Theory of Computing: A Gentle Introduction

209 Pages·2000·8.66 MB·English

by  Efim Kinber

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Theory of Computing
A Gentle Introduction
Theory of Computing
A Gentle Introduction
Efim Kinber
Sacred Heart University
Carl Smith
University of Maryland
PRENTICE HALL, Upper Saddle River, New Jersey 07458
Library of Congress Cataloging-in-Publication Data
Kinber, Efim
Theory of computing: a gentle introduction / Carl Smith, Efim Kinber,
p. cm.
Includes bibliological references and index.
ISBN: 0-13-027961-7
1. Electronic data processing. I. Smith, Carl, 1950 April 25 - II. Title.
QA76 .K475 2001
004-dc2l 00-024617
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Contents
Preface xiii
1 Introduction 1
1.1 Why Study the Theory of Computing? . . . .. . . . . . . . . . . . . .. . . . 3
1.2 What Is Computation? ................................. 3
1.3 The Contents of This Book ....... ....................... 4
1.4 Mathematical Preliminaries .............................. 5
Exercises ........... .................................... 9
2 Finite Automata 11
2.1 Deterministic Finite Automata ........................... 13
2.2 Nondeterministic Finite Automata ......................... 20
2.3 Determinism versus Nondeterminism ........................ 23
2.4 Regular Expressions ........ ........................... 29
2.5 Nonregular Languages ........ .......................... 43
2.6 Algorithms for Finite Automata ........................... 47
2.7 The State Minimization Problem .......................... 48
Exercises ............................................... 54
3 Context-Free Languages 67
3.1 Context-Free Grammars ................................ 69
3.2 Parsing .......... .................................. 76
3.3 Pushdown Automata .................................. 79
3.4 Languages and Automata ............................... 86
3.5 Closure Properties ........ ............................ 91
3.6 Languages That Are Not Context-Free ...................... 94
3.7 Chomsky Normal Form ........ ......................... 98
3.8 Determinism ........ ............................... 105
Exercises ............................................... 110
4 Turing Machines 119
4.1 Definition of a Turing Machine ........................... 121
vii
viii CONTENTS
4.2 Computations by Turing Machines ................... 130
4.3 Extensions of Turing Machines ..................... 134
4.3.1 Multiple Tapes ................................. 134
4.3.2 Multiple Heads ................................. 138
4.3.3 Two-Dimensional Tapes ........................... 139
4.3.4 Random Access Turing Machines ..................... 140
4.4 Nondeterministic Turing Machines ........................ 145
4.5 Turing Enumerable Languages ...... ..................... 148
Exercises ........... .................................... 151
5 Undecidability 157
5.1 The Church-Turing Thesis ............................... 159
5.2 Universal Turing Machines .............................. 161
5.3 The Halting Problem .................................. 162
5.4 Undecidable Problems ................................. 167
Exercises ........... .................................... 172
6 Computational Complexity 175
6.1 The Definition and the Class P ........................... 177
6.2 The Class AýP ......... .............................. 182
6.3 AHP-Completeness ........ ............................ 189
Exercises ........... .................................... 195
References 199
List of Symbols 202
Index 202
List of Figures
1.1 A Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
2.1 A Finite Automaton ........ ........................... 15
2.2 State Transition Diagram ....... ........................ 16
2.3 Recognizing Two Consecutive a's .......................... 16
2.4 Recognizing the Complement ....... ...................... 17
2.5 Mystery Automaton 1 ........ .......................... 17
2.6 Mystery Automaton 2 ........ .......................... 18
2.7 Recognizing an Even Number of a's and Odd Number of b's ..... .. 19
2.8 The Finite State Diagram of a Newspaper Vending Machine ..... .20
2.9 Combining Two Automata ............................... 21
2.10 Accepting All a's or All b's ....... ....................... 22
2.11 Start and End with a b ................................. 23
2.12 With Extra States ........ ............................ 24
2.13 Multiple Possible Transitions ....... ...................... 24
2.14 Replacement Transition ....... ......................... 25
2.15 A New Connection ........ ............................ 25
2.16 An e-Move ......... ................................ 26
2.17 An e-Move Eliminated ................................. 26
2.18 Segment of a Nondeterministic Finite Automaton ............... 27
2.19 Corresponding Deterministic Segment ...................... 27
2.20 Untransformed Automaton ....... ....................... 28
2.21 Transformed Automaton ................................ 29
2.22 Recognizing Strings Containing abbab ...................... 30
2.23 Deterministic Version of Figure 2.22 ........................ 30
2.24 Automaton A ....................................... 32
2.25 Automaton B ....................................... 32
2.26 The Union of A and B ................................. 33
2.27 Concatenation of L and M ...... ....................... 34
2.28 Kleene Star of A ........ ............................. 34
2.29 Singleton Automata ........ ........................... 37
2.30 Transformation Step 1 of 5 ....... ....................... 37
2.31 Transformation Step 2 of 5 ....... ....................... 37
ix
x LIST OF FIGURES
2.32 Transformation Step 3 of 5 ....................... 38
2.33 Transformation Step 4 of 5 ....... ....................... 38
2.34 Transformation Step 5 of 5 ....... ....................... 39
2.35 An Expression Diagram ....... ......................... 39
2.36 Initial Automaton ........ ............................ 40
2.37 Converted Automaton ........ .......................... 40
2.38 Automaton with One Accepting State ...................... 41
2.39 Steps 1, 2 and 3 ...................................... 41
2.40 Example Input for Algorithm B ........................... 42
2.41 Node 2 Deleted ........ .............................. 42
2.42 Node 3 deleted ........ .............................. 42
2.43 Final Expression Diagram ....... ........................ 43
2.44 Automaton with an Unreachable State ...................... 49
2.45 Without the Unreachable State ........................... 50
2.46 Distinguishability of r and r' Related to the Distinguishability of q
and q. ......... ................................... 50
2.47 A Minimal Deterministic Finite Automaton ................... 53
3.1 Parse Trees for John bought car ........................... 77
3.2 Parse Tree for the First Derivation of aa ...................... 79
3.3 Parse Tree for the Second Derivation of aa ................... 80
3.4 Pushdown Automaton ........ .......................... 82
3.5 Operation of the Automaton Accepting abbbaaaabb ............. 85
3.6 Computation of Automaton A on ababbaba ..... .............. 87
3.7 Schematic Parse Tree for w ....... ....................... 90
3.8 Schematic Parse Tree for w = uvxyz ...... .................. 96
4.1 A Turing Machine ........ ............................ 123
4.2 A Sequence of Turing Machines ........................... 126
4.3 Branching Turing Machines ...... ....................... 127
4.4 A Loop ......... .................................. 127
4.5 Writing b over a ........ ............................. 128
4.6 Moving Right and Writing a ............................. 128
4.7 Erasing the Input .................................... 128
4.8 Finding the First Blank on the Right ....................... 129
4.9 The Copying Machine Cp ....... ........................ 130
4.10 Computing the Function f(w) = wa ........................ 131
4.11 The Shifting Machine Shl ....... ........................ 132
4.12 Multitape Turing Machine ............................... 135
4.13 A Multitrack Tape ....... ............................ 136
4.14 Simulating k Tapes by 2k Tracks .......................... 136
4.15 Initial Configuration of Tracks ...... ..................... 137
4.16 Two-Dimensional Tape with Numbered Cells ................. 140
4.17 Simulation of a Two-Dimensional Tape in One Dimension ...... .140
LIST OF FIGURES xi
4.18 Random Access Thring Machine .................... 141
4.19 Commands of RAM Machine ...................... 142
4.20 Dovetailing . .... ...... .... .... ..... ..... ... 146
4.21 The Tree of Nondeterministic Computations .............. 147
5.1 U after Initialization on (M )(w) .................... 163
5.2 The Diagonal Function ................................. 164
5.3 The Program CONTR ................................. 165
6.1 An Undirected Weighted Graph .......................... 180
6.2 A Graph with a Hamiltonian Cycle ........................ 184
6.3 A Clique of Size 4 ........ ............................ 184
6.4 Algorithm A ........ ............................... 190
6.5 The Graph Derived from a Boolean Formula f ............ 193