Computability and Complexity Theory

This book surveys theoretical computer science, presenting fundamental concepts and results. Updated and revised, the new edition includes two new chapters on nonuniform complexity, circuit complexity and parallel complexity, and randomized complexity.

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Bibliographic Details
Main Authors Homer, Steven, Selman, Alan L
Format eBook Book
LanguageEnglish
Published New York Springer Nature 2011
Springer
Springer US
Edition2
SeriesTexts in Computer Science
Subjects
Algorithm Analysis and Problem Complexity
Computable functions
Computational complexity
Computer Science
Data processing Computer science
Theory of Computation
Online AccessGet full text
ISBN146140682X
9781461406822
1461406811
9781461406815
9781489989710
1489989714
ISSN1868-0941
1868-095X
DOI10.1007/978-1-4614-0682-2

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Table of Contents:
  • Intro -- Computability and Complexity Theory -- Preface to the First Edition 2001 -- Preface to the Second Edition 2011 -- Contents -- Chapter 1 Preliminaries -- Chapter 2 Introduction to Computability -- Chapter 3 Undecidability -- Chapter 4 Introduction to Complexity Theory -- Chapter 5 Basic Results of Complexity Theory -- Chapter 6 Nondeterminism and NP-Completeness -- Chapter 7 Relative Computability -- Chapter 8 Nonuniform Complexity -- Chapter 9 Parallelism -- Chapter 10 Probabilistic Complexity Classes -- Chapter 11 Introduction to Counting Classes -- Chapter 12 Interactive Proof Systems -- References -- Author Index -- Subject Index
  • 12.4 IP Is Included in PSPACE -- 12.5 PSPACE Is Included in IP -- 12.5.1 The Language ESAT -- 12.5.2 True Quantified Boolean Formulas -- 12.5.3 The Proof -- 12.6 Additional Homework Problems -- References -- Author Index -- Subject Index
  • Intro -- Computability and Complexity Theory -- Preface to the First Edition 2001 -- Preface to the Second Edition 2011 -- Contents -- Chapter 1 Preliminaries -- 1.1 Words and Languages -- 1.2 K-adic Representation -- 1.3 Partial Functions -- 1.4 Graphs -- 1.5 Propositional Logic -- 1.5.1 Boolean Functions -- 1.6 Cardinality -- 1.6.1 Ordered Sets -- 1.7 Elementary Algebra -- 1.7.1 Rings and Fields -- 1.7.2 Groups -- 1.7.2.1 Cosets -- 1.7.3 Number Theory -- 1.7.3.1 Polynomials -- Chapter 2 Introduction to Computability -- 2.1 Turing Machines -- 2.2 Turing Machine Concepts -- 2.3 Variations of Turing Machines -- 2.3.1 Multitape Turing Machines -- 2.3.2 Nondeterministic Turing Machines -- 2.4 Church's Thesis -- 2.5 RAMs -- 2.5.1 Turing Machines for RAMS -- Chapter 3 Undecidability -- 3.1 Decision Problems -- 3.2 Undecidable Problems -- 3.3 Pairing Functions -- 3.4 Computably Enumerable Sets -- 3.5 Halting Problem, Reductions, and Complete Sets -- 3.5.1 Complete Problems -- 3.5.1.1 Summary -- 3.6 S-m-n Theorem -- 3.7 Recursion Theorem -- 3.8 Rice's Theorem -- 3.9 Turing Reductions and Oracle Turing Machines -- 3.10 Recursion Theorem: Continued -- 3.11 References -- 3.12 Additional Homework Problems -- Chapter 4 Introduction to Complexity Theory -- 4.1 Complexity Classes and Complexity Measures -- 4.1.1 Computing Functions -- 4.2 Prerequisites -- Chapter 5 Basic Results of Complexity Theory -- 5.1 Linear Compression and Speedup -- 5.2 Constructible Functions -- 5.2.1 Simultaneous Simulation -- 5.3 Tape Reduction -- 5.4 Inclusion Relationships -- 5.4.1 Relations Between the Standard Classes -- 5.5 Separation Results -- 5.6 Translation Techniques and Padding -- 5.6.1 Tally Languages -- 5.7 Relations Between the Standard Classes: Continued -- 5.7.1 Complements of Complexity Classes: The Immerman-Szelepcsényi Theorem -- 5.8 Additional Homework Problems
  • Chapter 6 Nondeterminism and NP-Completeness -- 6.1 Characterizing NP -- 6.2 The Class P -- 6.3 Enumerations -- 6.4 NP-Completeness -- 6.5 The Cook-Levin Theorem -- 6.6 More NP-Complete Problems -- 6.6.1 The Diagonal Set Is NP-Complete -- 6.6.2 Some Natural NP-Complete Problems -- 6.7 Additional Homework Problems -- Chapter 7 Relative Computability -- 7.1 NP-Hardness -- 7.2 Search Problems -- 7.3 The Structure of NP -- 7.3.1 Composite Number and Graph Isomorphism -- 7.3.2 Reflection -- 7.4 The Polynomial Hierarchy -- 7.5 Complete Problems for Other Complexity Classes -- 7.5.1 PSPACE -- 7.5.1.1 Oracles for the P =? NP Question -- 7.5.2 Exponential Time -- 7.5.3 Polynomial Time and Logarithmic Space -- 7.5.4 A Note on Provably Intractable Problems -- 7.6 Additional Homework Problems -- Chapter 8 Nonuniform Complexity -- 8.1 Polynomial Size Families of Circuits -- 8.1.1 An Encoding of Circuits -- 8.1.2 Advice Classes -- 8.2 The Low and High Hierarchies -- Chapter 9 Parallelism -- 9.1 Alternating Turing Machines -- 9.2 Uniform Families of Circuits -- 9.3 Highly Parallelizable Problems -- 9.4 Uniformity Conditions -- 9.5 Alternating Turing Machines and Uniform Families of Circuits -- Chapter 10 Probabilistic Complexity Classes -- 10.1 The Class PP -- 10.2 The Class RP -- 10.2.1 The Class ZPP -- 10.3 The Class BPP -- 10.4 Randomly Chosen Hash Functions -- 10.4.1 Operators -- 10.5 The Graph Isomorphism Problem -- 10.6 Additional Homework Problems -- Chapter 11 Introduction to Counting Classes -- 11.1 Unique Satisfiability -- 11.2 Toda's Theorem -- 11.2.1 Results on BPP and P -- 11.2.2 The First Part of Toda's Theorem -- 11.2.3 The Second Part of Toda's Theorem -- 11.3 Additional Homework Problems -- Chapter 12 Interactive Proof Systems -- 12.1 The Formal Model -- 12.2 The Graph Non-Isomorphism Problem -- 12.3 Arthur-Merlin Games