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by Steven Homer and Alan L. Selman

Edition: 01Copyright: 2001

Publisher: Springer-Verlag New York

Published: 2001

International: No

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This volume introduces materials that are the core knowledge in the theory of computation. The book is self-contained, with a preliminary chapter describing key mathematical concepts and notations and subsequent chapters moving from the qualitative aspects of classical computability theory to the quantitative aspects of complexity theory. Dedicated chapters on undecidability, NP-completeness, and relative computability round off the work, which focuses on the limitations of computability and the distinctions between feasible and intractable.

**Topics and features:**

- Concise, focused materials cover the most fundamental concepts and results in the field of modern complexity theory, including the theory of NP-completeness, NP-hardness, the polynomial hierarchy, and complete problems for other complexity classes
- Contains information that otherwise exists only in research literature and presents it in a unified, simplified manner; for example, about complements of complexity classes, search problems, and intermediate problems in NP
- Provides key mathematical background information, including sections on logic and number theory and algebra
- Supported by numerous exercises and supplementary problems for reinforcement and self-study purposes
- With its accessibility and well-devised organization, this text/reference is an excellent resource and guide for those looking to develop a solid grounding in the theory of computing. Beginning graduates, advanced undergraduates, and professionals involved in theoretical computer science, complexity theory, and computability will find the book an essential and practical learning tool.

PRELIMINARIES

Words and Languages

K-adic Representation

Partial Functions

Graphs

Propositional Logic

Boolean Functions

Cardinality

Ordered Sets

Elementary Algebra

Rings and Fields

Groups

Number Theory

INTRODUCTION TO COMPUTABILITY

Turing Machines

Turing-Machine Concepts

Variations of Turing Machines

Multitape Turing Machines

Nondeterministic Turing Machines

Church's Thesis

RAMs

Turing Machines for RAMS

UNDECIDABILITY

Decision Problems

Undecidable Problems

Pairing Functions

Computably Enumerable Sets

Halting Problem, Reductions, and Complete Sets

Complete Problems

S-m-n Theorem

Recursion Theorem

Rice's Theorem

Turing Reductions and Oracle Turing Machines

Recursion Theorem, Continued

References

Additional Homework Problems

INTRODUCTION TO COMPLEXITY THEORY

Complexity Classes and Complexity Measures

Computing Functions

Prerequisites

BASIC RESULTS OF COMPLEXITY THEORY

Linear Compression and Speedup

Constructible Functions

Simultaneous Simulation

Tape Reduction

Inclusion Relationships

Relations between Standard Classes

Separation Results

Translation Techniques and Padding

Tally Languages

Relations between the Standard Classes--Continued

Complements of Complexity Classes: The Immerman-Szelepcsenyi Theorem

Additional Homework Problems

NONDETERMINISM AND NP-COMPLETENESS

Characterizing NP

The Class P

Enumerations

NP-Completeness

The Cook-Levin Theorem

More NP-Complete Problems

The Diagonal Set Is NP-Complete

Some Natural NP-Complete Problems

Additional Homework Problems

RELATIVE COMPUTABILITY

NP-Hardness

Search Problems

The Structure of NP

Composite Number and Graph Isomorphism

Reflection

The Polynomial Hierarchy

Complete Problems for Other Complexity Classes

PSPACE

Exponential Time

Polynomial Time and Logarithmic Space

A Note on Provably Intractable Problems

Additional Homework Problems

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Summary

This volume introduces materials that are the core knowledge in the theory of computation. The book is self-contained, with a preliminary chapter describing key mathematical concepts and notations and subsequent chapters moving from the qualitative aspects of classical computability theory to the quantitative aspects of complexity theory. Dedicated chapters on undecidability, NP-completeness, and relative computability round off the work, which focuses on the limitations of computability and the distinctions between feasible and intractable.

**Topics and features:**

- Concise, focused materials cover the most fundamental concepts and results in the field of modern complexity theory, including the theory of NP-completeness, NP-hardness, the polynomial hierarchy, and complete problems for other complexity classes
- Contains information that otherwise exists only in research literature and presents it in a unified, simplified manner; for example, about complements of complexity classes, search problems, and intermediate problems in NP
- Provides key mathematical background information, including sections on logic and number theory and algebra
- Supported by numerous exercises and supplementary problems for reinforcement and self-study purposes
- With its accessibility and well-devised organization, this text/reference is an excellent resource and guide for those looking to develop a solid grounding in the theory of computing. Beginning graduates, advanced undergraduates, and professionals involved in theoretical computer science, complexity theory, and computability will find the book an essential and practical learning tool.

Table of Contents

PRELIMINARIES

Words and Languages

K-adic Representation

Partial Functions

Graphs

Propositional Logic

Boolean Functions

Cardinality

Ordered Sets

Elementary Algebra

Rings and Fields

Groups

Number Theory

INTRODUCTION TO COMPUTABILITY

Turing Machines

Turing-Machine Concepts

Variations of Turing Machines

Multitape Turing Machines

Nondeterministic Turing Machines

Church's Thesis

RAMs

Turing Machines for RAMS

UNDECIDABILITY

Decision Problems

Undecidable Problems

Pairing Functions

Computably Enumerable Sets

Halting Problem, Reductions, and Complete Sets

Complete Problems

S-m-n Theorem

Recursion Theorem

Rice's Theorem

Turing Reductions and Oracle Turing Machines

Recursion Theorem, Continued

References

Additional Homework Problems

INTRODUCTION TO COMPLEXITY THEORY

Complexity Classes and Complexity Measures

Computing Functions

Prerequisites

BASIC RESULTS OF COMPLEXITY THEORY

Linear Compression and Speedup

Constructible Functions

Simultaneous Simulation

Tape Reduction

Inclusion Relationships

Relations between Standard Classes

Separation Results

Translation Techniques and Padding

Tally Languages

Relations between the Standard Classes--Continued

Complements of Complexity Classes: The Immerman-Szelepcsenyi Theorem

Additional Homework Problems

NONDETERMINISM AND NP-COMPLETENESS

Characterizing NP

The Class P

Enumerations

NP-Completeness

The Cook-Levin Theorem

More NP-Complete Problems

The Diagonal Set Is NP-Complete

Some Natural NP-Complete Problems

Additional Homework Problems

RELATIVE COMPUTABILITY

NP-Hardness

Search Problems

The Structure of NP

Composite Number and Graph Isomorphism

Reflection

The Polynomial Hierarchy

Complete Problems for Other Complexity Classes

PSPACE

Exponential Time

Polynomial Time and Logarithmic Space

A Note on Provably Intractable Problems

Additional Homework Problems

Publisher Info

Publisher: Springer-Verlag New York

Published: 2001

International: No

Published: 2001

International: No

This volume introduces materials that are the core knowledge in the theory of computation. The book is self-contained, with a preliminary chapter describing key mathematical concepts and notations and subsequent chapters moving from the qualitative aspects of classical computability theory to the quantitative aspects of complexity theory. Dedicated chapters on undecidability, NP-completeness, and relative computability round off the work, which focuses on the limitations of computability and the distinctions between feasible and intractable.

**Topics and features:**

- Concise, focused materials cover the most fundamental concepts and results in the field of modern complexity theory, including the theory of NP-completeness, NP-hardness, the polynomial hierarchy, and complete problems for other complexity classes
- Contains information that otherwise exists only in research literature and presents it in a unified, simplified manner; for example, about complements of complexity classes, search problems, and intermediate problems in NP
- Provides key mathematical background information, including sections on logic and number theory and algebra
- Supported by numerous exercises and supplementary problems for reinforcement and self-study purposes
- With its accessibility and well-devised organization, this text/reference is an excellent resource and guide for those looking to develop a solid grounding in the theory of computing. Beginning graduates, advanced undergraduates, and professionals involved in theoretical computer science, complexity theory, and computability will find the book an essential and practical learning tool.

PRELIMINARIES

Words and Languages

K-adic Representation

Partial Functions

Graphs

Propositional Logic

Boolean Functions

Cardinality

Ordered Sets

Elementary Algebra

Rings and Fields

Groups

Number Theory

INTRODUCTION TO COMPUTABILITY

Turing Machines

Turing-Machine Concepts

Variations of Turing Machines

Multitape Turing Machines

Nondeterministic Turing Machines

Church's Thesis

RAMs

Turing Machines for RAMS

UNDECIDABILITY

Decision Problems

Undecidable Problems

Pairing Functions

Computably Enumerable Sets

Halting Problem, Reductions, and Complete Sets

Complete Problems

S-m-n Theorem

Recursion Theorem

Rice's Theorem

Turing Reductions and Oracle Turing Machines

Recursion Theorem, Continued

References

Additional Homework Problems

INTRODUCTION TO COMPLEXITY THEORY

Complexity Classes and Complexity Measures

Computing Functions

Prerequisites

BASIC RESULTS OF COMPLEXITY THEORY

Linear Compression and Speedup

Constructible Functions

Simultaneous Simulation

Tape Reduction

Inclusion Relationships

Relations between Standard Classes

Separation Results

Translation Techniques and Padding

Tally Languages

Relations between the Standard Classes--Continued

Complements of Complexity Classes: The Immerman-Szelepcsenyi Theorem

Additional Homework Problems

NONDETERMINISM AND NP-COMPLETENESS

Characterizing NP

The Class P

Enumerations

NP-Completeness

The Cook-Levin Theorem

More NP-Complete Problems

The Diagonal Set Is NP-Complete

Some Natural NP-Complete Problems

Additional Homework Problems

RELATIVE COMPUTABILITY

NP-Hardness

Search Problems

The Structure of NP

Composite Number and Graph Isomorphism

Reflection

The Polynomial Hierarchy

Complete Problems for Other Complexity Classes

PSPACE

Exponential Time

Polynomial Time and Logarithmic Space

A Note on Provably Intractable Problems

Additional Homework Problems