CSC312 Automata Theory Lecture # 1 Introduction

Administrative Stuff Instructor: Muhammad Usman Akram musmanakram@ciitahore.edu.pk Room # C-11 WebLink: http://usman-blog.com/category/courses/theory-of-automata-fa15/ Office Hrs: Mon-Thu 11:30 – 13:00 hrs (or by appointment) Prerequisite: CSC102 - Discrete Structures

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.

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.

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%

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

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.

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.

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

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,…)

Computation memory CPU

temporary memory input memory CPU output memory Program memory

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

temporary memory input memory CPU output memory Program memory compute compute

temporary memory input memory CPU output memory Program memory compute compute

temporary memory input memory CPU Program memory output memory compute compute

Automaton temporary memory Automaton input memory CPU output memory Program memory

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

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

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

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

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

Mathematical Preliminaries Sets Functions Relations Graphs Proof Techniques

SETS A set is a collection of elements We write

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

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 }

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

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

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

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

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

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

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

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

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 )

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

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

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

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) }

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

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

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

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

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

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

Trees root parent leaf child Trees have no cycles

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

Binary Trees