Computability and Complexity Theory
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.
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accepting computation accepting configuration algorithm belongs to NP binary block Boolean c.e. set cells Church's thesis clean(fc co-NP complexity classes complexity theory computably enumerable computer science contains Corollary decidable set decision problem define definition denote deterministic Turing machine divisor effectively presentable encoding equivalent example executes exists finite alphabet function f Godel number graph G halt without accepting Hamiltonian circuit Homework induction infinite input of length input string input word j-complete language Lemma linear speedup theorem logspace machine that accepts moves multitape multitape Turing machine N simulates natural numbers nondeterministic Turing machine NP-complete NP-hard one-tape oracle Turing machine output partial computable function partial function polynomial hierarchy polynomial-time polynomial-time-bounded positive integer procedure proof of Theorem prove PSPACE query recursion relatively prime result satisfies scanned simulation space-bounded Turing machine subset Suppose time-bounded Turing machine total computable function transducer Turing-machine undecidable variables vertex cover words of length
Page 183 - T. Baker, J. Gill, and R. Solovay. Relativizations of the P = NP question.
Page 78 - A transducer T is a nondeterministic Turing machine with a read-only input tape, a write-only output tape, and accepting states in the usual manner. T computes a value y on an input string x if there is an accepting computation of T on x for which y is the final contents of T's output tape. Such transducers compute partial, multivalued functions.
Page 183 - JL Balcazar and U. Schoning. Bi-immune sets for complexity classes. Mathematical Systems Theory 18:1-10, 1985.