Preface to the Second Edition. Preface to the Third Edition

Transcription

1 C O N T E N T S Preface to the First Edition xi To the student xi To the educator xii The first edition xiii Feedback to the author xiii Acknowledgments xiv Preface to the Second Edition Preface to the Third Edition xvii xxi 0 Introduction Automata, Computability, and Complexity Complexity theory Computability theory Automata theory Mathematical Notions and Terminology Sets Sequences and tuples Functions and relations Graphs Strings and languages Boolean logic Summary of mathematical terms Definitions, Theorems, and Proofs Finding proofs Types of Proof Proof by construction Proof by contradiction Proof by induction Exercises, Problems, and Solutions v

2 vi CONTENTS Part One: Automata and Languages 29 1 Regular Languages Finite Automata Formal definition of a finite automaton Examples of finite automata Formal definition of computation Designing finite automata The regular operations Nondeterminism Formal definition of a nondeterministic finite automaton Equivalence of NFAs and DFAs Closure under the regular operations Regular Expressions Formal definition of a regular expression Equivalence with finite automata Nonregular Languages The pumping lemma for regular languages Exercises, Problems, and Solutions Context-Free Languages Context-Free Grammars Formal definition of a context-free grammar Examples of context-free grammars Designing context-free grammars Ambiguity Chomsky normal form Pushdown Automata Formal definition of a pushdown automaton Examples of pushdown automata Equivalence with context-free grammars Non-Context-Free Languages The pumping lemma for context-free languages Deterministic Context-Free Languages Properties of DCFLs Deterministic context-free grammars Relationship of DPDAs and DCFGs Parsing and LR(k) Grammars Exercises, Problems, and Solutions Part Two: Computability Theory The Church Turing Thesis Turing Machines Formal definition of a Turing machine

3 CONTENTS vii Examples of Turing machines Variants of Turing Machines Multitape Turing machines Nondeterministic Turing machines Enumerators Equivalence with other models The Definition of Algorithm Hilbert s problems Terminology for describing Turing machines Exercises, Problems, and Solutions Decidability Decidable Languages Decidable problems concerning regular languages Decidable problems concerning context-free languages Undecidability The diagonalization method An undecidable language A Turing-unrecognizable language Exercises, Problems, and Solutions Reducibility Undecidable Problems from Language Theory Reductions via computation histories A Simple Undecidable Problem Mapping Reducibility Computable functions Formal definition of mapping reducibility Exercises, Problems, and Solutions Advanced Topics in Computability Theory The Recursion Theorem Self-reference Terminology for the recursion theorem Applications Decidability of logical theories A decidable theory An undecidable theory Turing Reducibility A Definition of Information Minimal length descriptions Optimality of the definition Incompressible strings and randomness Exercises, Problems, and Solutions

4 viii CONTENTS Part Three: Complexity Theory Time Complexity Measuring Complexity Big-O and small-o notation Analyzing algorithms Complexity relationships among models The Class P Polynomial time Examples of problems in P The Class NP Examples of problems in NP The P versus NP question NP-completeness Polynomial time reducibility Definition of NP-completeness The Cook Levin Theorem Additional NP-complete Problems The vertex cover problem The Hamiltonian path problem The subset sum problem Exercises, Problems, and Solutions Space Complexity Savitch s Theorem The Class PSPACE PSPACE-completeness The TQBF problem Winning strategies for games Generalized geography The Classes L and NL NL-completeness Searching in graphs NL equals conl Exercises, Problems, and Solutions Intractability Hierarchy Theorems Exponential space completeness Relativization Limits of the diagonalization method Circuit Complexity Exercises, Problems, and Solutions Advanced Topics in Complexity Theory Approximation Algorithms

5 CONTENTS ix 10.2 Probabilistic Algorithms The class BPP Primality Read-once branching programs Alternation Alternating time and space The Polynomial time hierarchy Interactive Proof Systems Graph nonisomorphism Definition of the model IP = PSPACE Parallel Computation Uniform Boolean circuits The class NC P-completeness Cryptography Secret keys Public-key cryptosystems One-way functions Trapdoor functions Exercises, Problems, and Solutions Selected Bibliography 443 Index 448