Automata Theory, Languages, and Computation

Transcription

1 Introduction to S ECO IN O EDITION Automata Theory, Languages, and Computation Pearson Educatic ULB Darmstadt I Hill III River, N.J

2 Table of Contents 1 Automata: The Methods and the Madness Why Study Automata Theory? Introduction to Finite Automata Structural Representations Automata and Complexity Introduction to Formal Proof Deductive Proofs Reduction to Definitions Other Theorem Forms Theorems That Appear Not to Be If-Then Statements Additional Forms of Proof Proving Equivalences About Sets The Contrapositive Proof by Contradiction Counterexamples Inductive Proofs Inductions on Integers More General Forms of Integer Inductions Structural Inductions Mutual Inductions The Central Concepts of Automata Theory Alphabets Strings Languages Problems Summary of Chapter References for Chapter Finite Automata An Informal Picture of Finite Automata The Ground Rules The Protocol Enabling the Automata to Ignore Actions 41 vii

3 viii TABLE OF CONTENTS The Entire System as an Automaton Using the Product Automaton to Validate the Protocol Deterministic Finite Automata Definition of a Deterministic Finite Automaton How a DFA Processes Strings Simpler Notations for DFA's Extending the Transition Function to Strings The Language of a DFA Exercises for Section Nondeterministic Finite Automata An Informal View of Nondeterministic Finite Automata Definition of Nondeterministic Finite Automata The Extended Transition Function The Language of an NFA Equivalence of Deterministic and Nondeterministic Finite Automata A Bad Case for the Subset Construction Exercises for Section An Application: Text Search Finding Strings in Text Nondeterministic Finite Automata for Text Search A DFA to Recognize a Set of Keywords Exercises for Section Finite Automata With Epsilon-Transitions Uses of e-transitions The Formal Notation for an e-nfa Epsilon-Closures Extended Transitions and Languages for e-nfa's Eliminating e-transitions Exercises for Section Summary of Chapter References for Chapter Regular Expressions and Languages Regular Expressions The Operators of Regular Expressions Building Regular Expressions Precedence of Regular-Expression Operators Exercises for Section Finite Automata and Regular Expressions From DFA's to Regular Expressions Converting DFA's to Regular Expressions by Eliminating States Converting Regular Expressions to Automata Exercises for Section

4 TABLE OF CONTENTS ix 3.3 Applications of Regular Expressions Regular Expressions in UNIX Lexical Analysis Finding Patterns in Text Ill Exercises for Section Algebraic Laws for Regular Expressions Associativity and Commutativity Identities and Annihilators Distributive Laws The Idempotent Law Laws Involving Closures Discovering Laws for Regular Expressions The Test for a Regular-Expression Algebraic Law Exercises for Section Summary of Chapter References for Chapter Properties of Regular Languages Proving Languages not to be Regular The Pumping Lemma for Regular Languages Applications of the Pumping Lemma Exercises for Section Closure Properties of Regular Languages Closure of Regular Languages Under Boolean Operations Reversal Homomorphisms Inverse Homomorphisms Exercises for Section Decision Properties of Regular Languages Converting Among Representations Testing Emptiness of Regular Languages Testing Membership in a Regular Language Exercises for Section Equivalence and Minimization of Automata Testing Equivalence of States Testing Equivalence of Regular Languages Minimization of DFA's Why the Minimized DFA Can't Be Beaten Exercises for Section Summary of Chapter References for Chapter 4 166

5 TABLE OF CONTENTS Context-Free Grammars and Languages Context-Free Grammars An Informal Example Definition of Context-Free Grammars Derivations Using a Grammar Leftmost and Rightmost Derivations The Language of a Grammar Sentential Forms Exercises for Section Parse Trees Constructing Parse Trees The Yield of a Parse Tree Inference, Derivations, and Parse Trees From Inferences to Trees From Trees to Derivations From Derivations to Recursive Inferences Exercises for Section Applications of Context-Free Grammars Parsers The YACC Parser-Generator Markup Languages XML and Document-Type Definitions Exercises for Section Ambiguity in Grammars and Languages Ambiguous Grammars Removing Ambiguity From Grammars Leftmost Derivations as a Way to Express Ambiguity Inherent Ambiguity Exercises for Section Summary of Chapter References for Chapter Pushdown Automata Definition of the Pushdown Automaton Informal Introduction The Formal Definition of Pushdown Automata A Graphical Notation for PDA's Instantaneous Descriptions of a PDA Exercises for Section The Languages of a PDA Acceptance by Final State Acceptance by Empty Stack From Empty Stack to Final State From Final State to Empty Stack Exercises for Section

6 TABLE OF CONTENTS xi 6.3 Equivalence of PDA's and CFG's From Grammars to Pushdown Automata From PDA's to Grammars Exercises for Section Deterministic Pushdown Automata Definition of a Deterministic PDA Regular Languages and Deterministic PDA's DPDA's and Context-Free Languages DPDA's and Ambiguous Grammars Exercises for Section Summary of Chapter References for Chapter Properties of Context-Free Languages Normal Forms for Context-Free Grammars Eliminating Useless Symbols Computing the Generating and Reachable Symbols Eliminating e-productions Eliminating Unit Productions Chomsky Normal Form Exercises for Section The Pumping Lemma for Context-Free Languages The Size of Parse Trees Statement of the Pumping Lemma Applications of the Pumping Lemma for CFL's Exercises for Section Closure Properties of Context-Free Languages Substitutions Applications of the Substitution Theorem Reversal Intersection With a Regular Language Inverse Homomorphism Exercises for Section Decision Properties of CFL's Complexity of Converting Among CFG's and PDA's Running Time of Conversion to Chomsky Normal Form Testing Emptiness of CFL's Testing Membership in a CFL Preview of Undecidable CFL Problems Exercises for Section Summary of Chapter References for Chapter 7 304

7 xii TABLE OF CONTENTS 8 Introduction to Turing Machines Problems That Computers Cannot Solve Programs that Print "Hello, World" The Hypothetical "Hello, World" Tester Reducing One Problem to Another Exercises for Section The Turing Machine The Quest to Decide All Mathematical Questions Notation for the Turing Machine Instantaneous Descriptions for Turing Machines Transition Diagrams for Turing Machines The Language of a Turing Machine Turing Machines and Halting Exercises for Section Programming Techniques for Turing Machines Storage in the State Multiple Tracks Subroutines Exercises for Section Extensions to the Basic Turing Machine Multitape Turing Machines Equivalence of One-Tape and Multitape TM's Running Time and the Many-Tapes-to-One Construction Nondeterministic Turing Machines Exercises for Section Restricted Turing Machines Turing Machines With Semi-infinite Tapes Multistack Machines Counter Machines The Power of Counter Machines Exercises for Section Turing Machines and Computers Simulating a Turing Machine by Computer Simulating a Computer by a Turing Machine Comparing the Running Times of Computers and Turing Machines Summary of Chapter References for Chapter Undecidability A Language That Is Not Recursively Enumerable Enumerating the Binary Strings Codes for Turing Machines The Diagonalization Language Proof that L& is not Recursively Enumerable 372

8 TABLE OF CONTENTS xiii Exercises for Section An Undecidable Problem That is RE Recursive Languages Complements of Recursive and RE languages The Universal Language Undecidability of the Universal Language Exercises for Section Undecidable Problems About Turing Machines Reductions Turing Machines That Accept the Empty Language Rice's Theorem and Properties of the RE Languages Problems about Turing-Machine Specifications Exercises for Section Post's Correspondence Problem Definition of Post's Correspondence Problem The "Modified" PCP Completion of the Proof of PCP Undecidability Exercises for Section Other Undecidable Problems Problems About Programs Undecidability of Ambiguity for CFG's The Complement of a List Language Exercises for Section Summary of Chapter References for Chapter Intractable Problems The Classes V and MV Problems Solvable in Polynomial Time An Example: Kruskal's Algorithm Nondeterministic Polynomial Time An MV Example: The Traveling Salesman Problem Polynomial-Time Reductions NP-Complete Problems Exercises for Section An NP-Complete Problem The Satisfiability Problem Representing SAT Instances NP-Completeness of the SAT Problem Exercises for Section A Restricted Satisfiability Problem Normal Forms for Boolean Expressions Converting Expressions to CNF NP-Completeness of CSAT NP-Completeness of 3SAT Exercises for Section

9 xiv TABLE OF CONTENTS 10.4 Additional NP-Complete Problems Describing NP-complete Problems The Problem of Independent Sets The Node-Cover Problem The Directed Hamilton-Circuit Problem Undirected Hamilton Circuits and the TSP Summary of NP-Complete Problems Exercises for Section Summary of Chapter References for Chapter Additional Classes of Problems Complements of Languages in MV The Class of Languages Co-NV NP-Complete Problems and Co-AfV Exercises for Section Problems Solvable in Polynomial Space Polynomial-Space Turing Machines Relationship of VS and NVS to Previously Defined Classes Deterministic and Nondeterministic Polynomial Space A Problem That Is Complete for VS PS-Completeness Quantified Boolean Formulas Evaluating Quantified Boolean Formulas PS-Completeness of the QBF Problem Exercises for Section Language Classes Based on Randomization Quicksort: an Example of a Randomized Algorithm A Turing-Machine Model Using Randomization The Language of a Randomized Turing Machine The Class HV Recognizing Languages in 1ZV The Class ZVV Relationship Between TIV and ZVV Relationships to the Classes V and HV The Complexity of Primality Testing The Importance of Testing Primality Introduction to Modular Arithmetic The Complexity of Modular-Arithmetic Computations Random-Polynomial Primality Testing Nondeterministic Primality Tests Exercises for Section Summary of Chapter References for Chapter Index 513