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Computability and Complexity Theory

Computability and Complexity Theory - 01 edition

ISBN13: 978-0387950556

Cover of Computability and Complexity Theory 01 (ISBN 978-0387950556)
ISBN13: 978-0387950556
ISBN10: 0387950559
Cover type:
Edition/Copyright: 01
Publisher: Springer-Verlag New York
Published: 2001
International: No

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Computability and Complexity Theory - 01 edition

ISBN13: 978-0387950556

Steven Homer and Alan L. Selman

ISBN13: 978-0387950556
ISBN10: 0387950559
Cover type:
Edition/Copyright: 01
Publisher: Springer-Verlag New York

Published: 2001
International: No
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

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