## Table of Contents

(NOTE: *Each chapter concludes with Summary and References.*)

**1. Automata: The Methods and the Madness.**

Why Study Automata Theory?

Introduction to Finite Automata.

Structural Representations.

Automata and Complexity.

Introduction to Formal Proof.

Deduction 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.

**2. Finite Automata.**

An Informal Picture of Finite Automata.

The Ground Rules.

The Protocol.

Enabling the Automata to Ignore Actions.

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 2.2.

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 2.3.

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 2.4.

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.

**3. Regular Expressions and Languages.**

Regular Expressions.

The Operators of Regular Expressions.

Building Regular Expressions.

Precedence of Regular-Expression Operators.

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 3.2.

Applications of Regular Expressions.

Regular Expressions in UNIX.

Lexical Analysis.

Finding Patterns in Text.

Exercises for Section 3.3.

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 3.4.

**4. Properties of Regular Languages.**

Proving Languages not to be Regular.

The Pumping Lemma for Regular Languages.

Applications of the Pumping Lemma.

Exercises for Section 4.1.

Closure Properties of Regular Languages.

Closure of Regular Languages Under Boolean Operations.

Reversal.

Homomorphisms.

Inverse Homomorphisms.

Exercises for Section 4.2.

Decision Properties of Regular Languages.

Converting Among Representations.

Testing Emptiness of Regular Languages.

Testing Membership in a Regular Language.

Exercises for Section 4.3.

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 4.4.

**5. 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 5.1.

Parse Tress.

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 5.2.

Applications of Context-Free Grammars.

Parsers.

The YACC Parser-Generator.

Markup Languages.

XML and Document-Type Definitions.

Exercises for Section 5.3.

Ambiguity in Grammars and Languages.

Ambiguous Grammars.

Removing Ambiguity From Grammars.

Leftmost Derivations as a Way to Express Ambiguity.

Inherent Ambiguity.

Exercises for Section 5.4.

**6. 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 6.1.

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 6.2.

Equivalence of PDA's and CFG's.

From Grammar to Pushdown Automata.

From PDA's to Grammars.

Exercises for Section 6.3.

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 6.4.

**7. 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 7.1.

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 7.2.

Closure Properties of Context-Free Languages.

Substitutions.

Applications of the Substitution Theorem.

Reversal.

Intersection With a Regular Language.

Inverse Homomorphism.

Exercises for Section 7.3.

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 7.4.

**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 8.1.

The Turing Machine.

The Quest to Decide All Mathematical Questions.

Notation for the Turing Machine.

Instantaneous Descriptions for the Turing Machines.

Transition Diagrams for Turing Machines.

The Language of a Turing Machine.

Turing Machines and Halting.

Exercises for Section 8.2.

Programming Techniques for Turing Machines.

Storage in the State.

Multiple Tracks.

Shifting Over.

Subroutines.

Exercises for Section 8.3.

Extensions to the Basic Turing-Machines.

Multiple Turing Machines.

Equivalence of One-Tape and Multitape TM's.

Running Time of the Many-Tapes-to-One Construction.

Nondeterministic Turing Machines.

Exercises for Section 8.4.

Restricted Turing Machine

Turing Machines With Semi-infinite Tapes.

Multistack Machines.

Counter Machines.

The Power of Counter Machines.

Exercises for Section 8.5.

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.

**9. Undecidability.**

A Language That is Not Recursively Enumerable.

Enumerating the Binary Strings.

Codes for Turing Machines.

The Diagonalization Language.

Proof That *Ld* is Not Recursively Enumberable.

Exercises for Section 9.1.

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 9.2.

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 9.3.

Post's Correspondence Problem.

Definition of Post's Correspondence Problem.

The “Modified” PCP.

Completion of the Proof of PCP Undecidability.

Exercises for Section 9.4.

Other Undecidable Problems.

Problems About Programs.

Undecidability of Ambiguity for CFG's.

The Complement of a List Language.

Exercises for Section 9.5.

**10. Intractable Problems.**

The Classes *P* and *NP*.

Problems Solvable in Polynomial Time.

An Example: Kruskal's Algorithm.

Nondeterministic Polynomial Time.

An $\cal NP$ Example: The Traveling Salesman Problem.

Polynomial-Time Reductions.

NP-Complete Problems.

Exercises for Section 10.1

An NP-Complete Problem.

The Satisfiability Problem.

Representing SAT Instances.

NP-Completeness of the SAT Problem.

Exercises for Section 10.2.

A Restricted Satisfiability Problem.

Normal Forms for Boolean Expressions.

Converting Expressions to CNF.

NP-Completeness of CSAT.

NP-Completeness of 3SAT.

Exercises for Section 10.3.

Additional NP-Complete Problems.

Describing NP-Complete Problems.

The Problem of Independent Sets.

The Node-Cover Problems.

The Directed Hamilton-Circuit Problem.

Undirected Hamilton Circuits and the TSP.

Summary of NP-Complete Problems.

Exercises for Section 10.4.

**11. Additional Classes of Problems.**

Complements of Languages in *NP*.

The Class of Languages Co*NP*.

NP-Complete Problems and *NP*.

Exercises for Section 11.1.

Problems Solvable in Polynomial Space.

Polynomial-Space Turing Machines.

Relationship of *PS* and *NPS* to Previously Defined Classes.

Deterministic and Nondeterministic Polynomial Space.

A Problem That Is Complete for *PS*.

PS-Completeness.

Quantified Boolean Formulas.

Evaluating Quantified Boolean Formulas.

PS-Completeness of the QBF Problem.

Exercises for Section 11.3.

Language Classes Based on Randomized Algorithm.

Quicksort: an Example of a Randomized Algorithm.

A Turing-Machine Model Using randomization.

The Language of a Randomized Turing Machine.

The Class *RP*.

Recognizing Languages in *RP*.

The Class *ZPP*.

Relationship Between *RP* and *ZPP*.

Relationships to the Classes *P* and *NP*.

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 11.5.

**Index.**