CSC312 Automata Theory Lecture # 1 Introduction

Administrative Stuff Instructor: Dr. Mudasser Naseer mnaseer@ciitahore.edu.pk Cabin # 1, Faculty Room C7 Lectures: Sec-A: Tue 16:30 (C-6), Wed 16:30 (C-11) Sec-B: Mon 11:30 (C-13), Wed 15:00 (C-11) Office Hrs: Tue & Thu 1400 – 1600 hrs (or by appointment) Prerequisite: CSC102 - Discrete Structures

Course Objectives: 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, NFAs. Core concepts of Regular Languages and Finite Automata; Decidability for Regular Languages; Non-regular Languages; Context-free Languages and Pushdown Automata; Decidability for Context-free Languages; Non-context-free Languages; Turing Machines and Their Languages are important part of the course. Transducers (automata with output).

Course Organization Text Book: i) Denial I. A. Cohen Introduction to Computer Theory, Second Edition, John Wiley & Sons, 1997. 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 2~3 assignments, 4~5 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 Lect.# Topics/Contents 12 Kleene’s Theorem, Algorithm for turning TGs into REs 13 Kleene’s Theorem, Algorithm for turning REs into FAs 14 Nondeterminism, NFA, converting NFA into FA. 15 Finite Automata with output, Moore’s machines and Mealy machines with examples. 1’s Complement machine, Increment machine. 16 Theorems for Converting Moore machines into Mealy machines and vice versa. Transducers as models of sequential circuits. 17 Regular Languages, Closure properties (i.e. , Concatenation and Kleene closure) of Regular Languages with examples. 18 Complements and Intersections of Regular Languages, Theorems relating to regular languages and the related examples. 19 Non-Regular Languages, The pumping Lemma, Examples relating to Pumping Lemma. 20 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. 21 Sessional II

Schedule of Lectures Lect.# Topics/Contents 22 Context-Free Grammars, CFG’s for Regular Languages with examples. CFG’s for non-regular languages. 23 CFG’s of PALINDROME, EQUAL and EVEN-EVEN languages, Backus-Naur Form. 24 Parse Trees, Examples relating to Parse Trees, Lukasiewicz notation, Prefix and Postfix notations and their evaluation. 25 Ambiguous and Unambiguous CFG’s, Syntax tree, Total language tree. 26 Regular Grammars, Semi-word, Word, Working String, Converting FA’s into CFG’s. Constructing Transition Graphs from Regular Grammars. 27 Killing null productions. Killing unit productions, Chomsky Normal form with examples, Left most derivations. 28 Pushdown Automata, Constructing PDA’s for FA’s, Pushdown stack. 29 Examples related with PDA, PDA for Odd Palindrome, Even Palindrome, PalindromeX. 30 Nondeterministic PDA. Proving CFG = PDA with examples. 31 Constructing PDA from CFG in CNF with examples 32 Turing machines, Examples of Turing Machines with trace tables, Converting FA’s into Turing machines.

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/150s : 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,…)

Courtesy Costas Busch - RPI Computation memory CPU Courtesy Costas Busch - RPI

Courtesy Costas Busch - RPI temporary memory input memory CPU output memory Program memory Courtesy Costas Busch - RPI

Courtesy Costas Busch - RPI Example: temporary memory input memory CPU output memory Program memory compute compute Courtesy Costas Busch - RPI

Courtesy Costas Busch - RPI temporary memory input memory CPU output memory Program memory compute compute Courtesy Costas Busch - RPI

Courtesy Costas Busch - RPI temporary memory input memory CPU output memory Program memory compute compute Courtesy Costas Busch - RPI

Courtesy Costas Busch - RPI temporary memory input memory CPU Program memory output memory compute compute Courtesy Costas Busch - RPI

Courtesy Costas Busch - RPI Automaton temporary memory Automaton input memory CPU output memory Program memory Courtesy Costas Busch - RPI

Courtesy Costas Busch - RPI Different Kinds of Automata Automata are distinguished by the temporary memory Finite Automata: no temporary memory Pushdown Automata: stack Turing Machines: random access memory Courtesy Costas Busch - RPI

Courtesy Costas Busch - RPI Finite Automaton temporary memory input memory Finite Automaton output memory Example: Vending Machines (small computing power) Courtesy Costas Busch - RPI

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

Courtesy Costas Busch - RPI Turing Machine Random Access Memory input memory Turing Machine output memory Examples: Any Algorithm (highest computing power) Courtesy Costas Busch - RPI

Courtesy Costas Busch - RPI Power of Automata Finite Automata Pushdown Automata Turing Machine Less power More power Solve more computational problems Courtesy Costas Busch - RPI

Mathematical Preliminaries Sets Functions Relations Graphs Proof Techniques Courtesy Costas Busch - RPI

Courtesy Costas Busch - RPI SETS A set is a collection of elements We write Courtesy Costas Busch - RPI

Courtesy Costas Busch - RPI 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 Courtesy Costas Busch - RPI

Courtesy Costas Busch - RPI 1 2 3 4 5 A U 6 7 8 9 10 Universal Set: all possible elements U = { 1 , … , 10 } Courtesy Costas Busch - RPI

Courtesy Costas Busch - RPI 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 U 2 3 1 Venn diagrams Courtesy Costas Busch - RPI

Courtesy Costas Busch - RPI Complement Universal set = {1, …, 7} A = { 1, 2, 3 } A = { 4, 5, 6, 7} 4 A A 6 3 1 2 5 7 A = A Courtesy Costas Busch - RPI

Courtesy Costas Busch - RPI { even integers } = { odd integers } Integers 1 odd even 5 6 2 4 3 7 Courtesy Costas Busch - RPI

Courtesy Costas Busch - RPI DeMorgan’s Laws A U B = A B U A B = A U B U Courtesy Costas Busch - RPI

Courtesy Costas Busch - RPI Empty, Null Set: = { } S U = S S = S - = S - S = U = Universal Set Courtesy Costas Busch - RPI

Courtesy Costas Busch - RPI Subset A = { 1, 2, 3} B = { 1, 2, 3, 4, 5 } A B U Proper Subset: A B U B A Courtesy Costas Busch - RPI

Courtesy Costas Busch - RPI Disjoint Sets A = { 1, 2, 3 } B = { 5, 6} A B = U A B Courtesy Costas Busch - RPI

Courtesy Costas Busch - RPI Set Cardinality For finite sets A = { 2, 5, 7 } |A| = 3 (set size) Courtesy Costas Busch - RPI

Courtesy Costas Busch - RPI 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 ) Courtesy Costas Busch - RPI

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 Courtesy Costas Busch - RPI

Courtesy Costas Busch - RPI 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 Courtesy Costas Busch - RPI

Courtesy Costas Busch - RPI 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 Courtesy Costas Busch - RPI

Courtesy Costas Busch - RPI Equivalence Relations Reflexive: x R x Symmetric: x R y y R x Transitive: x R y and y R z x R z Example: R = ‘=‘ x = x x = y y = x x = y and y = z x = z Courtesy Costas Busch - RPI

Courtesy Costas Busch - RPI GRAPHS A directed graph e b node d a 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) } Courtesy Costas Busch - RPI

Courtesy Costas Busch - RPI Labeled Graph 2 6 e 2 b 1 3 d a 6 5 c Courtesy Costas Busch - RPI

Courtesy Costas Busch - RPI Walk a b c d e Walk is a sequence of adjacent edges (e, d), (d, c), (c, a) Courtesy Costas Busch - RPI

Courtesy Costas Busch - RPI Path a b c d e Path is a walk where no edge is repeated Simple path: no node is repeated Courtesy Costas Busch - RPI

Courtesy Costas Busch - RPI 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 Courtesy Costas Busch - RPI

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

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

Courtesy Costas Busch - RPI Trees root parent leaf child Trees have no cycles Courtesy Costas Busch - RPI

Courtesy Costas Busch - RPI root Level 0 Level 1 Height 3 leaf Level 2 Level 3 Courtesy Costas Busch - RPI

Courtesy Costas Busch - RPI Binary Trees Courtesy Costas Busch - RPI

Courtesy Costas Busch - RPI PROOF TECHNIQUES Proof by induction Proof by contradiction Courtesy Costas Busch - RPI

Courtesy Costas Busch - RPI Induction We have statements P1, P2, P3, … If we know for some b that P1, P2, …, Pb are true for any k >= b that P1, P2, …, Pk imply Pk+1 Then Every Pi is true Courtesy Costas Busch - RPI

Courtesy Costas Busch - RPI Proof by Induction Inductive basis Find P1, P2, …, Pb which are true Inductive hypothesis Let’s assume P1, P2, …, Pk are true, for any k >= b Inductive step Show that Pk+1 is true Courtesy Costas Busch - RPI

Courtesy Costas Busch - RPI Example Theorem: A binary tree of height n has at most 2n leaves. Proof by induction: let L(i) be the maximum number of leaves of any subtree at height i Courtesy Costas Busch - RPI

Courtesy Costas Busch - RPI We want to show: L(i) <= 2i Inductive basis L(0) = 1 (the root node) Inductive hypothesis Let’s assume L(i) <= 2i for all i = 0, 1, …, k Induction step we need to show that L(k + 1) <= 2k+1 Courtesy Costas Busch - RPI

Courtesy Costas Busch - RPI Induction Step height k k+1 From Inductive hypothesis: L(k) <= 2k Courtesy Costas Busch - RPI

Courtesy Costas Busch - RPI Induction Step height L(k) <= 2k k k+1 L(k+1) <= 2 * L(k) <= 2 * 2k = 2k+1 (we add at most two nodes for every leaf of level k) Courtesy Costas Busch - RPI

Courtesy Costas Busch - RPI Remark Recursion is another thing Example of recursive function: f(n) = f(n-1) + f(n-2) f(0) = 1, f(1) = 1 Courtesy Costas Busch - RPI

Courtesy Costas Busch - RPI Proof by Contradiction We want to prove that a statement P is true we assume that P is false then we arrive at an incorrect conclusion therefore, statement P must be true Courtesy Costas Busch - RPI

Courtesy Costas Busch - RPI Example Theorem: is not rational Proof: Assume by contradiction that it is rational = n/m n and m have no common factors We will show that this is impossible Courtesy Costas Busch - RPI

= n/m 2 m2 = n2 n is even n = 2 k Therefore, n2 is even m is even m = 2 p 2 m2 = 4k2 m2 = 2k2 Thus, m and n have common factor 2 Contradiction!