Introduction to Automata Theory, Languages, and Computation

Preface
Contents
1 Automata: The Methods and the Madness
1.1 Why Study Autornata Theory?
1.1.1 Introduction to Finite Automata
1.1.2 Structural Representations
1.1.3 Automata and Complexity
1.2 Introduction to Formal Proof
1.2.1 Deductive Proofs
1.2.2 Reduction to Definitions
1.2.3 Other Theorem Forms
1.2.4 Theorems That Appear Not to Be If-Then Statements
1.3 Additional Forms of Proof
1.3.1 Proving Equivalences About Sets
1.3.2 The Contrapositive
1.3.3 Proof by Contradiction
1.3.4 Counterexamples
1.4 Inductive Proofs
1.4.1 Inductions on Integers
1.4.2 More General Forms of Integer Inductions
1.4.3 Structural Inductions
1.4.4 Mutual Inductions
1.5 The Central Concepts of Automata Theory
1.5.1 Alphabets
1.5.2 Strings
1.5.3 Languages
1.5.4 Problems
1.6 Summary
1.7 Gradiance Problems
1.8 References
2 Finite Automata
2.1 An Informal Picture of Finite Automata
2.1.1 The Ground Rules
2.1.2 The Protocol
2.1.3 Enabling the Automata to Ignore Actions
2.1.4 The Entire System as an Automaton
2.1.5 Using the Product Automaton to Validate the Protocol
2.2 Deterministic Finite Automata
2.2.1 Definition of a Deterministic Finite Automaton
2.2.2 How a DFA Processes Strings
2.2.3 Simpler Notations for DFA's
2.2.4 Extending the Transition Function to Strings
2.2.5 The Language of a DFA
2.2.6 Exercises
2.3 Nondeterministic Finite Automata
2.3.1 An Informal View of Nondeterministic Finite Automata
2.3.2 Definition of Nondeterministic Finite Automata
2.3.3 The Extended Transition Function
2.3.4 The Language of an NFA
2.3.5 Equivalence of Deterministic and Nondeterministic Finite Automata
2.3.6 A Bad Case for the Subset Construction
2.3.7 Exercises
2.4 An Application: Text Search
2.4.1 Finding Strings in Text
2.4.2 Nondeterministic Finite Automata for Text Search
2.4.3 A DFA to Recognize a Set of Keywords
2.4.4 Exercises
2.5 Finite Automata With Epsilon-Transitions
2.5.1 Uses of ϵ-Transitions
2.5.2 The Formal Notation for an ϵ-NFA
2.5.3 Epsilon-Closures
2.5.4 Extended Transitions and Languages for ϵ-NFA's
2.5.5 Eliminating ϵ-Transitions
2.5.6 Exercises
2.6 Summary
2.7 Gradiance Problems
2.8 References
3 Regular Expressions and Languages
3.1 Regular Expressions
3.1.1 The Operators of Regular Expressions
3.1.2 Building Regular Expressions
3.1.3 Precedence of Regular-Expression Operators
3.1.4 Exercises
3.2 Finite Automata and Regular Expressions
3.2.1 From DFA's to Regular Expressions
3.2.2 Converting DFA's to Regular Expressions by Eliminating States
3.2.3 Converting Regular Expressions to Automata
3.2.4 Exercises
3.3 Applications of Regular Expressions
3.3.1 Regular Expressions in UNIX
3.3.2 Lexical Analysis
3.3.3 Finding Patterns in Text
3.3.4 Exercises
3.4 Algebraic Laws for Regular Expressions
3.4.1 Associativity and Commutativity
3.4.2 Identities and Annihilators
3.4.3 Distributive Laws
3.4.4 The Idempotent Law
3.4.5 Laws Involving Closures
3.4.6 Discovering Laws for Regular Expressions
3.4.7 The Test for a Regular-Expression Algebraic Law
3.4.8 Exercises
3.5 Summary
3.6 Gradiance Problems
3.7 References
4 Properties of Regular Languages
4.1 Proving Languages Not to Be Regular
4.1.1 The Pumping Lemma for Regular Languages
4.1.2 Applications of the Pumping Lemma
4.1.3 Exercises
4.2 Closure Properties of Regular Languages
4.2.1 Closure of Regular Languages Under Boolean Operations
4.2.2 Reversal
4.2.3 Homomorphisms
4.2.4 Inverse Homomorphisms
4.2.5 Exercises
4.3 Decision Properties of Regular Languages
4.3.1 Converting Among Representations
4.3.2 Testing Emptiness of Regular Languages
4.3.3 Testing Membership in a Regular Language
4.3.4 Exercises
4.4 Equivalence and Minimization of Automata
4.4.1 Testing Equivalence of States
4.4.2 Testing Equivalence of Regular Languages
4.4.3 Minimization of DFA's
4.4.4 Why the Minimized DFA Can't Be Beaten
4.4.5 Exercises
4.5 Summary
4.6 Gradiance Problems
4.7 References
5 Context-Free Grammars and Languages
5.1 Context-Free Grarnrnars
5.1.1 An Informal Example
5.1.2 Definition of Context-Free Grammars
5.1.3 Derivations Using a Grammar
5.1.4 Leftmost and Rightmost Derivations
5.1.5 The Language of a Grammar
5.1.6 Sentential Forms
5.1.7 Exercises
5.2 Parse Trees
5.2.1 Constructing Parse Trees
5.2.2 The Yield of a Parse Tree
5.2.3 Inference, Derivations, and Parse Trees
5.2.4 From Inferences to Trees
5.2.5 From Trees to Derivations
5.2.6 From Derivations to Recursive Inferences
5.2.7 Exercises
5.3 Applications of Context-Free Grarnrnars
5.3.1 Parsers
5.3.2 The YACC Parser-Generator
5.3.3 Markup Languages
5.3.4 XML and Document-Type Definitions
5.3.5 Exercises
5.4 Ambiguity in Grammars and Languages
5.4.1 Ambiguous Grammars
5.4.2 Removing Ambiguity From Grammars
5.4.3 Leftmost Derivations as a Way to Express Ambiguity
5.4.4 Inherent Ambiguity
5.4.5 Exercises
5.5 Summary
5.6 Gradiance Problems
5.7 References
6 Pushdown Automata
6.1 Definition of the Pushdown Automaton
6.1.1 Informal Introduction
6.1.2 The Formal Definition of Pushdown Automata
6.1.3 A Graphical Notation for PDA's
6.1.4 Instantaneous Descriptions of a PDA
6.1.5 Exercises
6.2 The Languages of a PDA
6.2.1 Acceptance by Final State
6.2.2 Acceptance by Empty Stack
6.2.3 From Empty Stack to Final State
6.2.4 From Final State to Empty Stack
6.2.5 Exercises
6.3 Equivalence of PDA's and CFG's
6.3.1 From Grammars to Pushdown Automata
6.3.2 From PDA's to Grammars
6.3.3 Exercises
6.4 Deteministic Pushdown Automata
6.4.1 Definition of a Deterministic PDA
6.4.2 Regular Languages and Deterministic PDA's
6.4.3 DPDA's and Context-Free Languages
6.4.4 DPDA's and Ambiguous Grammars
6.5 Summary
6.6 Gradiance Problems
6.7 References
7 Properties of Context-Free Languages
7.1 Normal Forms for Context-Free Grammars
7.1.1 Eliminating Useless Symbols
7.1.2 Computing the Generating and Reachable Symbols
7.1.3 Eliminating ϵ-Productions
7.1.4 Eliminating Unit Productions
7.1.5 Chomsky Normal Form
7.1.6 Exercises
7.2 The Pumping Lemma for Context-Free Languages
7.2.1 The Size of Parse Trees
7.2.2 Statement of the Pumping Lemma
7.2.3 Applications of the Pumping Lemma for CFL's
7.2.4 Exercises
7.3 Closure Properties of Context-Free Languages
7.3.1 Substitutions
7.3.2 Applications of the Substitution Theorem
7.3.3 Reversal
7.3.4 Intersection With a Regular Language
7.3.5 Inverse Homomorphism
7.3.6 Exercises
7.4 Decision Properties of CFL's
7.4.1 Complexity of Converting Among CFG's and PDA's
7.4.2 Running Time of Conversion to Chomsky Normal Form
7.4.3 Testing Emptiness of CFL's
7.4.4 Testing Membership in a CFL
7.4.5 Preview of Undecidable CFL Problems
7.4.6 Exercises
7.5 Summary
7.6 Gradiance Problems
7.7 References
8 Introduction to Turing Machines
8.1 Problems That Computers Cannot Solve
8.1.1 Programs that Print "Hello, World"
8.1.2 The Hypothetical "Hello, World" Tester
8.1.3 Reducing One Problem to Another
8.1.4 Exercises
8.2 The Turing Machine
8.2.1 The Quest to Decide All Mathematical Questions
8.2.2 Notation for the Turing Machine
8.2.3 Instantaneous Descriptions for Turing Machines
8.2.4 Transition Diagrams for Turing Machines
8.2.5 The Language of a Turing Machine
8.2.6 Turing Machine and Halting
8.2.7 Exercises
8.3 Programming Techniques for Turing Machines
8.3.1 Storage in the State
8.3.2 Multiple Tracks
8.3.3 Subroutines
8.3.4 Exercises
8.4 Extensions to the Basic Turing Machine
8.4.1 Multitape Turing Machines
8.4.2 Equivalence of One-Tape and Multitape TM's
8.4.3 Running Time and the Many-Tapes-to-One Construction
8.4.4 Nondeterministic Turing Machines
8.4.5 Exercises
8.5 Restricted Turing Machines
8.5.1 Turing Machines With Semi-infinite Tapes
8.5.2 Multistack Machines
8.5.3 Counter Machines
8.5.4 The Power of Counter Machines
8.6 Turing Machines and Cornputers
8.6.1 Simulating a Turing Machine by Computer
8.6.2 Simulating a Computer by a Turing Machine
8.6.3 Comparing the Running Times of Computers and Turing Machines
8.7 Summary
8.8 Gradiance Problems
8.9 References
9 Undecidability
9.1 A Language That Is Not Recursively Enumerable
9.1.1 Enumerating the Binary Strings
9.1.2 Codes for Turing Machines
9.1.3 The Diagonalization Language
9.1.4 Proof That Ld Is Not Recursively Enumerable
9.1.5 Exercises
9.2 An Undecidable Problem That Is RE
9.2.1 Recursive Languages
9.2.2 Complements of Recursive and RE languages
9.2.3 The Universal Language
9.2.4 Undecidability of the Universal Language
9.2.5 Exercises
9.3 Undecidable Problems About Turing Machines
9.3.1 Reductions
9.3.2 Turing Machines That Accept the Empty Language
9.3.3 Rice's Theorem and Properties of the RE Languages
9.3.4 Problems about Turing-Machine Specifications
9.3.5 Exercises
9.4 Post's Correspondence Problem
9.4.1 Definition of Post's Correspondence Problem
9.4.2 The "Modified" PCP
9.4.3 Completion of the Proof of PCP Undecidability
9.4.4 Exercises
9.5 Other Undecidable Problems
9.5.1 Problems About Programs
9.5.2 Undecidability of Ambiguity for CFG's
9.5.3 The Complement ofa List Language
9.5.4 Exercises
9.6 Summary
9.7 Gradiance Problems
9.8 References
10 Intractable Problems
10.1 The Classes P and NP
10.1.1 Problems Solvable in Polynomial Time
10.1.2 An Example: Kruskal's Algorithm
10.1.3 Nondeterministic Polynomial Time
10.1.4 An NP Example: The Traveling Salesman Problem
10.1.5 Polynomial-Time Reductions
10.1.6 NP-Complete Problems
10.1.7 Exercises
10.2 An NP-Complete Problem
10.2.1 The Satisfiability Problem
10.2.2 Representing SAT Instances
10.2.3 NP-Completeness of the SAT Problem
10.2.4 Exercises
10.3 A Restricted Satisfiability Problem
10.3.1 Normal Forms for Boolean Expressions
10.3.2 Converting Expressionsto CNF
10.3.3 NP-Completeness of CSAT
10.3.4 NP-Completeness of 3SAT
10.3.5 Exercises
10.4 Additional NP-Cornplete Problems
10.4.1 Describing NP-complete Problems
10.4.2 The Problem of Independent Sets
10.4.3 The Node-Cover Problem
10.4.4 The Directed Hamilton-Circuit Problem
10.4.5. Undirected Hamilton Circuits and the TSP
10.4.6 Summary of NP-Complete Problems
10.4.7 Exercises
10.5 Summary
10.6 Gradiance Problems
10.7 References
11 Additional Classes of Problems
11.1 Complements of Languages in NP
11.1.1 The Class of Languages Co-NP
11.1.2 NP-Complete Problems and Co-NP
11.1.3 Exercises
11.2 Problems Solvable in Polynomial Space
11.2.1 Polynomial-Space Turing Machines
11.2.2 Relationship of PS and NPS to Previously Defined Classes
11.2.3 Deterministic and Nondeterministic Polynomial Space
11.3 A Problem That Is Complete for PS
11.3.1 PS-Completeness
11.3.2 Quantified Boolean Formulas
11.3.3 Evaluating Quantified Boolean Formulas
11.3.4 PS-Completeness of the QBF Problem
11.3.5 Exercises
11.4 Language Classes Based on Randomization
11.4.1 Quicksort: an Example of a Randomized Algorithm
11.4.2 A Turing-Machine Model Using Randomization
11.4.3 The Language of a Randomized Turing Machine
11.4.4 The Class RP
11.4.5 Recognizing Languages in RP
11.4.6 The Class ZPP
11.4.7 Relationship Between RP and ZPP
11.4.8 Relationships to the Classes P and NP
11.5 The Complexity of Primality Testing
11.5.1 The Importance of Testing Primality
11.5.2 Introduction to Modular Arithmetic
11.5.3 The Complexity of Modular-Arithmetic Computations
11.5.4 Random-Polynomial Primality Testing
11.5.5 Nondeterministic Primality Tests
11.5.6 Exercises
11.6 Summary
11.8 References
Index