Course pages 2015–16
No. of lectures: 12
Suggested hours of supervisions: 3
Prerequisite course: Discrete Mathematics
This course is a prerequisite for Complexity Theory (Part IB).
The aim of this course is to introduce several apparently different formalisations of the informal notion of algorithm; to show that they are equivalent; and to use them to demonstrate that there are uncomputable functions and algorithmically undecidable problems.
- Introduction: algorithmically undecidable problems. Decision problems. The informal notion of algorithm, or effective procedure. Examples of algorithmically undecidable problems. [1 lecture]
- Register machines. Definition and examples; graphical notation. Register machine computable functions. Doing arithmetic with register machines. [1 lecture]
- Universal register machine. Natural number encoding of pairs and lists. Coding register machine programs as numbers. Specification and implementation of a universal register machine. [2 lectures]
- Undecidability of the halting problem. Statement and proof. Example of an uncomputable partial function. Decidable sets of numbers; examples of undecidable sets of numbers. [1 lecture]
- Turing machines. Informal description. Definition and examples. Turing computable functions. Equivalence of register machine computability and Turing computability. The Church-Turing Thesis. [2 lectures]
- Primitive and partial recursive functions. Definition and examples. Existence of a recursive, but not primitive recursive function. A partial function is partial recursive if and only if it is computable. [2 lectures]
- Lambda-Calculus. Alpha and beta conversion. Normalization. Encoding data. Writing recursive functions in the lambda-calculus. The relationship between computable functions and lambda-definable functions. [3 lectures]
At the end of the course students should
- be familiar with the register machine, Turing machine and lambda-calculus models of computability;
- understand the notion of coding programs as data, and of a universal machine;
- be able to use diagonalisation to prove the undecidability of the Halting Problem;
- understand the mathematical notion of partial recursive function and its relationship to computability.
* Hopcroft, J.E., Motwani, R. & Ullman, J.D. (2001). Introduction to automata theory, languages, and computation. Addison-Wesley (2nd ed.).
* Hindley, J.R. & Seldin, J.P. (2008). Lambda-calculus and combinators, an introduction. Cambridge University Press (2nd ed.).
Cutland, N.J. (1980). Computability: an introduction to recursive function theory. Cambridge University Press.
Davis, M.D., Sigal, R. & Weyuker, E.J. (1994). Computability, complexity and languages. Academic Press (2nd ed.).
Sudkamp, T.A. (2005). Languages and machines. Addison-Wesley (3rd ed.).