## Theory of computational complexityA complete treatment of fundamentals and recent advances in complexity theory Complexity theory studies the inherent difficulties of solving algorithmic problems by digital computers. This comprehensive work discusses the major topics in complexity theory, including fundamental topics as well as recent breakthroughs not previously available in book form. Theory of Computational Complexity offers a thorough presentation of the fundamentals of complexity theory, including NP-completeness theory, the polynomial-time hierarchy, relativization, and the application to cryptography. It also examines the theory of nonuniform computational complexity, including the computational models of decision trees and Boolean circuits, and the notion of polynomial-time isomorphism. The theory of probabilistic complexity, which studies complexity issues related to randomized computation as well as interactive proof systems and probabilistically checkable proofs, is also covered. Extraordinary in both its breadth and depth, this volume: * Provides complete proofs of recent breakthroughs in complexity theory * Presents results in well-defined form with complete proofs and numerous exercises * Includes scores of graphs and figures to clarify difficult material An invaluable resource for researchers as well as an important guide for graduate and advanced undergraduate students, Theory of Computational Complexity is destined to become the standard reference in the field. |

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### Contents

JVFCompleteness | 43 |

The PolynomialTime Hierarchy and Polynomial Space | 77 |

Structure of NP | 113 |

Copyright | |

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### Common terms and phrases

approximation Arthur assume binary Boolean assignment Boolean circuit Boolean formula Boolean function branching program circuit class of sets clauses clique complexity classes computation path computation tree configuration consider constant construct contains Cook's Theorem Corollary decision tree define denote deterministic edges elusive encoding error probability Exercise exists a polynomial-time fanin finite formula F function f gate graph G graph property halts hence induction input integer interactive proof system isomorphic JVP-complete label Lemma linear Merlin multilinear node nondeterministic Note notion O(logn one-way functions oracle machine oracle NTM output P-complete P/poly paddable permutation polynomial function polynomial-time computable polynomial-time hierarchy predicate probabilistic problem Prove PSPACE PSPACE-complete quantifier random bits randomized algorithms recursive rejects relativized satisfies simplicial complex simulation space stage strings of length subset symbol tape head total number truth assignment Turing machine Turing reducibility variables verifier vertices