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Mathematics for Computation Theory

*Lecturer: Dr J.K.M. Moody*

*No. of lectures and practical classes:* 12 + 6

*This course is a prerequisite for Introduction to Security, Computation Theory, Artificial Intelligence I, Quantum Computing.*

**Aims**

The aims of this course are to introduce the basic notation and constructs of set theory. The course will cover functions and relations, and will treat sufficient of the theory of partial orders to handle well-founded induction. The ideas are illustrated by developing the theory of finite automata and regular languages, including a proof of Kleene's Theorem.

**Lectures**

Part A. Discrete Mathematics

**Modelling discrete systems.**Computation as a discrete process. Sets: membership, subsets, union, intersection, complement, difference. Symmetric difference. Boolean algebra. Venn diagrams. Propositions and predicates.**Constructions on sets.**Cartesian product. Disjoint union (connection with datatypes). Relations as a subset of a product. Binary relations. Functions and partial functions.**Algebra of relations.**Product and inverse relations. Composition of functions. Bijections. Function spaces. Some natural bijections between sets. Power sets. Predicates, characteristic functions. Multi-sets.*n*-ary predicates.**Relations on a set.**Reflexive, symmetric and transitive properties of a relation on a set. Equivalence relations. Orders, partial and total. Examples.**Well-founded relations and the principle of induction.**Well-founded relations. Their derived partial orderings. Minimal elements. Well-ordering. Examples. Mathematical induction. Well-founded induction via minimal counterexamples.**Algebraic languages (``free algebras'') and structural induction.**Languages of algebraic expressions. Free algebras. Examples. Parse trees. Well-foundedness in free algebras. Principle of structural induction for free algebras. Languages of regular expressions.

Part B. Finite Automata and Regular Languages

**Deterministic finite automata (DFA).**Mechanical recognition of strings in a language. Stimulus-response behaviour, event detection. Mealey machines, Moore machines. Examples. Accepting strings via some output of a DFA. Examples.**Using automata to recognise languages.**Events/languages as sets of strings. Non-deterministic finite automata (NDFA). Proof that the language classes accepted by DFAs and NDFAs coincide.**Algebraic operations on languages.**Finite union and intersection, concatenation, and Kleene closure. Regular expressions and the languages they denote. Examples of regular languages, including recognisers.**Regular languages are representable events.**Arden's rule as a recursive definition. Its proof by induction. Extension to matrices of events. Statement of Kleene's Theorem, including an overview of the proof structure. Regular events can be represented: proof by induction on the structure of regular expressions using NDFAs.**Representable events have regular expressions.**Representable events are regular: proof by induction on the number of states using event transition matrices and Arden's rule.**Properties of regular languages.**Strings accepted by an*n*-state machine. The Pumping Lemma. Examples of languages that are not regular, including palindromes. Complements and intersections of regular languages are regular. Some decidable problems for regular languages. Extended regular expressions. A non-elementary problem. (Cook's tour)

**Objectives**

At the end of the course students should

- be familiar with the notation of set theory, and be able to use it to
express simple mathematical constructs
- know how to translate between relational algebra and relational
calculus
- be able to carry out simple proofs by induction, including structural
induction over partially ordered (well-founded) sets
- understand how to derive a regular expression to describe the event
accepted by a deterministic finite automaton
- understand how to use the Pumping Lemma to show that a given language
cannot be regular

**Recommended reading**

Part A:

* Devlin, K. (2003). *Sets, functions, and logic: an introduction to abstract mathematics*. Chapman and Hall/CRC Mathematics (3rd ed.). ISBN 1-584-88449-5

Rosen, K.H. (2003). *Discrete mathematics and its applications*. McGraw-Hill (5th ed.). ISBN 0-072-93033-0

Kolman, B., Busby, R.C. & Ross, S.C. (2003). *Discrete mathematical structures for computer science*. Prentice Hall (5th ed.). ISBN 0-130-45797-3

Forster, T.E. (2003). *Logic, induction and sets*. LMS Student Text 56. Cambridge University Press. ISBN 0-521-53361-9

Part B:

Hopcroft, J.E., Motwani, R. & Ullman, J.D. (2001). *Introduction to automata theory, languages and computation*. Addison-Wesley (2nd ed.). ISBN 0-201-44124-1

Sudkamp, T.A. (1998). *Languages and machines: an introduction to the theory of computer science*. Addison-Wesley (2nd ed.). ISBN 0-201-15768-3

Hopcroft, J.E. & Ullman, J.D. (1974). *The design and analysis of computer algorithms*. Addison-Wesley. ISBN 0-201-00029-6

Closely related to parts of the course but out of print:

Conway, J.H. (1971). *Regular algebra and finite machines*. Chapman and Hall.

Stanat, D.F. & McAllister, D.F. (1977). *Discrete mathematics in computer science*. Prentice Hall.

Stone, H.S. (1973). *Discrete mathematical structures and their applications*. SRA.

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