach the right subtree. OCaml can represent infinite trees by the means discussed in the lecture on laziness. Another tree representation (suitable for solving solitaire, for example) is by a function next : pos -> pos list, which maps a board position to a list of the positions possible after the next move. For simplicity, the examples below use the OCaml datatype tree, which has only finite trees.A breadth-first traversal explores the nodes horizontally rather than vertically. When visiting a node, it does not traverse the subtrees until it has visited all other nodes at the current depth. This is easily implemented by keeping a list of trees to visit. Initially, this list consists of one element: the entire tree. Each iteration removes a tree from the head of the list and adds its subtrees after the end of the list.Anil Madhavapeddyhttps://www.cl.cam.ac.uk/teaching/2627/FoundsCS/forest/focs-equality/focs-equality/teaching/2627/FoundsCS/forest/focs-equality/Equality Tests false | y::l -> if x = y then true else member x l val member : 'a -> 'a list -> bool = ]]>All the list functions we have encountered up to now have been “polymorphic”, working for lists of any type. Function member uses linear search to report whether or not x occurs in l.To do this generically, it uses a special feature of OCaml known as “polymorphic equality”, which manifests itself via the =, >=, <=, > and < operators. These operators inspect the structure of the values using a consistent order. Types you can legitimately compare this way include integers, strings, booleans, and tuples or lists of primitive types.More complex types can be compared this way within careful limits: recursive structures or function values will not work (we will cover function values in the Currying lecture later). For now, it is sufficient to use these magic polymorphic equality operators. As you get more familiar with OCaml and the use of higher order functions (also covered in a later lecture), you will encounter the use of explicit compare functions that are used to provide more complex equality tests.The presence of polymorphic equality is a contentious feature in OCaml. While it provides a great ease of use in smaller codebases, it starts to become more dangerous when building larger OCaml-based systems. Most large-scale users of OCaml tend towards not using it in important code, but it is just fine for our purposes while learning the beginning steps of computer science.Anil Madhavapeddyhttps://www.cl.cam.ac.uk/teaching/2627/FoundsCS/forest/focs-iterative-deepening/focs-iterative-deepening/teaching/2627/FoundsCS/forest/focs-iterative-deepening/Iterative deepening: Another Exhaustive SearchBreadth-first search examines nodes: Recompute nodes at depth instead of storing them Time factor is if 1]]>; complexity is still Space required at depth drops from to Breadth-first search is not practical for infinite trees: it uses too much space. Large parts of the tree have to be stored. Consider the slightly more general problem of searching trees whose branching factor is (for binary trees, ). Then breadth-first search to depth examines nodes, which is , ignoring the constant factor of . Since all nodes that are examined are also stored, the space and time requirements are both .Depth-first iterative deepening combines the space efficiency of depth-first with the “nearest-first” property of breadth-first search. It performs repeated depth-first searches with increasing depth bounds, each time discarding the result of the previous search. Thus it searches to depth 1, then to depth 2, and so on until it finds a solution. We can afford to discard previous results because the number of nodes is growing exponentially. There are nodes at level ; if , this number actually exceeds the total number of nodes of all previous levels put together, namely .Korf shows that the time needed for iterative deepening to reach depth is only times that for breadth-first search, if 1]]>. This is a constant factor; both algorithms have the same time complexity, . In typical applications where the extra factor of is quite tolerable. The reduction in the space requirement is exponential, from for breadth-first to for iterative deepening. Of course, this assumes that the tree itself is not stored in memory.Anil Madhavapeddyhttps://www.cl.cam.ac.uk/teaching/2627/FoundsCS/forest/focs-linear-search/focs-linear-search/teaching/2627/FoundsCS/forest/focs-linear-search/Linear Searchfind in list by comparing with each element obviously time simple & general ordered searching needs only indexed lookup needs only Linear search is the obvious way to find a desired item in a collection: simply look through all the items, one at a time. If is in the list, then it will be found in steps on average, and even the worst case is obviously .Large collections of data are usually ordered or indexed so that items can be found in time, which is exponentially better than . Even is achievable (using a hash table), though subject to the usual proviso that machine limits are not exceeded.Efficient indexing methods are of prime importance: consider Web search engines. Nevertheless, linear search is often used to search small collections because it is so simple and general, and it is the starting point for better algorithms.Anil Madhavapeddyhttps://www.cl.cam.ac.uk/teaching/2627/FoundsCS/forest/focs-predicate-functionals/focs-predicate-functionals/teaching/2627/FoundsCS/forest/focs-predicate-functionals/List Functionals for Predicates false | x::xs -> (p x) || (exists p xs) val exists : ('a -> bool) -> 'a list -> bool = # let rec filter p = function | [] -> [] | x::xs -> if p x then x :: filter p xs else filter p xs val filter : ('a -> bool) -> 'a list -> 'a list = ]]>A predicate is a boolean-valued function.The functional exists transforms a predicate into a predicate over lists. Given a list, exists p tests whether or not some list element satisfies p (making it return true). If it finds one, it stops searching immediately, thanks to the behaviour of the lazy || operator.Dually, we have a functional to test whether all list elements satisfy the predicate. If it finds a counterexample then it, too, stops searching. true | x::xs -> (p x) && all p xs val all : ('a -> bool) -> 'a list -> bool = ]]>The filter functional, like map, transforms lists. It applies a predicate to all the list elements, but instead of returning the resulting values (which could only be true or false), it returns the list of elements satisfying the predicate.Anil Madhavapeddyhttps://www.cl.cam.ac.uk/teaching/2627/FoundsCS/forest/focs-change-exceptions/focs-change-exceptions/teaching/2627/FoundsCS/forest/focs-change-exceptions/Making Change with Exceptions [] | [], _ -> raise Change | c::till, amt -> if amt < 0 then raise Change else try c :: change (c::till) (amt - c) with Change -> change till amt exception Change val change : int list -> int -> int list = ]]>In the Lists lectures, we considered the problem of making change. The greedy algorithm presented there could not express “6 using 5 and 2” because it always took the largest coin. Returning the list of all possible solutions avoids that problem rather expensively: we only need one solution.Using exceptions, we can code a backtracking algorithm: one that can undo past decisions if it comes to a dead end. The exception Change is raised if we run out of coins (with a non-zero amount) or if the amount goes negative. We always try the largest coin, but enclose the recursive call in an exception handler, which undoes the choice if it goes wrong.Carefully observe how exceptions interact with recursion. The exception handler always undoes the most recent choice, leaving others possibly to be undone later. If making change really is impossible, then eventually exception Change will be raised with no handler to catch it, and it will be reported at top level.Anil MadhavapeddyAnil Madhavapeddyhttps://www.cl.cam.ac.uk/teaching/2627/FoundsCS/forest/focs-ex-10-5/focs-ex-10-5/teaching/2627/FoundsCS/forest/focs-ex-10-5/Exercises › An implicit infinite treeExerciseConsider the following OCaml function.If we regard it as representing a tree, where the subtrees are computed from the current label, what tree does next 1 represent?Anil MadhavapeddyAnil Madhavapeddyhttps://www.cl.cam.ac.uk/teaching/2627/FoundsCS/forest/focs-ex-10-2/focs-ex-10-2/teaching/2627/FoundsCS/forest/focs-ex-10-2/Exercises › Array-based circular queuesExerciseThe traditional way to implement queues uses a fixed-length array. Two indices into the array indicate the start and end of the queue, which wraps around from the end of the array to the start. How appropriate is such a data structure for implementing breadth-first search?Anil MadhavapeddyAnil Madhavapeddyhttps://www.cl.cam.ac.uk/teaching/2627/FoundsCS/forest/focs-ex-10-4/focs-ex-10-4/teaching/2627/FoundsCS/forest/focs-ex-10-4/Exercises › Search with a branching factor near oneExerciseIterative deepening is inappropriate if , where is the branching factor. What search strategy is appropriate in this case?Anil MadhavapeddyAnil Madhavapeddyhttps://www.cl.cam.ac.uk/teaching/2627/FoundsCS/forest/focs-ex-4-1/focs-ex-4-1/teaching/2627/FoundsCS/forest/focs-ex-4-1/Exercises › Set union without repetitionsExerciseSets can be represented in OCaml using lists containing no duplicated items (i.e. where no item is equal to another using polymorphic comparison).Using the member function defined above, code a function to implement set union. It should avoid introducing repetitions, for example the union of the lists [4; 7; 1] and [6; 4; 7] should be [1; 6; 4; 7] (though the order does not matter).Anil Madhavapeddyhttps://www.cl.cam.ac.uk/teaching/2627/FoundsCS/forest/focs-topic-imperative/focs-topic-imperative/teaching/2627/FoundsCS/forest/focs-topic-imperative/ImperativeAnil Madhavapeddyhttps://www.cl.cam.ac.uk/teaching/2627/FoundsCS/forest/focs-stacks/focs-stacks/teaching/2627/FoundsCS/forest/focs-stacks/Another Abstract Data Type: Stacksempty is the empty stack null tests whether a stack is empty top returns the element at the top of a stack pop discards the element at the top of a stack push adds an element at the top of a stackA stack is a sequence such that items can be added or removed from the head only. A stack obeys a Last-In-First-Out (LIFO) discipline: the item next to be removed is the one that has been on the stack for the shortest time. Lists can easily implement stacks because both cons and hd affect the head. But unlike lists, stacks are often regarded as an imperative data structure: the effect of push or pop is to change an existing stack, not return a new one.In conventional programming languages, a stack is often implemented by storing the elements in an array, using a variable (the “stack pointer”) to count them. Most language processors keep track of recursive function calls using an internal stack.Anil Madhavapeddyhttps://www.cl.cam.ac.uk/teaching/2627/FoundsCS/forest/focs-array-examples/focs-array-examples/teaching/2627/FoundsCS/forest/focs-array-examples/Array ExamplesIn the following session, the identifier ar is bound to an array of 20 elements, which are initially set to the squares of their subscripts. The array’s third element (which actually has subscript 2) is inspected and found to be four. The second call to Array.get supplies a subscript that is out of range, so OCaml rejects it. i * i) val ar : int array = [|0; 1; 4; 9; 16; 25; 36; 49; 64; 81; 100; 121; 144; 169; 196; 225; 256; 289; 324; 361|] # Array.get ar 2 - : int = 4 # Array.get ar 20 Exception: Invalid_argument "index out of bounds". # Array.set ar 2 33; ar - : int array = [|0; 1; 33; 9; 16; 25; 36; 49; 64; 81; 100; 121; 144; 169; 196; 225; 256; 289; 324; 361|]]]>By calling Array.set, we then modify the element with subscript 2. Note however that we cannot modify the array’s length. If we outgrow the array, we have to create a new one, copy the data into it, and then forget the old array. Typically the new array would be double the size of the old one, so that the cost of copying is insignificant.OCaml provides numerous operators for modifying, computing over and searching in arrays. Many are analogous to functions on lists. For example, Array.exists takes a boolean-valued function and returns true if an array element satisfies it. i > 200) ar - : bool = true # Array.exists (fun i -> i < 0) ar - : bool = false]]>Anil Madhavapeddyhttps://www.cl.cam.ac.uk/teaching/2627/FoundsCS/forest/focs-arrays/focs-arrays/teaching/2627/FoundsCS/forest/focs-arrays/ArraysA conventional array is an indexed storage area. It is updated in place by the command a.(k) <- x The concept is inherently imperative. A functional array is a finite map from integers to data. Updating implies copying to return update(A, k, x) The new array equals A except that A.(k) = x. Can we do updates efficiently?The elements of a list can only be reached by counting from the front. Elements of a tree are reached by following a path from the root. An array hides such structural matters; its elements are uniformly designated by number. Immediate access to arbitrary parts of a data structure is called random access.Arrays are the dominant data structure in conventional programming languages. The ingenious use of arrays is the key to many of the great classical algorithms, such as Hoare’s original quicksort (the partition step) and Warshall’s transitive-closure algorithm.The drawback is that subscripting is a chief cause of programmer error. That is why arrays play little role in this introductory course.Functional arrays are described below in order to illustrate another way of using trees to organise data. Here is a summary of basic dictionary data structures in order of decreasing generality and increasing efficiency:Linear search: Most general, needing only equality on keys, but inefficient: linear time. Binary search: Needs an ordering on keys. Logarithmic access time in the average case, but our binary search trees are linear in the worst case. Array subscripting: Least general, requiring keys to be integers, but even worst-case time is logarithmic.Anil Madhavapeddyhttps://www.cl.cam.ac.uk/teaching/2627/FoundsCS/forest/focs-commands/focs-commands/teaching/2627/FoundsCS/forest/focs-commands/Commands: Expressions with EffectsBasic commands update references, write to files, etc. causes a series of expressions to be evaluated and returns the value of . A typical command returns the empty tuple: () if then else behaves like the traditional control structure if and have effects. Other OCaml constructs behave naturally with commands, including match expressions and recursive functions.We use the term command informally to refer to an expression that has an effect on the state. All expressions denote some value, but they can return (), which conveys no actual information.We need a way to execute one command after another. The construct evaluates the expressions to in the order given and returns the value of . The values of the other expressions are discarded; their only purpose is to change the state.Commands may be used with if and match much as in conventional languages. OCaml functions play the role of procedures.Other languages that combine the functional and imperative programming paradigms include Lisp (and its dialect Scheme), F#, Scala, Swift and Kotlin. Rust is a notable modern example: a systems programming language whose type system and pattern matching draw heavily on the ML family (its first compiler was written in OCaml), yet which gives the programmer precise control over mutable state.Anil Madhavapeddyhttps://www.cl.cam.ac.uk/teaching/2627/FoundsCS/forest/focs-goals-of-programming/focs-goals-of-programming/teaching/2627/FoundsCS/forest/focs-goals-of-programming/Goals of Programmingto describe a computation so that it can be done mechanically: Expressions compute values. Commands cause effects. to do so efficiently and correctly, giving the right answers quickly to allow easy modification as needs change Through an orderly structure based on abstraction principles Such as modules or classesProgramming in-the-small concerns the writing of code to do simple, clearly defined tasks. Programs provide expressions for describing mathematical formulae and so forth. This was the original contribution of FORTRAN, the FORmula TRANslator. Commands describe how control should flow from one part of the program to the next.As we code layer upon layer, we eventually find ourselves programming in the large : joining large modules to solve some messy task. Programming languages have used various mechanisms to allow one part of the program to provide interfaces to other parts. Modules encapsulate a body of code, allowing outside access only through a programmer-defined interface. Abstract Data Types are a simpler version of this concept, which implement a single concept such as dates or floating-point numbers.Object-oriented programming is the most complicated approach to modularity. Classes define concepts, and they can be built upon other classes. Operations can be defined that work in appropriately specialised ways on a family of related classes. Objects are instances of classes and hold the data that is being manipulated.This course does not cover OCaml’s sophisticated module system, which can do many of the same things as classes. You will learn all about objects when you study Java. OCaml includes a powerful object system, although this is not used as much as its module system.Anil Madhavapeddyhttps://www.cl.cam.ac.uk/teaching/2627/FoundsCS/forest/focs-while/focs-while/teaching/2627/FoundsCS/forest/focs-while/Iteration: the while command None | _::xs -> Some xs val tlopt : 'a list -> 'a list option = # let length xs = let lp = ref xs in (* list of uncounted elements *) let np = ref 0 in (* accumulated count *) let fin = ref false in while not !fin do match tlopt !lp with | None -> fin := true | Some xs -> lp := xs; np := 1 + !np done; !np (* the final count is returned *) val length : 'a list -> int = ]]>Once we can change the state, we need to do so repeatedly. Recursion can serve this purpose, but having to declare a procedure for every loop is clumsy, and compilers for conventional languages seldom exploit tail-recursion.Early programming languages provided little support for repetition. The programmer had to set up loops using goto commands, exiting the loop using another goto controlled by an if. Modern languages provide a confusing jumble of looping constructs, the most fundamental of which is while B do C. The boolean expression is evaluated, and if true, command is executed and the command repeats. If evaluates to false then the while command terminates, perhaps without executing even once.OCaml’s main looping construct is while, which returns the value (). The function length declares references to hold the list under examination (lp) and number of elements counted so far (np) as well as whether the end of the list has been reached (the boolean reference fin). While the list is non-empty, we skip over one more element (by setting it to its tail) and count that element.The body of the while loop first checks to see if the end of the list has been reached, in which case it sets the fin variable to true. If there is a tail value, then two assignments are executed in sequence. The lp reference is set to the tail of the list, and the np reference integer is incremented by one. When the while loop terminates due to the fin variable being set to true, the expression !np returns the computed length as the function’s result.Anil Madhavapeddyhttps://www.cl.cam.ac.uk/teaching/2627/FoundsCS/forest/focs-def-mutable-reference/focs-def-mutable-reference/teaching/2627/FoundsCS/forest/focs-def-mutable-reference/Mutable referenceDefinitionA reference is a cell in the store, also called a “location”: ref E creates a reference with initial contents the value of E, !P returns the current contents of reference P (an operation called dereferencing), and P := E updates the contents of P to the value of E. If is some OCaml type, then ref is the type of references to cells that can hold values of . Introduced in OCaml Primitives for References.Part of the course glossary.Anil Madhavapeddyhttps://www.cl.cam.ac.uk/teaching/2627/FoundsCS/forest/focs-array-primitives/focs-array-primitives/teaching/2627/FoundsCS/forest/focs-array-primitives/OCaml Primitives for Arrays i * 10) (* array[5] initialised to (fun i) *) val aa : int array = [|0; 10; 20; 30; 40|] # Array.get aa 3 (* retrieve the 4th cell in the array *) - : int = 30 # Array.set aa 3 42 (* set the 4th cell's value to 42 *) - : unit = ()]]>There are many other array operations in the Array module in the OCaml standard library. 'a -> 'a array = # Array.init - : int -> (int -> 'a) -> 'a array = # Array.get - : 'a array -> int -> 'a = # Array.set - : 'a array -> int -> 'a -> unit = ]]>OCaml arrays are like references that hold several elements instead of one. The elements of an -element array are designated by the integers from 0 to . The th array element is usually written . If is a type then array is the type of arrays (of any size) with elements from .Calling Array.init n f creates an array of the size specified in by function . Initially, element holds the value of for , …, . Like ref, it allocates mutable storage to hold the specified values.Calling Array.get A i returns the contents of .Calling Array.set A i E modifies the array by storing the value of as the new contents of ; it returns () as its value.OCaml’s arrays are much safer than C’s. In C, an array is nothing more than an address indicating the start of a storage area. Nothing indicates the size of the area. Therefore C programs are vulnerable to buffer overrun attacks: an attacker sends more data than the receiving program expects, overrunning the area of storage set aside to hold it. The attack eventually overwrites the program itself, replacing it with code controlled by the attacker.Anil Madhavapeddyhttps://www.cl.cam.ac.uk/teaching/2627/FoundsCS/forest/focs-references/focs-references/teaching/2627/FoundsCS/forest/focs-references/OCaml Primitives for References SyntaxEffect ref Ecreate a reference with initial contents = value of E !Preturn current contents of reference P P := Eupdate contents of P to value of E The above text presents the OCaml primitives, but most languages have analogues of them, often heavily disguised. We need a means of creating references (or allocating storage), getting at the current contents of a reference cell, and updating that cell.The function ref creates references (also called “locations”). Calling ref allocates a new location in memory. Initially, this location holds the value given by expression .The function !, when applied to a reference, returns its contents. This operation is called dereferencing. Clearly ! is not a mathematical function; its result depends upon the store.The assignment P:=E evaluates expression , which must return a reference , and . It stores at address the value of . Syntactically, := is a function and P:=E is an expression, even though it updates the store. Like many functions that change the state, it returns the value () of type unit.If is some OCaml type, then ref is the type of references to cells that can hold values of . Please do not confuse the type ref with the function ref. This table of the primitive functions and their types might be useful: SyntaxOCaml Type ref'a -> 'a ref !'a ref -> 'a :='a ref -> 'a -> unit Anil Madhavapeddyhttps://www.cl.cam.ac.uk/teaching/2627/FoundsCS/forest/focs-private-references/focs-private-references/teaching/2627/FoundsCS/forest/focs-private-references/Private, Persistent References !balance then raise (TooMuch (amt - !balance)) else begin balance := !balance - amt; !balance end in withdraw val makeAccount : int -> int -> int = ]]>As you may have noticed, OCaml’s programming style looks clumsy compared with that of languages like C. OCaml omits the defaults and abbreviations they provide to shorten programs. However, OCaml’s explicitness makes it ideal for teaching the fine points of references and arrays. OCaml’s references are more flexible than those found in other languages.The function makeAccount models a bank. Calling the function with a specified initial balance creates a new reference (balance) to maintain the account balance and returns a function (withdraw) having sole access to that reference. Calling withdraw reduces the balance by the specified amount and returns the new balance. You can pay money in by withdrawing a negative amount. The if-construct prevents the account from going overdrawn, raising an exception.Look at the sequence inside begin … end in the else part above. The first expression updates the account balance and returns the trivial value (). The second expression, !balance, returns the current balance but does not return the reference itself: that would allow unauthorised updates.This example is based on one by Dr A C Norman.Anil Madhavapeddyhttps://www.cl.cam.ac.uk/teaching/2627/FoundsCS/forest/focs-procedural-programming/focs-procedural-programming/teaching/2627/FoundsCS/forest/focs-procedural-programming/Procedural ProgrammingProcedural programs can change the machine state. They can interact with its environment They use control structures like branching, iteration and procedures.They use data abstractions of the computer’s memory:references to memory cells arrays that are blocks of memory cells linked structures such as linked listsProcedural programming is programming in the traditional sense of the word. A program state is repeatedly transformed by the execution of commands or statements. A state change might be local to the machine and consist of updating a variable or array. A state change might consist of sending data to the outside world. Even reading data counts as a state change, since this act normally removes the data from the environment.Procedural programming languages provide primitive commands and control structures for combining them. The primitive commands include assignment for updating variables, and various input/output commands for communication. Control structures include if and match constructs for conditional execution, and repetitive constructs such as while. Programmers can package up their own commands as procedures taking arguments. The need for such “subroutines” was evident from the earliest days; they represent one of the first examples of abstraction in programming languages.OCaml makes no distinction between commands and expressions. OCaml provides built-in ‘functions’ to perform assignment and communication, and these can be used in the traditional (procedural) style. OCaml programmers often follow a functional style for most internal computations and use imperative features mainly for communication with the outside world.Anil Madhavapeddyhttps://www.cl.cam.ac.uk/teaching/2627/FoundsCS/forest/focs-recursion-vs-iteration/focs-recursion-vs-iteration/teaching/2627/FoundsCS/forest/focs-recursion-vs-iteration/Recursion vs Iteration“Iterative” normally refers to a loop, coded using while for example (see the final lecture) Tail-recursion is only efficient if the compiler detects it Mainly it saves space (memory), though iterative code can also run faster Do not make programs iterative unless the gain is worth itA classic book by Abelson and Sussman, which describes the Lisp dialect known as Scheme, used iterative to mean tail-recursive. Iterative functions produce computations resembling those that can be done using while-loops in conventional languages.Many algorithms can be expressed naturally using recursion, but only awkwardly using iteration. There is a story that Dijkstra sneaked recursion into Algol-60 by inserting the words “any other occurrence of the procedure name denotes execution of the procedure.” By not using the word “recursion”, he managed to slip this amendment past sceptical colleagues.Obsession with tail recursion leads to a coding style in which functions have many more arguments than necessary. Write straightforward code first, avoiding only gross inefficiency. If the program turns out to be too slow, tools are available for pinpointing the cause. Always remember KISS (Keep It Simple, Stupid).I hope you have all noticed by now that the summation can be done even more efficiently using the arithmetic progression formula:Anil Madhavapeddyhttps://www.cl.cam.ac.uk/teaching/2627/FoundsCS/forest/focs-references-comparison/focs-references-comparison/teaching/2627/FoundsCS/forest/focs-references-comparison/References: OCaml vs conventional languagesWe must write !p to get the contents of p We write just p for the address of p We can store private reference cells (like balance) in functions—analogous to elements of object-oriented programming OCaml’s assignment syntax is instead of = OCaml has few control structures: while, match, if and for (the latter is not covered in this course) OCaml has syntax for updating an array via the a.(i) <- v syntax which is the same as Array.set a i v.Conventional syntax for variables and assignments has hardly changed since Fortran, the first high-level language. In conventional languages, virtually all variables can be updated. We declare something like p: int, mentioning no reference type even if the language provides them. If we do not specify an initial value, we may get whatever bits were previously at that address. Illegal values arising from uninitialised variables can cause errors that are almost impossible to diagnose.Dereferencing operators (like OCaml’s !) are especially unpopular, because they clutter the program text. Virtually all programming languages make dereferencing implicit (that is, automatic).It is generally accepted these days that a two-dimensional array is nothing but an array of arrays. An assignment to such an array is typically written something like ; in C, the syntax is A[i][j] = x. Higher dimensions are treated analogously. The corresponding OCaml code can either declare an array of arrays, or use the A.(i) syntax to calculate the linear offset into a single array.You can use the constructs we have learnt to easily create linked (mutable) lists as an alternative to arrays.It is worth mentioning that OCaml’s references fully suffice for coding the sort of linked data structures taught in algorithms courses, as the mlist type above illustrates. The programming style is a little different from the usual, but the principles are the same. OCaml also provides comprehensive input/output primitives for various types of file and operating system.OCaml’s system of modules include structures, which can be seen as encapsulated groups of declarations, and signatures, which are specifications of structures listing the name and type of each component. Finally, there are functors, which are analogous to functions that combine a number of argument structures, and which can be used to plug program components together. These primitives are useful for managing large programming projects.Anil Madhavapeddyhttps://www.cl.cam.ac.uk/teaching/2627/FoundsCS/forest/focs-references-examples/focs-references-examples/teaching/2627/FoundsCS/forest/focs-references-examples/Trying Out ReferencesThe first line declares p to hold a reference to an integer, initially 5. Its type is int ref, not just int, so it admits assignment. Assignment never changes let bindings: they are immutable. The identifier p will always denote the reference mentioned in its declaration unless superseded by a new usage of p. Only the contents of the reference is mutable.OCaml displays a reference value as , where value is the contents. This notation is readable but gives us no way of telling whether two references holding the same value are actually the same reference. To display a reference as a machine address has obvious drawbacks!In the first assignment, the expression !p yields the reference’s current contents, namely 5. The assignment changes the contents of p to 6. Most languages do not have an explicit dereferencing operator (like !) because of its inconvenience. Instead, by convention, occurrences of the reference on the left-hand side of the := denote locations and those on the right-hand side denote the contents. A special ‘address of’ operator may be available to override the convention and make a reference on the right-hand side to denote a location. Logically this is a mess, but it makes programs shorter.The list ps is declared to hold a new reference (initially containing 77) as well as p. Then the new reference is updated to hold 3. The assignment to hd ps does not update ps, only the contents of a reference in that list.Anil Madhavapeddyhttps://www.cl.cam.ac.uk/teaching/2627/FoundsCS/forest/focs-bank-accounts/focs-bank-accounts/teaching/2627/FoundsCS/forest/focs-bank-accounts/Two Bank Accounts int = # let director = makeAccount 4000000 val director : int -> int = # student 5 (* coach fare *) - : int = 495 # director 150000 (* Tesla *) - : int = 3850000 # student 500 (* oh oh *) Exception: TooMuch 5.]]>Each call to makeAccount returns a copy of withdraw holding a fresh instance of the reference balance. As with a real bank pass-book, there is no access to the account balance except via the corresponding withdraw function. If that function is discarded, the reference cell becomes unreachable; the computer will eventually reclaim it, just as banks close down dormant accounts.Here we see two people managing their accounts. For better or worse, neither can take money from the other.We could generalise makeAccount to return several functions that jointly manage information held in shared references. The functions might be packaged using OCaml records, which are not discussed in this course. Most procedural languages do not properly support the concept of private references, although object-oriented languages take them as a basic theme.Anil MadhavapeddyAnil Madhavapeddyhttps://www.cl.cam.ac.uk/teaching/2627/FoundsCS/forest/focs-ex-11-3/focs-ex-11-3/teaching/2627/FoundsCS/forest/focs-ex-11-3/Exercises › Effects in a while conditionExerciseWhat is the effect of ?Anil MadhavapeddyAnil Madhavapeddyhttps://www.cl.cam.ac.uk/teaching/2627/FoundsCS/forest/focs-ex-11-4/focs-ex-11-4/teaching/2627/FoundsCS/forest/focs-ex-11-4/Exercises › Exchanging two referencesExerciseWrite a function to exchange the values of two references, xr and yr.Anil MadhavapeddyAnil Madhavapeddyhttps://www.cl.cam.ac.uk/teaching/2627/FoundsCS/forest/focs-ex-11-2/focs-ex-11-2/teaching/2627/FoundsCS/forest/focs-ex-11-2/Exercises › Iterative power with whileExerciseWrite a version of function power (Lecture ) using while instead of recursion.Anil MadhavapeddyAnil Madhavapeddyhttps://www.cl.cam.ac.uk/teaching/2627/FoundsCS/forest/focs-ex-11-1/focs-ex-11-1/teaching/2627/FoundsCS/forest/focs-ex-11-1/Exercises › Lists of referencesExerciseComment, with examples, on the differences between an int ref list and an int list ref.Anil Madhavapeddyhttps://www.cl.cam.ac.uk/teaching/2627/FoundsCS/forest/focs-topic-types/focs-topic-types/teaching/2627/FoundsCS/forest/focs-topic-types/TypesAnil Madhavapeddyhttps://www.cl.cam.ac.uk/teaching/2627/FoundsCS/forest/focs-constructors-args/focs-constructors-args/teaching/2627/FoundsCS/forest/focs-constructors-args/A Datatype whose Constructors have ArgumentsConstructors with arguments (like Lorry) are distinct values. (So Car true is distinct from Car false). Different kinds of vehicle can belong to one list: [Bike; Car true; Motorbike 450]OCaml generalises the notion of enumeration type to allow data to be associated with each constructor. The constructor Bike is a vehicle all by itself, but the other three constructors create vehicles from arguments.Since we might find it hard to remember what the various int and bool components are for, it is wise to include comments in complex declarations. In OCaml, comments are enclosed in the brackets and . Programmers should comment their code to explain design decisions and key features of the algorithms (sometimes by citing a reference work).The list [Bike; Car true; Motorbike 450] represents a bicycle, a Reliant Robin and a large motorbike. It can be almost seen as a mixed-type list containing integers and booleans. It is actually a list of vehicles; datatypes lessen the impact of the restriction that all list elements must have the same type.Anil Madhavapeddyhttps://www.cl.cam.ac.uk/teaching/2627/FoundsCS/forest/focs-first-session/focs-first-session/teaching/2627/FoundsCS/forest/focs-first-session/A first session with OCamlThe first line of this simple session is a value declaration. It makes the name pi stand for the floating point number 3.14159. (Such names are called identifiers.) OCaml echoes the name (pi) and type (float) of the declared identifier.The second line computes the area of the circle with radius 1.5 using the formula . We use pi as an abbreviation for 3.14159. Multiplication is expressed using *., which is called an infix operator because it is written between its two operands.OCaml replies with the computed value (about 7.07) and its type (again float). float = ]]>To work abstractly, we should provide the service “compute the area of a circle,” so that we no longer need to remember the formula. This sort of encapsulated computation is called a function. The third line declares the function area. Given any floating point number r, it returns another floating point number computed using the area formula; note that the function has type float -> float.The fourth line calls the function area supplying 2.0 as the argument. A circle of radius 2 has an area of about 12.6. Note that brackets around a function argument are not necessary.The function uses pi to stand for 3.14159. Unlike what you may have seen in other programming languages, pi cannot be "assigned to" or otherwise updated. Its meaning within area will persist even if we issue a new let declaration for pi afterwards.Anil Madhavapeddyhttps://www.cl.cam.ac.uk/teaching/2627/FoundsCS/forest/focs-def-abstraction-barrier/focs-def-abstraction-barrier/teaching/2627/FoundsCS/forest/focs-def-abstraction-barrier/Abstraction barrierDefinitionLarge systems can only be understood in levels: the interface to the higher level should supply the advertised services and, just as important, block access to the means by which those services are implemented. This abstraction barrier allows one level to be changed without affecting levels above. Introduced in Basic Concepts in Computer Science.Part of the course glossary.Anil Madhavapeddyhttps://www.cl.cam.ac.uk/teaching/2627/FoundsCS/forest/focs-def-algebraic-datatype/focs-def-algebraic-datatype/teaching/2627/FoundsCS/forest/focs-def-algebraic-datatype/Algebraic datatypeDefinitionOCaml generalises the notion of enumeration type to allow data to be associated with each constructor: some constructors are values all by themselves, while others create values from arguments. Constructors with arguments are distinct values, and values built from different constructors of one datatype can belong to a single list. Introduced in A Datatype whose Constructors have Arguments.Part of the course glossary.Anil Madhavapeddyhttps://www.cl.cam.ac.uk/teaching/2627/FoundsCS/forest/focs-enumeration/focs-enumeration/teaching/2627/FoundsCS/forest/focs-enumeration/An Enumeration TypeWe will now learn how to define more expressive types than the basic ones supplied with the core OCaml language.We have declared a new type named vehicle. along with four new constants. They are the constructors of the datatype.The type declaration adds a new type to our OCaml session. Type vehicle is as good as any built-in type and even admits pattern-matching (as we used with the built-in list types earlier). The four new identifiers of type vehicle are called constructors.We could represent the various vehicles by the numbers 0–3. However, the code would be hard to read and even harder to maintain. Consider adding Tricycle as a new vehicle. If we wanted to add it before Bike, then all the numbers would have to be changed. Using type, such additions are trivial and the compiler can (at least sometimes) warn us when it encounters a function declaration that doesn’t yet have a case for Tricycle.Representing vehicles by strings like "Bike", "Car", etc., is also bad. Comparing string values is slow and the compiler can’t warn us of misspellings like "MOtorbike": they will make our code fail.Most programming languages allow the declaration of types like vehicle. Because they consist of a series of identifiers, they are called enumeration types. Other common examples are days of the week or colours. The compiler chooses the integers for us; type-checking prevents us from confusing Bike with Red or Sunday.Anil Madhavapeddyhttps://www.cl.cam.ac.uk/teaching/2627/FoundsCS/forest/focs-basic-concepts/focs-basic-concepts/teaching/2627/FoundsCS/forest/focs-basic-concepts/Basic Concepts in Computer ScienceComputers: a child can use them; nobody can fully understand them! We can master complexity through levels of abstraction. Focus on 2 or 3 levels at most!Recurring issues:what services to provide at each level how to implement them using lower-level services the interface that defines how the two levels should communicateA basic concept in computer science is that large systems can only be understood in levels, with each level further subdivided into functions or services of some sort. The interface to the higher level should supply the advertised services. Just as important, it should block access to the means by which those services are implemented. This abstraction barrier allows one level to be changed without affecting levels above. For example, when a manufacturer designs a faster version of a processor, it is essential that existing programs continue to run on it. Any differences between the old and new processors should be invisible to the program.Modern processors have elaborate specifications, which still sometimes leave out important details. In the old days, you then had to consult the circuit diagrams. Anil Madhavapeddyhttps://www.cl.cam.ac.uk/teaching/2627/FoundsCS/forest/focs-dates/focs-dates/teaching/2627/FoundsCS/forest/focs-dates/Example 1: DatesAbstract level: dates over a certain interval Concrete level: could be 6 characters: YYMMDD (where each character is represented by 8 bits) Date crises caused by inadequate internal formats: Digital’s PDP-10: using 12-bit dates (good for at most 11 years) 2000 crisis: 48 bits could be good for lifetime of universe!Digital Equipment Corporation’s date crisis occurred in 1975. The PDP-10 was a 36-bit mainframe computer. It represented dates using a 12-bit format designed for the tiny PDP-8. With 12 bits, one can distinguish days or 11 years.Last century, the most common industry format for dates used six characters: two for the year, two for the month and two for the day. The most common “solution” to the year 2000 crisis is to add two further characters, thereby altering file sizes. Others have noticed that the existing six characters consist of 48 bits, already sufficient to represent all dates over the projected lifetime of the universe: = days = years!Mathematicians think in terms of unbounded ranges, but the representation we choose for the computer usually imposes hard limits. A good programming language like OCaml lets one easily change the representation used in the program. But if files in the old representation exist all over the place, there will still be conversion problems. The need for compatibility with older systems causes problems across the computer industry. Anil Madhavapeddyhttps://www.cl.cam.ac.uk/teaching/2627/FoundsCS/forest/focs-floating-point/focs-floating-point/teaching/2627/FoundsCS/forest/focs-floating-point/Example 2: Floating Point NumbersComputers have integers like 1066 and floats like . A floating-point number is represented by two integers. The concept of data type involves:how a value is represented inside the computer the suite of operations given to programmers valid and invalid (or exceptional) results, such as “infinity”Computer arithmetic can yield incorrect answers!In science, numbers written with finite precision and a decimal exponent are said to be in standard form. The computational equivalent is the floating point number. These are familiar to anybody who has used a scientific calculator. Internally, a float consists of two integers.Because of its finite precision, floating-point computations are potentially inaccurate. To see an example, use your nearest electronic calculator to compute . I get ! With certain computations, the errors spiral out of control. Many programming languages fail to check whether even integer computations fall within the allowed range: you can add two positive integers and get a negative one!Most computers give us a choice of precisions. In 32-bit precision, integers typically range from (namely 2 147 483 647) to ; floats are accurate to about six decimal places and can get as large as or so. For floats, 64-bit precision is often preferred. Early languages like Fortran required variables to be declared as INTEGER, REAL or COMPLEX and barred programmers from mixing numbers in a computation. Nowadays, programs handle many different kinds of data, including text and symbols. The concept of a data type can ensure that different types of data are not combined in a senseless way.Inside the computer, all data are stored as bits. In most programming languages, the compiler uses types to generate correct machine code, and types are not stored during program execution. In this course, we focus almost entirely on programming in a high-level language: OCaml. Anil Madhavapeddyhttps://www.cl.cam.ac.uk/teaching/2627/FoundsCS/forest/focs-vehicles/focs-vehicles/teaching/2627/FoundsCS/forest/focs-vehicles/Declaring a Function on Vehicles 2 | Motorbike -> 2 | Car -> 4 | Lorry -> 18 val wheels : vehicle -> int = ]]>Datatype constructors can be used in patterns. Pattern-matching is fast, even complicated nested patterns. Notice the type of the argument is vehicle, which we defined earlier.The beauty of datatype declarations is that the new types behave as if they were built into OCaml. Type-checking catches common errors, such as mixing up different datatypes in a function like wheels, as well as missing and redundant patterns.Anil Madhavapeddyhttps://www.cl.cam.ac.uk/teaching/2627/FoundsCS/forest/focs-dictionary-adt/focs-dictionary-adt/teaching/2627/FoundsCS/forest/focs-dictionary-adt/Dictionarieslookup: find an item in the dictionary update (insert): replace (store) an item in the dictionary delete: remove an item from the dictionary empty: the null dictionary Missing: exception for errors in lookup and deleteIdeally, an abstract type should provide these operations and hide the internal data structures.A dictionary attaches values to identifiers, called “keys”. Before choosing the internal representation for a data structure, you need to specify the full set of operations. In fact, here we only consider update (associating a value with an identifier) and lookup (retrieving such a value). Deletion is more difficult and would limit our choices. Some applications may need additional operations, such as merge (combining two dictionaries). We shall see that update can be done efficiently in a functional style, without excessive copying.An abstract type provides specified operations while hiding low-level details, such as the data structure used to represent dictionaries. Abstract types can be declared in any modern programming language. Java’s objects serve this role, as do OCaml’s modules. This course does not cover modules, and we simply declare the dictionary operations individually.An association list (a list of pairs) is the simplest dictionary representation. Lookup is by linear search, and therefore slow: . Association lists are only usable if there are few keys in use. However, they are general in that the keys do not need a concept of ordering, only equality. raise Missing | (x, y) :: pairs -> if a = x then y else lookup a pairs val lookup : 'a -> ('a * 'b) list -> 'b = # let update (l, b, y) = (b, y) :: l val update : ('a * 'b) list * 'a * 'b -> ('a * 'b) list = ]]>To enter a new (key, value) pair, simply “cons” it to the list with update. This takes constant time, which is the best we could hope for. But the space requirement is huge: linear in the number of updates, not in the number of distinct keys. Obsolete entries are never deleted: that would require first finding them, increasing the update time from to .Anil Madhavapeddyhttps://www.cl.cam.ac.uk/teaching/2627/FoundsCS/forest/focs-equality/focs-equality/teaching/2627/FoundsCS/forest/focs-equality/Equality Tests false | y::l -> if x = y then true else member x l val member : 'a -> 'a list -> bool = ]]>All the list functions we have encountered up to now have been “polymorphic”, working for lists of any type. Function member uses linear search to report whether or not x occurs in l.To do this generically, it uses a special feature of OCaml known as “polymorphic equality”, which manifests itself via the =, >=, <=, > and < operators. These operators inspect the structure of the values using a consistent order. Types you can legitimately compare this way include integers, strings, booleans, and tuples or lists of primitive types.More complex types can be compared this way within careful limits: recursive structures or function values will not work (we will cover function values in the Currying lecture later). For now, it is sufficient to use these magic polymorphic equality operators. As you get more familiar with OCaml and the use of higher order functions (also covered in a later lecture), you will encounter the use of explicit compare functions that are used to provide more complex equality tests.The presence of polymorphic equality is a contentious feature in OCaml. While it provides a great ease of use in smaller codebases, it starts to become more dangerous when building larger OCaml-based systems. Most large-scale users of OCaml tend towards not using it in important code, but it is just fine for our purposes while learning the beginning steps of computer science.Anil Madhavapeddyhttps://www.cl.cam.ac.uk/teaching/2627/FoundsCS/forest/focs-dates/focs-dates/teaching/2627/FoundsCS/forest/focs-dates/Example 1: DatesAbstract level: dates over a certain interval Concrete level: could be 6 characters: YYMMDD (where each character is represented by 8 bits) Date crises caused by inadequate internal formats: Digital’s PDP-10: using 12-bit dates (good for at most 11 years) 2000 crisis: 48 bits could be good for lifetime of universe!Digital Equipment Corporation’s date crisis occurred in 1975. The PDP-10 was a 36-bit mainframe computer. It represented dates using a 12-bit format designed for the tiny PDP-8. With 12 bits, one can distinguish days or 11 years.Last century, the most common industry format for dates used six characters: two for the year, two for the month and two for the day. The most common “solution” to the year 2000 crisis is to add two further characters, thereby altering file sizes. Others have noticed that the existing six characters consist of 48 bits, already sufficient to represent all dates over the projected lifetime of the universe: = days = years!Mathematicians think in terms of unbounded ranges, but the representation we choose for the computer usually imposes hard limits. A good programming language like OCaml lets one easily change the representation used in the program. But if files in the old representation exist all over the place, there will still be conversion problems. The need for compatibility with older systems causes problems across the computer industry.Anil Madhavapeddyhttps://www.cl.cam.ac.uk/teaching/2627/FoundsCS/forest/focs-floating-point/focs-floating-point/teaching/2627/FoundsCS/forest/focs-floating-point/Example 2: Floating Point NumbersComputers have integers like 1066 and floats like . A floating-point number is represented by two integers. The concept of data type involves:how a value is represented inside the computer the suite of operations given to programmers valid and invalid (or exceptional) results, such as “infinity”Computer arithmetic can yield incorrect answers!In science, numbers written with finite precision and a decimal exponent are said to be in standard form. The computational equivalent is the floating point number. These are familiar to anybody who has used a scientific calculator. Internally, a float consists of two integers.Because of its finite precision, floating-point computations are potentially inaccurate. To see an example, use your nearest electronic calculator to compute . I get ! With certain computations, the errors spiral out of control. Many programming languages fail to check whether even integer computations fall within the allowed range: you can add two positive integers and get a negative one!Most computers give us a choice of precisions. In 32-bit precision, integers typically range from (namely 2 147 483 647) to ; floats are accurate to about six decimal places and can get as large as or so. For floats, 64-bit precision is often preferred. Early languages like Fortran required variables to be declared as INTEGER, REAL or COMPLEX and barred programmers from mixing numbers in a computation. Nowadays, programs handle many different kinds of data, including text and symbols. The concept of a data type can ensure that different types of data are not combined in a senseless way.Inside the computer, all data are stored as bits. In most programming languages, the compiler uses types to generate correct machine code, and types are not stored during program execution. In this course, we focus almost entirely on programming in a high-level language: OCaml.Anil Madhavapeddyhttps://www.cl.cam.ac.uk/teaching/2627/FoundsCS/forest/focs-anonymous-functions/focs-anonymous-functions/teaching/2627/FoundsCS/forest/focs-anonymous-functions/Functions Without NamesIf functions are to be regarded as computational values, then we need a notation for them. The fun notation expresses a non-recursive function value without giving the function a name. is the function such that . The function fun n -> n*2 is a doubling function. n * 2 - : int -> int = # (fun n -> n * 2) 17 - : int = 34]]>The main purpose of fun-notation is to package up small expressions that are to be applied repeatedly using some other function. The expression fun n -> n*2 has the same value as the identifier double, declared as follows: int = ]]>The fun notation can also do pattern matching, and the function keyword adds an anonymous variable name to pattern match against. The following functions are all equivalent, with the latter definitions bound to the is_zero value and the earlier ones anonymous: match x with 0 -> true | _ -> false - : int -> bool = # function 0 -> true | _ -> false - : int -> bool = # let is_zero = fun x -> match x with 0 -> true | _ -> false val is_zero : int -> bool = # let is_zero = function 0 -> true | _ -> false val is_zero : int -> bool = ]]>Anil Madhavapeddyhttps://www.cl.cam.ac.uk/teaching/2627/FoundsCS/forest/focs-head-tail/focs-head-tail/teaching/2627/FoundsCS/forest/focs-head-tail/Getting at the Head and Tail true | x :: l -> false val null : 'a list -> bool = # null [] - : bool = true # null [1; 2; 3] - : bool = false # let hd (x::l) = x Line 1, characters 8-14: Warning 8 [partial-match]: this pattern-matching is not exhaustive. Here is an example of a case that is not matched: [] val hd : 'a list -> 'a = # hd [1; 2; 3] - : int = 1 # let tl (x::l) = l Line 1, characters 8-14: Warning 8 [partial-match]: this pattern-matching is not exhaustive. Here is an example of a case that is not matched: [] val tl : 'a list -> 'a list = # tl [7; 6; 5] - : int list = [6; 5]]]>The empty list has neither head nor tail. Applying hd or tl to [] is an error—strictly speaking, an “exception”. The function null can be used to check for the empty list beforehand. Taking a list apart using combinations of hd and tl is hard to get right. Fortunately, it is seldom necessary because of pattern-matching.The declaration of null introduces a new concept known as "pattern matching", which we will explore more in subsequent lectures. For now, it is sufficient to observe that let null = function allows for matching on the two possible values that might be passed in as argument to null here: one for the empty list (for which it returns true) and one for non-empty lists (for which it returns false).The declaration of hd above has only one clause, for non-empty lists. They have the form x::l and the function returns x, which is the head. If you compile this program, OCaml also prints a warning to tell us that calling the function could raise an exception because not all possible inputs are handled, including a counter-example (in this case, the empty list []). The declaration of tl is similar to hd.These three primitive functions are polymorphic and allow flexibility in the types of their arguments and results. Note their types! bool = # hd - : 'a list -> 'a = # tl - : 'a list -> 'a list = ]]>Symbols 'a and 'b are called type variables and stand for any types. Code written using these functions is checked for type correctness at compile time. And this guarantees strong properties at run time, for example that the elements of any list all have the same type. They are usually read as their corresponding greek characters; 'a is "alpha", 'b is "beta", and so on.Anil Madhavapeddyhttps://www.cl.cam.ac.uk/teaching/2627/FoundsCS/forest/focs-goals-of-programming/focs-goals-of-programming/teaching/2627/FoundsCS/forest/focs-goals-of-programming/Goals of Programmingto describe a computation so that it can be done mechanically: Expressions compute values. Commands cause effects. to do so efficiently and correctly, giving the right answers quickly to allow easy modification as needs change Through an orderly structure based on abstraction principles Such as modules or classesProgramming in-the-small concerns the writing of code to do simple, clearly defined tasks. Programs provide expressions for describing mathematical formulae and so forth. This was the original contribution of FORTRAN, the FORmula TRANslator. Commands describe how control should flow from one part of the program to the next.As we code layer upon layer, we eventually find ourselves programming in the large : joining large modules to solve some messy task. Programming languages have used various mechanisms to allow one part of the program to provide interfaces to other parts. Modules encapsulate a body of code, allowing outside access only through a programmer-defined interface. Abstract Data Types are a simpler version of this concept, which implement a single concept such as dates or floating-point numbers.Object-oriented programming is the most complicated approach to modularity. Classes define concepts, and they can be built upon other classes. Operations can be defined that work in appropriately specialised ways on a family of related classes. Objects are instances of classes and hold the data that is being manipulated.This course does not cover OCaml’s sophisticated module system, which can do many of the same things as classes. You will learn all about objects when you study Java. OCaml includes a powerful object system, although this is not used as much as its module system.Anil Madhavapeddyhttps://www.cl.cam.ac.uk/teaching/2627/FoundsCS/forest/focs-lazy-ocaml/focs-lazy-ocaml/teaching/2627/FoundsCS/forest/focs-lazy-ocaml/Lazy Lists in OCamlThe empty tuple () and its type unit Delayed version of is fun () -> E 'a seq) type 'a seq = Nil | Cons of 'a * (unit -> 'a seq) # let head (Cons (x, _)) = x Line 1, characters 10-23: Warning 8 [partial-match]: this pattern-matching is not exhaustive. Here is an example of a case that is not matched: Nil val head : 'a seq -> 'a = # let tail (Cons (_, xf)) = xf () Line 1, characters 10-24: Warning 8 [partial-match]: this pattern-matching is not exhaustive. Here is an example of a case that is not matched: Nil val tail : 'a seq -> 'a seq = ]]> has head and tail function The primitive OCaml type unit has one element, which is written (). This element may be regarded as a 0-tuple, and unit as the nullary Cartesian product. (Think of the connection between multiplication and the number 1.)The empty tuple serves as a placeholder in situations where no information is required. It may:appear in a data structure. For example, a unit-valued dictionary represents a set of keys. be the argument of a function, where its effect is to delay evaluation. be the argument or result of a procedure. (see the Procedural Programming section)The empty tuple, like all tuples, is a constructor and is allowed in patterns; for example: let f () = ...In particular is the function that takes an argument of type unit and returns the value of as its result. Expression is not evaluated until the function is called, even though the only possible argument is (). The function simply delays the evaluation of .Anil Madhavapeddyhttps://www.cl.cam.ac.uk/teaching/2627/FoundsCS/forest/focs-def-mutable-reference/focs-def-mutable-reference/teaching/2627/FoundsCS/forest/focs-def-mutable-reference/Mutable referenceDefinitionA reference is a cell in the store, also called a “location”: ref E creates a reference with initial contents the value of E, !P returns the current contents of reference P (an operation called dereferencing), and P := E updates the contents of P to the value of E. If is some OCaml type, then ref is the type of references to cells that can hold values of . Introduced in OCaml Primitives for References.Part of the course glossary.Anil Madhavapeddyhttps://www.cl.cam.ac.uk/teaching/2627/FoundsCS/forest/focs-references/focs-references/teaching/2627/FoundsCS/forest/focs-references/OCaml Primitives for References SyntaxEffect ref Ecreate a reference with initial contents = value of E !Preturn current contents of reference P P := Eupdate contents of P to value of E The above text presents the OCaml primitives, but most languages have analogues of them, often heavily disguised. We need a means of creating references (or allocating storage), getting at the current contents of a reference cell, and updating that cell.The function ref creates references (also called “locations”). Calling ref allocates a new location in memory. Initially, this location holds the value given by expression .The function !, when applied to a reference, returns its contents. This operation is called dereferencing. Clearly ! is not a mathematical function; its result depends upon the store.The assignment P:=E evaluates expression , which must return a reference , and . It stores at address the value of . Syntactically, := is a function and P:=E is an expression, even though it updates the store. Like many functions that change the state, it returns the value () of type unit.If is some OCaml type, then ref is the type of references to cells that can hold values of . Please do not confuse the type ref with the function ref. This table of the primitive functions and their types might be useful: SyntaxOCaml Type ref'a -> 'a ref !'a ref -> 'a :='a ref -> 'a -> unit Anil Madhavapeddyhttps://www.cl.cam.ac.uk/teaching/2627/FoundsCS/forest/focs-def-pattern-matching/focs-def-pattern-matching/teaching/2627/FoundsCS/forest/focs-def-pattern-matching/Pattern matchingDefinitionPattern matching takes a value apart by matching it against the forms that value can have, letting one function declaration handle several cases, separated by a vertical bar |. Taking a list apart using combinations of hd and tl is hard to get right; fortunately, it is seldom necessary because of pattern-matching. Introduced in Getting at the Head and Tail.Part of the course glossary.Anil Madhavapeddyhttps://www.cl.cam.ac.uk/teaching/2627/FoundsCS/forest/focs-def-polymorphism/focs-def-polymorphism/teaching/2627/FoundsCS/forest/focs-def-polymorphism/PolymorphismDefinitionA polymorphic function allows flexibility in the types of its arguments and results: the list primitives null, hd and tl work whatever the element type is. Symbols 'a and 'b are called type variables and stand for any types, and code written using them is still checked for type correctness at compile time. Introduced in Getting at the Head and Tail.Part of the course glossary.Anil Madhavapeddyhttps://www.cl.cam.ac.uk/teaching/2627/FoundsCS/forest/focs-npower/focs-npower/teaching/2627/FoundsCS/forest/focs-npower/Raising a Number to a Power int -> float = ]]>Our new npower definition can now take additional arguments, reflected in the arrows present in the type of npower; these represent parameters that can be passed to the new value being defined, with the final segment being the resulting type. Thus our npower type can be read as "pass in a float and integer to return a float".Mathematical Justification (for ):The function npower raises its float argument x to the power n, a non-negative integer. The function is recursive: it calls itself. You can spot a recursive function due to the rec keyword in the definition: this indicates that any invocation of the function name within the function body should call itself. This concept should be familiar from mathematics, since exponentiation is defined by the rules shown above. You may also have seen recursion in the product rule for differentiation: . In finding the derivative of , we recursively find the derivatives of and , combining them to obtain the desired result. The recursion is meaningful because it terminates: we reduce the problem to two smaller problems, and this cannot go on forever. The OCaml programmer uses recursion heavily. For , the equation yields an obvious computation:The equation clearly holds even for negative . However, the corresponding computation runs forever:Note that the function npower contains both an integer constant (0) and a floating point constant (1.0). The decimal point makes all the difference. OCaml will notice and ascribe different meaning to each type of constant. float = ]]>Now for a tiresome but necessary aside. In most languages, the types of arguments and results must always be specified. OCaml is unusual that it normally infers the types itself. However, sometimes it is useful to supply a hint to help you debug and develop your program. OCaml will still infer the types even if you don’t specify them, but in some cases it will use a more inefficient function than a specialised one. Some languages have just one type of number, converting automatically between different formats; this is slow and could lead to unexpected rounding errors. Type constraints are allowed almost anywhere. We can put one on any occurrence of x in the function. float = ]]>Or we can constrain the type of the function’s result: float = ]]>OCaml treats the equality and comparison test specially. Expressions like if x = y then … are allowed provided x and y have the same type and equality testing is possible for that type. (We discuss equality further in a later lecture.) Note that x <> y is OCaml for .A characteristic feature of the computer is its ability to test for conditions and act accordingly. In the early days, a program might jump to a given address depending on the sign of some number. Later, John McCarthy defined the conditional expression to satisfy if true then x else y = x and if false then x else y = y.OCaml evaluates the expression if then else by first evaluating . If the result is true then OCaml evaluates and otherwise . Only one of the two expressions and is evaluated! If both were evaluated, then recursive functions like npower above would run forever.The if-expression is governed by an expression of type bool, whose two values are true and false. In modern programming languages, tests are not built into “conditional branch” constructs but can just be part of normal expressions. Tests, or Boolean expressions, can be expressed using relational operators such as < and =. They can be combined using the Boolean operators for negation (not), conjunction (written as &&) and disjunction (written as ||). New properties can be declared as functions: here, to test whether an integer is even, for example: bool = ]]>Anil Madhavapeddyhttps://www.cl.cam.ac.uk/teaching/2627/FoundsCS/forest/focs-references-examples/focs-references-examples/teaching/2627/FoundsCS/forest/focs-references-examples/Trying Out ReferencesThe first line declares p to hold a reference to an integer, initially 5. Its type is int ref, not just int, so it admits assignment. Assignment never changes let bindings: they are immutable. The identifier p will always denote the reference mentioned in its declaration unless superseded by a new usage of p. Only the contents of the reference is mutable.OCaml displays a reference value as , where value is the contents. This notation is readable but gives us no way of telling whether two references holding the same value are actually the same reference. To display a reference as a machine address has obvious drawbacks!In the first assignment, the expression !p yields the reference’s current contents, namely 5. The assignment changes the contents of p to 6. Most languages do not have an explicit dereferencing operator (like !) because of its inconvenience. Instead, by convention, occurrences of the reference on the left-hand side of the := denote locations and those on the right-hand side denote the contents. A special ‘address of’ operator may be available to override the convention and make a reference on the right-hand side to denote a location. Logically this is a mess, but it makes programs shorter.The list ps is declared to hold a new reference (initially containing 77) as well as p. Then the new reference is updated to hold 3. The assignment to hd ps does not update ps, only the contents of a reference in that list.Anil Madhavapeddyhttps://www.cl.cam.ac.uk/teaching/2627/FoundsCS/forest/focs-why-ocaml/focs-why-ocaml/teaching/2627/FoundsCS/forest/focs-why-ocaml/Why Program in OCaml?Why program in OCaml at all?It is interactive. It has a flexible notion of data type. It hides the underlying hardware: no crashes. Programs can easily be understood mathematically. It distinguishes naming something from updating memory. It manages storage for us.Programming languages matter. They affect the reliability, security, and efficiency of the code you write, as well as how easy it is to read, refactor, and extend. The languages you know can also change how you think, influencing the way you design software even when you’re not using them.What makes OCaml special is that it occupies a sweet spot in the space of programming language designs. It provides a combination of efficiency, expressiveness and practicality that is difficult to find matched by any other language. “ML” was originally the meta language of the LCF (Logic for Computable Functions) proof assistant released by Robin Milner in 1972 (at Stanford, and later at Cambridge). ML was turned into a compiler in order to make it easier to use LCF on different machines, and it was gradually turned into a full-fledged system of its own by the 1980s.The modern OCaml emerged in 1996, and the past twenty five years have seen OCaml attract a significant user base with language improvements being steadily added to support the growing commercial and academic codebases. OCaml is therefore the outcome of years of research into programming languages, and a good base to begin our journey into learning the foundations of computer science.Because of its connection to mathematics, OCaml programs can be designed and understood without thinking in detail about how the computer will run them. Although a program can abort, it cannot crash: it remains under the control of the OCaml system. It still achieves respectable efficiency and provides lower-level primitives for those who need them. Most other languages allow direct access to the underlying machine and even try to execute illegal operations, causing crashes.The only way to learn programming is by writing and running programs. This web notebook provides an interactive environment where you can modify the example fragments and see the results for yourself. You should also consider installing OCaml on your own computer so that you try more advanced programs locally.Anil MadhavapeddyAnil Madhavapeddyhttps://www.cl.cam.ac.uk/teaching/2627/FoundsCS/forest/focs-ex-4-3/focs-ex-4-3/teaching/2627/FoundsCS/forest/focs-ex-4-3/Exercises › A non-exhaustive zipExerciseHow does this version of zip differ from the one above? (x, y) :: zip xs ys | ([], []) -> []]]>Anil MadhavapeddyAnil Madhavapeddyhttps://www.cl.cam.ac.uk/teaching/2627/FoundsCS/forest/focs-ex-6-4/focs-ex-6-4/teaching/2627/FoundsCS/forest/focs-ex-6-4/Exercises › A type for arithmetic expressionsExerciseGive the declaration of an OCaml type for arithmetic expressions that have the following possibilities: floating-point numbers, variables (represented by strings), or expressions of the form , , .Anil MadhavapeddyAnil Madhavapeddyhttps://www.cl.cam.ac.uk/teaching/2627/FoundsCS/forest/focs-ex-1-2/focs-ex-1-2/teaching/2627/FoundsCS/forest/focs-ex-1-2/Exercises › Comparing and adding yearsExerciseUsing the date representation of the previous exercise, code OCaml functions to (a) compare two years (b) add/subtract some given number of years from another year.Anil MadhavapeddyAnil Madhavapeddyhttps://www.cl.cam.ac.uk/teaching/2627/FoundsCS/forest/focs-ex-6-1/focs-ex-6-1/teaching/2627/FoundsCS/forest/focs-ex-6-1/Exercises › Days of the week typeExerciseGive the declaration of an OCaml type for the days of the week. Comment on the practicality of such a type in a calendar application.Anil MadhavapeddyAnil Madhavapeddyhttps://www.cl.cam.ac.uk/teaching/2627/FoundsCS/forest/focs-ex-1-5/focs-ex-1-5/teaching/2627/FoundsCS/forest/focs-ex-1-5/Exercises › Floating-point error by repeated additionExerciseBecause computer arithmetic is based on binary numbers, simple decimals such as 0.1 often cannot be represented exactly. Write a function mul that performs the computationwhere has type float. (It is essential to use repeated addition rather than multiplication!)The value computed with n = 10000 and x = 0.1 may print as 1000.0, which looks exact. If that happens, then evaluate the expression mul 0.1 10000 -. 1000.0An error of this type has been blamed for the failure of an American Patriot Missile battery to intercept an incoming Iraqi missile during the first Gulf War. The missile hit an American Army barracks, killing 28.Anil MadhavapeddyAnil Madhavapeddyhttps://www.cl.cam.ac.uk/teaching/2627/FoundsCS/forest/focs-ex-1-6/focs-ex-1-6/teaching/2627/FoundsCS/forest/focs-ex-1-6/Exercises › Golden ratio instabilityExerciseAnother example of the inaccuracy of floating-point arithmetic takes the golden ratio as its starting point:In theory, it is easy to prove that for all 0]]>. Code this computation in OCaml and report the value of . Hint: in OCaml, is expressed as sqrt 5.0.Anil MadhavapeddyAnil Madhavapeddyhttps://www.cl.cam.ac.uk/teaching/2627/FoundsCS/forest/focs-ex-11-1/focs-ex-11-1/teaching/2627/FoundsCS/forest/focs-ex-11-1/Exercises › Lists of referencesExerciseComment, with examples, on the differences between an int ref list and an int list ref.Anil MadhavapeddyAnil Madhavapeddyhttps://www.cl.cam.ac.uk/teaching/2627/FoundsCS/forest/focs-ex-3-4/focs-ex-3-4/teaching/2627/FoundsCS/forest/focs-ex-3-4/Exercises › Polymorphic types of id and loopExerciseConsider the polymorphic types in these two function declarations: 'a = # let rec loop x = loop x val loop : 'a -> 'b = ]]>Explain why these types make logical sense, preventing run time type errors, even for expressions like id [id [id 0]] or loop true / loop 3. (/ is the integer division operator in OCaml)Anil MadhavapeddyAnil Madhavapeddyhttps://www.cl.cam.ac.uk/teaching/2627/FoundsCS/forest/focs-ex-4-6/focs-ex-4-6/teaching/2627/FoundsCS/forest/focs-ex-4-6/Exercises › Reasoning from polymorphic typesExerciseWe know nothing about the functions f and g other than their polymorphic types: val f : 'a * 'b -> 'b * 'a and val g : 'a -> 'a list. Suppose that f (1, true) and g 0 are evaluated and return their results. State, with reasons, what you think the resulting values will be.Anil MadhavapeddyAnil Madhavapeddyhttps://www.cl.cam.ac.uk/teaching/2627/FoundsCS/forest/focs-ex-1-3/focs-ex-1-3/teaching/2627/FoundsCS/forest/focs-ex-1-3/Exercises › Redundant conditionalsExerciseWhy would no experienced programmer write an expression of the form if … then true else false? What about expressions of the form if … then false else true?Anil MadhavapeddyAnil Madhavapeddyhttps://www.cl.cam.ac.uk/teaching/2627/FoundsCS/forest/focs-ex-1-1/focs-ex-1-1/teaching/2627/FoundsCS/forest/focs-ex-1-1/Exercises › Two-digit year windowingExerciseOne solution to the year 2000 bug involves storing years as two digits, but interpreting them such that 50 means 1950 and 49 means 2049. Comment on the merits and demerits of this approach.Anil MadhavapeddyAnil Madhavapeddyhttps://www.cl.cam.ac.uk/teaching/2627/FoundsCS/forest/focs-ex-1-4/focs-ex-1-4/teaching/2627/FoundsCS/forest/focs-ex-1-4/Exercises › Type inference in powerExerciseFunctions npower and power both return a float. The definition of npower returns the float value 1.0 in its base case. The definition of power does not, so how does the OCaml type checker know that power returns a float?Anil Madhavapeddyhttps://www.cl.cam.ac.uk/teaching/2627/FoundsCS/forest/focs-topic-strings/focs-topic-strings/teaching/2627/FoundsCS/forest/focs-topic-strings/StringsAnil Madhavapeddyhttps://www.cl.cam.ac.uk/teaching/2627/FoundsCS/forest/focs-currying/focs-currying/teaching/2627/FoundsCS/forest/focs-currying/Curried FunctionsA curried function returns another function as its result. We use the string concatenation operator (^) to illustrate how this works. string -> string = # let prefix = fun a -> fun b -> a ^ b val prefix : string -> string -> string = # let promote = prefix "Senior " val promote : string -> string = # prefix "Junior " "Professor" - : string = "Junior Professor" # promote "Professor" - : string = "Senior Professor"]]>A short form for the definition of prefix is simply to pass multiple arguments to the function definition. The following two definitions are equivalent in OCaml: fun b -> a ^ b val prefix : string -> string -> string = # let prefix a b = a ^ b val prefix : string -> string -> string = ]]>Currying is the technique of expressing a function taking multiple arguments as nested functions, each taking a single argument. The fun-notation lets us package n*2 as the function fun n -> n * 2, but what if there are several variables, as in fun n -> n * 2 + k? A function of two arguments could be coded using pattern-matching on pairs, writing fun (n, k) -> n * 2 + k.Currying is an alternative, where we nest the fun-notation: fun n -> n * 2 + k - : int -> int -> int = ]]>Applying this curried function to the argument 1 yields another function, in which k has been replaced by 1: fun n -> n * 2 + k val fn : int -> int -> int = # let fn' = fn 1 (* n * 2 + 1 *) val fn' : int -> int = # fn' 3 (* 3 * 2 + 1 *) - : int = 7]]>And this function, when applied to 3, yields the result 7. The two arguments are supplied one after another.The earlier prefix example is similar and uses the expression a ^ b, where ^ is the infix operator for string concatenation. Function promote binds the first argument of prefix to "Senior "; the resulting function prefixes that title to any string to which it is applied.Anil Madhavapeddyhttps://www.cl.cam.ac.uk/teaching/2627/FoundsCS/forest/focs-strings-chars/focs-strings-chars/teaching/2627/FoundsCS/forest/focs-strings-chars/Lists, Strings and CharactersStrings are provided in most programming languages to allow text processing. Strings are essential for communication with users. Even a purely numerical program formats its results ultimately as strings.In a few programming languages, strings simply are lists of characters. In OCaml they are a separate type, unrelated to lists, reflecting the fact that strings are an abstract concept in themselves.Similarly, characters are not strings of size one, but are a primitive concept. Character constants in OCaml have the form 'c', where is any character. For example, the comma character is ','.Special characters are coded in strings using escape sequences involving the backslash character; among many others, a double quote is written and the newline character is written . For example, the string represents three text lines.In addition to the operators described above, the relations <, <=, >, and >= work for strings and yield alphabetic order (more precisely, lexicographic order with respect to ASCII character codes).Anil Madhavapeddyhttps://www.cl.cam.ac.uk/teaching/2627/FoundsCS/forest/focs-topic-matrices/focs-topic-matrices/teaching/2627/FoundsCS/forest/focs-topic-matrices/MatricesAnil Madhavapeddyhttps://www.cl.cam.ac.uk/teaching/2627/FoundsCS/forest/focs-transpose/focs-transpose/teaching/2627/FoundsCS/forest/focs-transpose/Example: Matrix Transpose [] | rows -> (map List.hd rows) :: (transp (map List.tl rows)) val transp : 'a list list -> 'a list list = ]]>A matrix can be viewed as a list of rows, each row a list of matrix elements. This representation is not especially efficient compared with the conventional one (using arrays). Lists of lists turn up often, though, and we can see how to deal with them by taking familiar matrix operations as examples. ML for the Working Programmer goes as far as Gaussian elimination, which presents surprisingly few difficulties.The transpose of the matrix is , which in OCaml corresponds to the following transformation on lists of lists:The workings of function transp are simple. If rows is the matrix to be transposed, then map hd extracts its first column and map tl extracts its second column:A recursive call transposes the latter matrix, which is then given the column [a; d] as its first row. The two functions expressed using map would otherwise have to be declared separately.Anil Madhavapeddyhttps://www.cl.cam.ac.uk/teaching/2627/FoundsCS/forest/focs-matmul-ocaml/focs-matmul-ocaml/teaching/2627/FoundsCS/forest/focs-matmul-ocaml/Matrix Multiplication in OCamlDot product of two vectors—a curried function 0.0 | x::xs, y::ys -> (x *. y) +. (dotprod xs ys) Lines 2-4, characters 5-51: Warning 8 [partial-match]: this pattern-matching is not exhaustive. Here is an example of a case that is not matched: ([], _::_) val dotprod : float list -> float list -> float = ]]>Matrix product map (dotprod row) cols) arows val matprod : float list list -> float list list -> float list list = ]]>The transp brows converts into a list of columns, yielding a list whose elements are the columns of . Each row of is obtained by multiplying a row of by the columns of .Because dotprod is curried, it can be applied to a row of . The resulting function is applied to all the columns of . We have another example of currying and partial application.The outer map applies dotprod to each row of . The inner map, using fun-notation, applies dotprod row to each column of . Compare with the version in ML for the Working Programmer (page 89) which does not use map and requires two additional function declarations.In the dot product function, the two vectors must have the same length. Otherwise, exception Match_failure is raised.Anil Madhavapeddyhttps://www.cl.cam.ac.uk/teaching/2627/FoundsCS/forest/focs-matrix-multiplication/focs-matrix-multiplication/teaching/2627/FoundsCS/forest/focs-matrix-multiplication/Review of Matrix MultiplicationThe right side is the vector dot product . Repeat for each row of and column of .The dot product of two vectors isA simple case of matrix multiplication is when consists of a single row and consists of a single column. Provided and contain the same number of elements, multiplying them yields a matrix whose single element is the dot product shown above.If is an matrix and is a matrix then is an matrix. For each and , the element of is the dot product of row of with column of .The (1, 1) element above is computed byCoding matrix multiplication in a conventional programming language usually involves three nested loops. It is hard to avoid mistakes in the subscripting, which often runs slowly due to redundant internal calculations.Anil MadhavapeddyAnil Madhavapeddyhttps://www.cl.cam.ac.uk/teaching/2627/FoundsCS/forest/focs-ex-11-5/focs-ex-11-5/teaching/2627/FoundsCS/forest/focs-ex-11-5/Exercises › Matrix operations on arraysExerciseArrays of multiple dimensions are represented in OCaml by arrays of arrays. Write functions to (a) create an identity matrix, given , and (b) to transpose an matrix. Identity matrices have the following form:Contributions