1. CSC312
Automata Theory
Lecture # 1
Introduction

2. Administrative Stuff Instructor: Muhammad Usman Akram
Room # C-11
WebLink:
Office Hrs: Mon-Thu 11:30 – 13:00 hrs
(or by appointment)
Prerequisite: CSC102 - Discrete Structures

3. Course Objectives:
This course is designed to enable the students to study and mathematically model various abstract computing machines that serve as models for computations and examine the relationship between these automata and formal languages.

4. Course Outline
Regular expressions, TG, DFA and NFAs. Core concepts of Regular Languages and Finite Automata; Moore and Mealy Machines, Decidability for Regular Languages; Non-regular Languages; Transducers (automata with output). Context-free Grammar and Languages; Decidability for Context-free Languages; Non-context-free Languages; Pushdown Automata, If time allows we will also have a short introduction to Turing machines.

5. Course Organization
Text Book: i) Denial I. A. Cohen Introduction to Computer Theory, Second Edition, John Wiley & Sons.
Reference Books:
i) J. E. Hopcroft, R. Motwani, & J. D. Ullman Introduction to Automata Theory, Languages, and Computation, Third Edition, Pearson, 2008.
Instruments: There will be 3~4 assignments, 3~4 quizzes, Weights: Assignments 10% Quizzes 15% S-I 10%
S-II 15% Final Exam 50%

6. Schedule of Lectures
Lect.#
Topics/Contents 1
Introduction to Automata theory, Its background, Mathematical Preliminaries, Sets, Functions, Graphs, Proof Techniques 2
Formal Languages, Introduction to defining languages, alphabet, language, word, null string, length of a string, reverse of a string, Palindrome, Kleene closure. 3
Formal definition of Regular Expressions, Defining languages with regular expressions, Languages associated with regular expressions. 4
Equality of Regular Expressions, Introducing the language EVEN-EVEN. 5
More examples related to regular expressions. 6
Introducing Finite Automata., Defining languages using Finite Automata. Constructing Finite Automata for different languages. 7
Recognizing the language defined by the given Finite Automata. 8
More examples related to Finite Automata. 9
Transition Graphs with examples, Generalized Transition Graphs, Non-determinism in case of Transition Graphs. 10
Non-deterministic FA’s. Differences between FA, TG and NFA. 11
Sessional I

7. Schedule of Lectures 12 Discussion on the solution of S-I 13
Topics/Contents 12
Discussion on the solution of S-I 13
Kleene’s Theorem, Algorithm for turning TGs into REs 14
Kleene’s Theorem, Algorithm for turning REs into FAs 15
Different Rules for turning Res into FAs 16
Nondeterminism, NFA, converting NFA into FA. Union of two FAs using NFA. 17
Finite Automata with output, Moore’s machines and Mealy machines with examples. 1’s Complement machine, Increment machine. 18
Theorems for Converting Moore machines into Mealy machines and vice versa. 19
Transducers as models of sequential circuits. 20
Regular Languages, Closure properties (i.e. , Concatenation and Kleene closure) of Regular Languages with examples. 21
Complements and Intersections of Regular Languages, Theorems relating to regular languages and the related examples. 22
Non-Regular Languages, The pumping Lemma, Examples relating to Pumping Lemma.

8. Schedule of Lectures 23 Sessional-II 24
Decidability, decision procedure, Blue-paint method, Effective decision procedure to prove whether two given RE’s or FA’s are equivalent. Myhill-Nerode theorem, Related Examples. 25
Context-Free Grammars, CFG’s for Regular Languages with examples. CFG’s for non-regular languages. 26
CFG’s of PALINDROME, EQUAL and EVEN-EVEN languages, Backus-Naur Form. 27
Parse Trees, Examples relating to Parse Trees, Lukasiewicz notation, Prefix and Postfix notations and their evaluation. 28
Ambiguous and Unambiguous CFG’s, Total language tree. Regular Grammars, Semi-word, Word, Working String, Converting FA’s into CFG’s. 29
Regular Grammars, Constructing Transition Graphs from Regular Grammars. Killing null productions. Killing unit productions, 30
Chomsky Normal form with examples, Left most derivations.
Pushdown Automata, Constructing PDA’s for FA’s, Pushdown stack.

9. Some basics
Automaton = A self-operating machine or mechanism (Dictionary definition), plural is Automata.
Automata = abstract computing devices
Automata theory = the study of abstract machines (or more appropriately, abstract 'mathematical' machines or systems), and the computational problems that can be solved using these machines.
Mathematical models of computation
Finite automata
Push-down automata
Turing machines

10. History
1930s : Alan Turing defined machines more powerful than any in existence, or even any that we could imagine – Goal was to establish the boundary between what was and was not computable.
1940s/1950s : In an attempt to model “Brain function” researchers defined finite state machines.
Late 1950s : Linguist Noam Chomsky began the study of Formal Grammars.
1960s : A convergence of all this into a formal theory of computer science, with very deep philosophical implications as well as practical applications (compilers, web searching, hardware, A.I., algorithm design, software engineering,…)

11. Computation
memory
CPU

12. temporary memory
input memory
CPU
output memory
Program memory

13. Example: temporary memory input memory CPU output memory
Program memory
compute
compute

14. temporary memory
input memory
CPU
output memory
Program memory
compute
compute

15. temporary memory
input memory
CPU
output memory
Program memory
compute
compute

16. temporary memory
input memory
CPU
Program memory
output memory
compute
compute

17. Automaton temporary memory Automaton input memory CPU output memory
Program memory

18. Different Kinds of Automata
Automata are distinguished by the temporary memory
Finite Automata: no temporary memory
Pushdown Automata: stack
Turing Machines: random access memory

19. Finite Automaton temporary memory input memory Finite Automaton
output memory
Example: Automatic Door, Vending Machines
(small computing power)

20. Pushdown Automaton Stack Push, Pop input memory Pushdown Automaton
output memory
Example: Compilers for Programming Languages
(medium computing power)

21. Turing Machine Random Access Memory input memory Turing Machine
output memory
Examples: Any Algorithm
(highest computing power)

22. Power of Automata Finite Automata Pushdown Automata Turing Machine
Less power
More power
Solve more
computational problems

23. Mathematical Preliminaries
Sets
Functions
Relations
Graphs
Proof Techniques

24. SETS
A set is a collection of elements
We write

25. Set Representations C = { a, b, c, d, e, f, g, h, i, j, k }
C = { a, b, …, k }
S = { 2, 4, 6, … }
S = { j : j > 0, and j = 2k for some k>0 }
S = { j : j is nonnegative and even }
finite set
infinite set

26. A = { 1, 2, 3, 4, 5 } 1 2 3 4 5 A U 6 7 8 9 10
Universal Set: all possible elements
U = { 1 , … , 10 }

27. Set Operations A = { 1, 2, 3 } B = { 2, 3, 4, 5} Union
A U B = { 1, 2, 3, 4, 5 }
Intersection
A B = { 2, 3 }
Difference
A - B = { 1 }
B - A = { 4, 5 } A B 2 4 1 3 5 2 U 3 1 4
Venn diagrams 5

28. A A Complement Universal set = {1, …, 7}4 A A 6 3 1 2 5 7
A = A

29. { even integers } = { odd integers }1
odd
even 5 6 2 4 3 7

30. DeMorgan’s Laws
A U B = A B U
A B = A U B U

31. Empty, Null Set:
= { }
S U = S
S =
S = S
- S = U
= Universal Set

32. Subset A = { 1, 2, 3} B = { 1, 2, 3, 4, 5 } A B U Proper Subset: A B U

33. Disjoint Sets
A = { 1, 2, 3 } B = { 5, 6}
A B = U A B

34. Set Cardinality
For finite sets
A = { 2, 5, 7 }
|A| = 3
(set size)

35. Powersets A powerset is a set of sets S = { a, b, c }
Powerset of S = the set of all the subsets of S
2S = { , {a}, {b}, {c}, {a, b}, {a, c}, {b, c}, {a, b, c} }
Observation: | 2S | = 2|S| ( 8 = 23 )

36. Generalizes to more than two sets
Cartesian Product
A = { 2, 4 } B = { 2, 3, 5 }
A X B = { (2, 2), (2, 3), (2, 5), ( 4, 2), (4, 3), (4, 5) }
|A X B| = |A| |B|
Generalizes to more than two sets
A X B X … X Z

37. FUNCTIONS domain range 4 A B f(1) = a a 1 2 b c 3 5 f : A -> B
If A = domain
then f is a total function
otherwise f is a partial function

38. RELATIONS Let A & B be sets. A binary relation “R” from A to B
R = {(x1, y1), (x2, y2), (x3, y3), …}
Where and
R ⊆ A x B
xi R yi to denote
e. g. if R = ‘>’: 2 > 1, 3 > 2, 3 > 1

39. GRAPHS A directed graph e b d a c Nodes (Vertices)
edge c
Nodes (Vertices)
V = { a, b, c, d, e }
Edges
E = { (a,b), (b,c), (b,e),(c,a), (c,e), (d,c), (e,b), (e,d) }

40. Labeled Graph 2 6 e 2 b 1 3 d a 6 5 c

41. Walk e b d a c Walk is a sequence of adjacent edges
(e, d), (d, c), (c, a)

42. Path e b d a c Path is a walk where no edge is repeated
Simple path: no node is repeated

43. Cycle e base b 3 1 d a 2 c Cycle: a walk from a node (base) to itself
Simple cycle: only the base node is repeated

44. Euler Tour 8
base e 7 1 b 4 6 5 d a 2 3 c
A cycle that contains each edge once

45. Hamiltonian Cycle 5 base e 1 b 4 d a 2 3 c
A simple cycle that contains all nodes

46. Trees
root
parent
leaf
child
Trees have no cycles

47. root
Level 0
Level 1
Height 3
leaf
Level 2
Level 3

48. Binary Trees