Computability and Complexity Theory

Steven Homer and Alan L. Selman

Springer Verlag New York, 2001

ISBN 0-387-95055-9

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

1. PRELIMINARIES
   • Words and Languages
   • K-adic Representation
   • Partial Functions
   • Graphs
   • Propositional Logic
   • Cardinality
   • Elementary Algebra
2. INTRODUCTION TO COMPUTABILITY
   • Turing Machines
   • Turing-Machine Concepts
   • Variations of Turing Machines
   • Church’s Thesis
   • RAMs
3. UNDECIDABILITY
   • Decision Problems
   • Undecidable Problems
   • Pairing Functions
   • Computably Enumerable Sets
   • Halting Problem, Reductions, and Complete Sets
   • S-m-n Theorem
   • Recursion Theorem
   • Rice’s Theorem
   • Turing Reductions and Oracle Turing Machines
   • Recursion Theorem, Continued
   • References
   • Additional Homework Problems
4. INTRODUCTION TO COMPLEXITY THEORY
   • Complexity Classes and Complexity Measures
   • Prerequisites
5. BASIC RESULTS OF COMPLEXITY THEORY
   • Linear Compression and Speedup
   • Constructible Functions
   • Tape Reduction
   • Inclusion Relationships
     • Relations between the Standard Classes
   • Separation Results
   • Translation Techniques and Padding
   • Relations between the Standard Classes--Continued
     • Complements of Complexity Classes: The Immerman-Szelepcsenyi Theorem
   • Additional Homework Problems
6. NONDETERMINISM AND NP-COMPLETENESS
   • Characterizing NP
   • The Class P
   • Enumerations
   • NP-Completeness
   • The Cook-Levin Theorem
   • More NP-Complete Problems
   • Additional Homework Problems
7. RELATIVE COMPUTABILITY
   • NP-Hardness
   • Search Problems
   • The Structure of NP
     • Composite Number and Graph Isomorphism
   • The Polynomial Hierarchy
   • Complete Problems for Other Complexity Classes
   • Additional Homework Problems

Front matter with Preface and Table of Contents

Important Links:

• List of Corrections of minor errors
• Corrections of Technical Problems and Discussion of Pedagogical Issues
• Publisher's Web page
• Amazon.com
• Book Review