Computational complexity theory
著者
書誌事項
Computational complexity theory
（IAS/Park City mathematics series / [Dan Freed, series editor], v. 10）
American Mathematical Society, c2004
- : hardcover
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注記
"Volume contains the lecture notes from the Graduate Summer School program on Computational Complexity Theory held in Princeton in the summer of 2000" -- T.p. verso
Includes bibliographical references
内容説明・目次
内容説明
Computational complexity theory is a major research area in mathematics and computer science, the goal of which is to set the formal mathematical foundations for efficient computation. There has been significant development in the nature and scope of the field in the last thirty years. It has evolved to encompass a broad variety of computational tasks by a diverse set of computational models, such as randomized, interactive, distributed, and parallel computations. These models can include many computers, which may behave cooperatively or adversarially. Each summer the IAS/Park City Mathematics Institute Graduate Summer School gathers some of the best researchers and educators in the field to present diverse sets of lectures. This volume presents three weeks of lectures given at the Summer School on Computational Complexity Theory. Topics are structured as follows: Week One: Complexity Theory: From Godel to Feynman. This section of the book gives a general introduction to the field, with the main set of lectures describing basic models, techniques, results, and open problems. Week Two: Lower Bounds on Concrete Models. Topics discussed in this section include communication and circuit complexity, arithmetic and algebraic complexity, and proof complexity. Week Three: Randomness in Computation. Lectures are devoted to different notions of pseudorandomness, interactive proof systems and zero knowledge, and probabilistically checkable proofs (PCPs). The volume is recommended for independent study and is suitable for graduate students and researchers interested in computational complexity. Information for our distributors: Members of the Mathematical Association of America (MAA) and the National Council of Teachers of Mathematics (NCTM) receive a 20% discount from list price.
目次
Week One: Complexity theory: From Godel to Feynman Complexity theory: From Godel to Feynman History and basic concepts Resources, reductions and P vs. NP Probabilistic and quantum computation Complexity classes Space complexity and circuit complexity Oracles and the polynomial time hierarchy Circuit lower bounds "Natural" proofs of lower bounds Bibliography Average case complexity Average case complexity Bibliography Exploring complexity through reductions Introduction PCP theorem and hardness of computing approximate solutions Which problems have strongly exponential complexity? Toda's theorem: $PH\subseteq P^{\ No. P}$ Bibliography Quantum computation Introduction Bipartite quantum systems Quantum circuits and Shor's factoring algorithm Bibliography Lower bounds: Circuit and communication complexity Communication complexity Lower bounds for probabilistic communication complexity Communication complexity and circuit depth Lower bound for directed $st$-connectivity Lower bound for $FORK$ (continued) Bibliography Proof complexity An introduction to proof complexity Lower bounds in proof complexity Automatizability and interpolation The restriction method Other research and open problems Bibliography Randomness in computation Pseudorandomness Preface Computational indistinguishability Pseudorandom generators Pseudorandom functions and concluding remarks Appendix Bibliography Pseudorandomness-Part II Introduction Deterministic simulation of randomized algorithms The Nisan-Wigderson generator Analysis of the Nisan-Wigderson generator Randomness extractors Bibliography Probabilistic proof systems-Part I Interactive proofs Zero-knowledge proofs Suggestions for further reading Bibliography Probabilistically checkable proofs Introduction to PCPs NP-hardness of PCS A couple of digressions Proof composition and the PCP theorem Bibliography.
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