gate theory of computation book

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THEORY OF COMPUTATION

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Computer Science

Information Technology

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Syllabus Theory of Computation

THE GATE ACADEMY PVT.LTD. H.O.: #74, Keshava Krupa (third Floor), 30thCross, 10thMain, Jayanagar 4thBlock, Bangalore-11 080 65700750 i f @th t d C i ht d W b th t d

Syllabus for Theory Of Computation

Regular languages and finite automata, Context free languages and Push-down automata, Recursively

enumerable sets and Turing machines, Undecidability; NP completeness.

Analysis of GATE Papers

(Theory Of Computation)

Year

Percentage of marks

Overall Percentage

2013

8.00

8.09

2012

5.00

2011

9.00

2010

7.00

2009 6.66

2008

10.00

2007

8.00

2006

9.33

2005

9.33

2004

6.00

2003

10.67


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Contents Theory of Computation

THE GATE ACADEMY PVT.LTD. H.O.: #74, KeshavaKrupa (third Floor), 30thCross, 10thMain, Jayanagar 4thBlock, Bangalore-11 080 65700750 i f @th t d C i ht d W b th t d P I

C O N T E N T S

Chapter Page No

#1. Introduction/ Preliminaries 1 3 Preliminaries 1 2

Properties of Relations 2 3

#2.Finite Automata 4 30

Finite Automata Introduction 4 5

Deterministic Finite Automata 5 6

Transition Diagram 6 8

Non-Deterministic Finite Automata 8 10 Construction of DFA From NFA 10 11

NFA with -Transitions 11 14

Eliminating -Transitions 14 16

Assignment 1 17 21

Assignment 2 21 24

Answer keys 25

Explanations 25 30

#3.Regular Expression 31 74

Definitions 31 32 Languages Associated with Regular Expressions 32 33

Algebraic Laws for Regular Expressions 33 34

Converting Regular Expression to Automata (-NFA) 34 36

Construction of Regular Expression From Finite

Automata

37 40

Ordering the Elimination of States 40 42

Finite Automata with Output 42 45

Regular Grammars 45 50

Properties of Regular Languages 50 55

Myhill-Nerode Theorem 55 57 Pumping Lemma for Regular Languages 57 - 62

Finding (in) Distiguishable States 63 65

Assignment 1 66 67

Assignment 2 68 69

Answer keys 70

Explanations 70 74

#4.Context Free Grammar

75 123

Introduction 75 76

Context Free Language 76 78 Leftmost and Rightmost Derivations Ambiguity 78 80


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Contents Theory of Computation

THE GATE ACADEMY PVT.LTD. H.O.: #74, KeshavaKrupa (third Floor), 30thCross, 10thMain, Jayanagar 4thBlock, Bangalore-11 080 65700750 i f @th t d C i ht d W b th t d P II

Simplification of Context Free Grammars 80

Elimination of Useless Symbols 80 84

Normal Forms 85 91 Pushdown Automata 91 92

Definition of PDAs 92 98

Consruction of CFG from a PDA With Empty Stack 98 100

Properties of Context Free Languages 100 104

Decision Algorithms for CFLs 104 105

Non-Context Free Languages 106 108

The Pumping Lemma for CFLs 108 110

Membership Algorithm for Context -Free Grammars 111 112

Assignment 1 113 115

Assignment 2 115 119 Answer keys 120

Explanations 120 - 123

#5. Turing Machines 124 161 Introduction 124 126

Modification of Turing Machines 126 128

Post Machine 128 130

Two-Pushdown Stack Machine 130 131

Counter Machines 131

Storage in the State 132 Multiple Tracks 133 139

Universal Turing Machine 139 140

Context Sensitive Grammar 140 141

Linear Bounded Automata 141 142

Hierarchy of Formal Languages 142 143

Undesidability 143 144

The Universal Language 144 151

The Classes P & NP 151 153

Some of the NP-Hard Problems 153 154

Assignment 1 155 156 Assignment 2 157 158

Answer Keys 159

Explanations 159 161

Module Test 162 177

Test Questions 162 170

Answer Keys 171

Explanations 171 177

Reference Books 178


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Chapter-1 Theory of Computation

THE GATE ACADEMY PVT.LTD. H.O.: #74, KeshavaKrupa (third Floor), 30th

Cross, 10th

Main, Jayanagar 4th

Block, Bangalore-11

080 65700750 i f @th t d C i ht d W b th t d P 1

CHAPTER 1

Introduction/ Preliminaries

(String, Alphabet, Set, Relation, Equivalence of Relation etc.)

Preliminaries:

1. String:A string is a finite sequence of symbols put together.

Ex: bbvl

The length of this string is '4'.Note: The length of empty string denoted by , is the string consisting of zero symbols. Thus

I I = 0.

2. Alphabet:An alphabet is a finite set of symbols

Ex: {a, b, 0, 1, B}

3. Formal language:A formal language is a set of strings of symbols from some alphabet.

Ex: The language consisting of all strings formed by the alphabet {0, 1}Note:

1.

The empty set, , is a formal language. The cardinality (size) of this language is zero.2.

The set consisting of empty string, {e} is a formal language. The cardinality (size) of this

language is one.

4. Set:A set is a collection of objects (members of the set) without repetition.i.

Finite Set: A set which contains finite number of elements is said to be finite set.

Ex:- {1,2,3,4}

ii.

Countably Infinite Set:Sets that can be placed in one-to-one correspondence with the

integers are said to be countably infinite or countable or denumerable.

Ex:-The set * of the finite-length strings from an alphabet are countably infinite, (if : {0, 1} then *: {0,01,1,10,011.....} i.e all possible strings with l0' and'1')

iii. Uncountable set:

Sets that can't be placed in one-to-one correspondence with the

integers are said to be uncountable sets.

Ex:-The set of real numbers.


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5. Relations: A(binary) relation is a set of ordered tuple. The first component of each tuple is

chosen from a set called the domain and the second component of each pair is chosen from a

(possibly different) set called the range.

Ex:

Let A {1, 2, 3, 4} be a set. A relation R, on 'A' can be defined as R: {(1,2), (1,3), (1,1), (2,3)}

Properties of Relations:We say a relation R on set S is

1.

Reflexiveif aRa for all a in S.2.

Irreflexive

if aRa is false for all a is S.

3. Transitiveif aRb and bRc implies aRc.4.

Symmetricif aRb implies bRa.

5. Asymmetric

if aRb implies that bRa is false.

6. Anti-symmetricif aRb and bRa implies a = b.

quivalence Relation: A relation is said to be equivalence relation if it satisfies the following

properties.

1.

Reflexive

2. Symmetric3. Transitive

Animportant property of equivalence relation 'R' on set 'S' is that R partitions 'S' into disjointnonempty equivalence classes (may be infinite)

i.e., S = S1 S2....., where for each i and j, with i j

1. Si Sj= 2. For each 'a' and 'b' in S jaRb is true.

3.

For each 'a' in Sj and 'b' in Sj, aRb is false.

The Si's are called equivalence classes.

Ex: The relation 'R' on people defined by pRq if and only if p and q were born at the same hour

of the same day of some year.

The number of equivalence classes are: 24 (no. of hours) 7 (no. of days in a week).

Closure of relations: Suppose P is a set of properties of relations. The P-closure of a relation Ris the smallest relation R that includes all the pairs of R and possesses the properties in P.

Ex:- Let R = {(1,2), (2,3), (3,1)} be a relation on set {l,2,3}

i. The reflexive-transitive closure of R is denoted by R* is

Added by reflexivity

(1,2), (2,3), (3,1), (1,1), (2,2), (3,3), (1,3), (2,1), (3,2)}

By transitivity

ii.

The symmetric closure of R is

{(1,2), (2,3), (3,1), (2,1),(3,2), (1,3)}


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TERM DEFINITION

prefix of s A string obtained by removing zero or more trailingsymbols of string s; e.g., ban is a prefix of banana.

suffix of s A string formed by deleting zero or more of the leading

symbols of s; e.g., nana is a suffix of banana.

substring of s A string obtained by deleting a prefix and a suffix from s;

e.g., nan is a substring of banana. Every prefix and every

suffix of s is a substring of s, but not every substring of sis a prefix or a suffix of s. For every string s, both s and s

are prefixes, suffixes, and substrings of s.

proper prefix, suffix, or substring

of s

Any nonempty string x that is, respectively, a prefix,

suffix, or substring of s such that s x.

subsequence of s Any string formed by deleting zero or more notnecessarily contiguous symbols from s; e.g., baaa is a

subsequence of banana.


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CHAPTER 2

Finite Automata

Finite Automata Introduction

A finite automaton involves states and transitions among states in response to inputs. They are

useful for building several different kinds of software, including the lexical analysis componentof a complier and systems for verifying the correctness of circuits or protocols. They also serve

as the control unit in many physical systems including: vending machines, elevators, automatictraffic signals, and computer microprocessors. Also network protocol stacks.

Deterministic Finite Automata

A DFA captures the basic elements of an abstract machine: it reads in a string, and depending on

the input and the way the machine was designed, it outputs either true or false. A DFA is alwaysis in one of N states, which we usually name 0 through N-1. Each state is labeled true or false.

The DFA begins in a designated state called the start state .As the input characters are read inone at a time, the DFA changes from one state to another in a pre-specified way. The new state is

completely determined by the current state and the character just read in. When the input isexhausted, the DFA outputs true or false according to the label of the state it is currently in.

A Deterministic finite automaton is represented by a Quintuple

(5-tuple: Q,,,q0,F)whereQ : Finite set of states

: Finite set of input symbols called the alphabet. : Q X Q is a transition function from Q X to Qq0: A start state, one of the states in QF : A set of final states, such that F Q.

Induction

Suppose is a string of the form that ism a is the last symbol of , and is the stringconsisting of all but the last symbol. For example is broken into and .Then

, (

, , )

Now (2.1) may seem like a lot to take in, but the idea is simple. To compute , first compute, the state that the automaton is in after processing all but the last symbol of . Supposethis state is p; that is. , . Then , is what we get by making a transition from state pon input a, the last symbol of w. That is, , = ,

Example 1

Let us design a DFA to accept the language

L = {| has both an even number of 0's and an even number of 1's}


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It should not be surprising that the job of the states of this DFA is to count both the number of

0's and the number of l's, but count them modulo 2. That is, the state is used to remember

whether the number of 0's seen so far is even or odd, and also to remember whether the numberof l's seen so far is even or odd. There are thus four states, which can be given the followinginterpretations:

: Both the number of 0's seen so far and the number of l's seen so far are even.

: The number of 0's seen so far is even, but the number of l's seen so far is odd.

: The number of l's seen so far is even, but the number of 0's seen so far is odd.

: Both the number of 0's seen so far and the number of l's seen so far are odd.

State is both the start state and the alone the accepting state. It is the start state, becausebefore reading any inputs, the numbers of 0's and l's seen so far are both zero, and zero is even.It is the only accepting state, because it describes exactly the condition for a sequence of 0's and

l's to be in language L.

We now know almost how to specify the DFA for language L. It is

*, , , +, *,+, , , *+

Deterministic Finite Automata

Figure 2 1: Transition diagram for the DFA of Example 1

where the transition function S is described by the transition diagram of Fig. 2.1. Notice howeach input 0 causes the state to cross the horizontal, dashed line. Thus, after seeing an even

number of 0's we are always above the line, in state go or qi while after seeing an odd number of

0's we are always below the line, in state q-z or q%. Likewise, every 1 causes the state to crossthe vertical, dashed line. Thus, after seeing an even number of l's. we are always to the left, in

state q0or q2. while after seeing an odd number of l's we are to the right, in state q\ or g3. These

observations are an informal proof that the four states have the interpretations attributed to

them. However, one could prove the correctness of our claims about the states formally, by amutual induction with respect to example.

We can also represent this DFA by a transition table. Figure 2.2 shows this table. However, weare not just concerned with the design of this DFA; we want to use it to illustrate the

construction of 5 from its transition function 8. Suppose the input is 110101. Since this string


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has even numbers of 0's and l's both, we expect it is in the language. Thus, we expect that S( ,110101) = since is the only accepting state. Let us now verify that claim.

Figure 2 2: Transition table for the DFA of Example 1

The check involves computing 8(q0,w) for each prefix w of 110101, starting at e and going inincreasing size. The summary of this calculation is:

, ., ( , , ) (,) , ( , , ) (,) , ( , , ) (,) , ( ,, ) (,) , ( ,, ) (,) , ( ,, ) (,)

Acceptance by an Automata :

A string x is said to be accepted by a finite automaton M Q, , , q0, F if q0, x) = P forsome p in F. The language accepted by M, designated L M, is the set *x | q0,x) is in F}.

A language is a regular set (or just regular) if it is the set accepted by some automaton.

There are two preferred notations for describing Automata

1 Transition diagram

2 Transition table

Example

1. Give DFA for accepting the set of all strings containing as substring.

Transition diagram

Transition Table:

0 1

q0 q0 q1

q1 q0 q2

q2 q0 q3

*q3 q3 q3

1 1q1 q2 q3q0

1

0/10

Start

0

0


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