CSC312 Automata Theory Lecture # 1 Introduction

CSC312 Automata Theory Lecture # 1 Introduction

Administrative Stuff • Instructor: Muhammad Usman Akram musmanakram@cuilahore.edu.pk Room # C-11 WebLink: http://usmanlive.com • 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. UllmanIntroduction 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

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 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

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

B A 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 } 2 4 1 3 5 2 U 3 1 4 Venn diagrams 5

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

{ even integers } = { odd integers } Integers 1 odd 0 5 even 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

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

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

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 B 4 A 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 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) }

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

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

e b d a c Path 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

CSC312 Automata Theory Lecture # 1 Introduction