1. CSC312
Automata Theory
Lecture # 1
Introduction

2. Administrative Stuff Instructor: Dr. Mudasser Naseer
Cabin # 1, Faculty Room C7
Lectures: Sec-B: Wed 1630, Thu 1800 hrs.
Sec-C: Wed 1800, Thu 1630 hrs.
Office Hrs: Tue & Thu 1400 – 1600 hrs
(or by appointment)
Prerequisite: CSC102 - Discrete Structures

3. Course Organization
Text Book: i) J. E. Hopcroft, R. Motwani, & J. D. Ullman Introduction to Automata Theory, Languages, and Computation, Third Edition, Pearson, 2008.
ii) Denial I. A. Cohen Introduction to Computer Theory, First Edition, John Wiley & Sons, 1986.
Instruments: There will be 2~3 assignments, 4~5 quizzes, Weights: Assignments 10% Quizzes 15% S-I 10%
S-II 15% Final Exam 50%

4. Course Outline
Study and mathematically model various abstract computing machines that serve as models for computations and examine the relationship between these automata and formal languages. 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).

5. Schedule of Lectures
Lect.#
Topics/Contents 1
Introduction to Automata theory, Its background, Mathematical Preliminaries, Sets, Functions, Relations, 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
Defining languages recursively, Recursive definitions of languages like EVEN, INTEGERS, POSITIVE, POLYNOMIALS, ARITHMETIC EXPRESSIONS 4
Theorems relating to recursive definitions and their proofs. 5
Formal definition of Regular Expressions, Defining languages with regular expressions, Languages associated with regular expressions. 6
Equality of Regular Expressions, Introducing the language EVEN-EVEN. 7
More examples related to regular expressions. 8
Introducing Finite Automata., Defining languages using Finite Automata. Constructing Finite Automata for different languages. 9
Recognizing the language defined by the given Finite Automata. 10
More examples related to Finite Automata. 11
Sessional I

6. Schedule of Lectures (Cont…)
Topics/Contents 12
Transition Graphs with examples, Generalized Transition Graphs, Non-determinism in case of Transition Graphs. 13
Non-deterministic FA’s. Differences between FA, TG and NFA. 14
Finite Automata with output, machines and Mealy machines with examples. 1’s Complement machine, Increment machine. 15
Transducers as models of sequential circuits, Theorems for Converting Moore machines into Mealy machines and vice versa. 16
Regular Languages, Closure properties (i.e. , Concatenation and Kleene closure) of Regular Languages with examples. Complements and Intersections of Regular Languages, Theorems relating to regular languages and the related examples. 17
Decidability, decision procedure, Blue-paint method, Effective decision procedure to prove whether two given RE’s or FA’s are equivalent. 18
Non-Regular Languages, The pumping Lemma, Examples relating to Pumping Lemma. 19
Myhill-Nerode theorem, Related Examples, Quotient Languages. 20
Context-Free Grammars, CFG’s for Regular Languages with examples. 21
CFG’s for non-regular languages, CFG’s of PALINDROME, EQUAL and EVEN-EVEN languages, Backus-Naur Form. 22
Parse Trees, Examples relating to Parse Trees, Lukasiewicz notation, Prefix and Postfix notations and their evaluation. 23
Sessional II

7. Schedule of Lectures (Cont…)
Topics/Contents 24
Ambiguous and Unambiguous CFG’s, Total language tree. Regular Grammars, Semi-word, Word, Working String, Converting FA’s into CFG’s. 25
Regular Grammars, Constructing Transition Graphs from Regular Grammars. Killing null productions. 26
Killing unit productions, Chomsky Normal form with examples, Left most derivations. 27
Pushdown Automata, Constructing PDA’s for FA’s, Pushdown stack. Nondeterministic PDA. 28
Examples related with PDA, PDA for Odd Palindrome, Even Palindrome, PalindromeX. 29
Proving CFG = PDA with examples. Constructing PDA from CFG in CNF and making trace table for any word. 30
Context Free Languages, their closure properties, , Concatenation and Kleene Closures using CFG’s and PDA’s. 31
Top down parsing, Bottom up parsing with examples, Turing machines, Examples of Turing Machines with trace tables, Converting FA’s into Turing machines. 32
The subprogram Insert, The subprogram Delete, Constructing Turing machine for the language EQUAL using Insert subprogram.

8. 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

9. 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,…)

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Computation
memory
CPU
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temporary memory
input memory
CPU
output memory
Program memory
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Example:
temporary memory
input memory
CPU
output memory
Program memory
compute
compute
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temporary memory
input memory
CPU
output memory
Program memory
compute
compute
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temporary memory
input memory
CPU
output memory
Program memory
compute
compute
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temporary memory
input memory
CPU
Program memory
output memory
compute
compute
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Automaton
temporary memory
Automaton
input memory
CPU
output memory
Program memory
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Different Kinds of Automata
Automata are distinguished by the temporary memory
Finite Automata: no temporary memory
Pushdown Automata: stack
Turing Machines: random access memory
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Finite Automaton
temporary memory
input memory
Finite
Automaton
output memory
Example: Vending Machines
(small computing power)
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Pushdown Automaton
Stack
Push, Pop
input memory
Pushdown
Automaton
output memory
Example: Compilers for Programming Languages
(medium computing power)
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Turing Machine
Random Access Memory
input memory
Turing
Machine
output memory
Examples: Any Algorithm
(highest computing power)
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Power of Automata
Finite
Automata
Pushdown
Automata
Turing
Machine
Less power
More power
Solve more
computational problems
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22. Mathematical Preliminaries
Sets
Functions
Relations
Graphs
Proof Techniques
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SETS
A set is a collection of elements
We write
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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
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Universal Set: all possible elements
U = { 1 , … , 10 }
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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
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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
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{ even integers } = { odd integers }
Integers 1
odd
even 5 6 2 4 3 7
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DeMorgan’s Laws
A U B = A B U
A B = A U B U
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Empty, Null Set:
= { }
S U = S
S =
S = S
- S = U
= Universal Set
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Subset
A = { 1, 2, 3} B = { 1, 2, 3, 4, 5 }
A B U
Proper Subset:
A B U B A
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Disjoint Sets
A = { 1, 2, 3 } B = { 5, 6}
A B = U A B
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Set Cardinality
For finite sets
A = { 2, 5, 7 }
|A| = 3
(set size)
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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 )
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35. 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
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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
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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
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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
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Equivalence Classes
Given an equivalence relation R on a set A and an element , the equivalence class of is the set
Example:
R = { (1, 1), (2, 2), (1, 2), (2, 1),
(3, 3), (4, 4), (3, 4), (4, 3) }
Equivalence class of 1 = {1, 2}
Equivalence class of 3 = {3, 4}
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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) }
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Labeled Graph 2 6 e 2 b 1 3 d a 6 5 c
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Walk a b c d e
Walk is a sequence of adjacent edges
(e, d), (d, c), (c, a)
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Path a b c d e
Path is a walk where no edge is repeated
Simple path: no node is repeated
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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
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Euler Tour 8
base e 7 1 b 4 6 5 d a 2 3 c
A cycle that contains each edge once
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Hamiltonian Cycle 5
base e 1 b 4 d a 2 3 c
A simple cycle that contains all nodes
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Finding All Simple Paths e b d a c
origin
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Step 1 e b d a c
origin
(c, a)
(c, e)
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Step 2 e b d a
(c, a)
(c, a), (a, b)
(c, e)
(c, e), (e, b)
(c, e), (e, d) c
origin
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Step 3 e b d a
(c, a)
(c, a), (a, b)
(c, a), (a, b), (b, e)
(c, e)
(c, e), (e, b)
(c, e), (e, d) c
origin
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Step 4 e b d a
(c, a)
(c, a), (a, b)
(c, a), (a, b), (b, e)
(c, a), (a, b), (b, e), (e,d)
(c, e)
(c, e), (e, b)
(c, e), (e, d) c
origin
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Trees
root
parent
leaf
child
Trees have no cycles
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root
Level 0
Level 1
Height 3
leaf
Level 2
Level 3
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Binary Trees
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PROOF TECHNIQUES
Proof by induction
Proof by contradiction
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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
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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
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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
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We want to show: L(i) <= 2i
Inductive basis
L(0) = (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
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Induction Step
height k
k+1
From Inductive hypothesis: L(k) <= 2k
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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)
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Remark
Recursion is another thing
Example of recursive function:
f(n) = f(n-1) + f(n-2)
f(0) = 1, f(1) = 1
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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
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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
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= n/m 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!
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