Theory of Computation Lecture Notes

Theory of Computation
Dr Mohamed BAHACHE
February 21, 2025
Computer Science Department

Contents
1 Mathematical concepts(Sets,Relations,...) 1
1.1 Sets ........................................ 1
1.1.1 Set specification ............................. 1
1.1.2 Set operators ............................... 1
1.1.3 Sequences and Tuples .......................... 2
1.1.4 Cartesian product ............................ 2
1.2 Relations ..................................... 2
1.2.1 Few relation types ............................ 2
1.2.2 Relation properties ........................... 2
1.3 Monoïd ...................................... 3
1.4 Homomorphism .................................. 3
1.5 Function ...................................... 3
1.6 Graphs ....................................... 4
1.6.1 Terminology ............................... 4
1.6.2 Tree .................................... 5
1.7 Mathematical proofs ............................... 5
1.7.1 Types of proofs .............................. 6
2 Alphabets,Words and Languages 7
2.1 Alphabets .................................... 7
2.2 The word (string) ................................ 7
2.3 Operation on words ............................... 7
2.3.1 Length of a word ............................. 7
2.3.2 Mirror of word .............................. 8
2.3.3 Factorization of word .......................... 9
2.4 Quotient of word ................................. 9
2.4.1 Right quotient .............................. 9
2.5 Languages ..................................... 10
2.5.1 Operations on languages ........................ 10
2.5.2 Arden’s equation ............................. 11
3 Generator system of language(Grammar) 13
3.1 Grammar ..................................... 13
3.2 The derivation in a grammar .......................... 14
3.2.1 The derivation in one step ........................ 14
3.2.2 Successive derivation .......................... 14
3.2.3 Generated language by a grammar ................... 14
3.3 Chomsky hierarchy ................................ 14
3.3.1 Grammar of type0 ............................ 15
3.3.2 Grammar of type 1 ............................ 16
3.3.3 Context free Grammar(type 2) ..................... 16
3.3.4 Regular grammar(type 3) ........................ 17
4 Regular languages and regular expressions 19
4.1 Regular languages ................................ 19
4.1.1 Defintion ................................. 19
4.1.2 Reminder-Regular grammar(type 3) .................. 19
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ii Abstract
4.1.3 Definition: ................................. 20
4.1.4 Definition: ................................. 21
4.2 Pumping lemma ................................. 22
Bibliography 23
A Code A.1

List of Tables
List of Figures
1.1 Principle of a function .............................. 3
1.2 Example of a graph ................................ 4
1.3 Terminology of graph .............................. 5
1.4 Example of tree .................................. 5
3.1 Ambiguous grammar ............................... 17
3.2 Inclusion relation between formal grammars. ................. 18
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