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Description :
DESCRIPTION
Mathematics and Computation provides a broad, conceptual overview of computational complexity theory—the mathematical study of efficient computation. With important practical applications to computer science and industry, computational complexity theory has evolved into a highly interdisciplinary field, with strong links to most mathematical areas and to a growing number of scientific endeavors.
Avi Wigderson takes a sweeping survey of complexity theory, emphasizing the field’s insights and challenges. He explains the ideas and motivations leading to key models, notions, and results. In particular, he looks at algorithms and complexity, computations and proofs, randomness and interaction, quantum and arithmetic computation, and cryptography and learning, all as parts of a cohesive whole with numerous cross-influences. Wigderson illustrates the immense breadth of the field, its beauty and richness, and its diverse and growing interactions with other areas of mathematics. He ends with a comprehensive look at the theory of computation, its methodology and aspirations, and the unique and fundamental ways in which it has shaped and will further shape science, technology, and society. For further reading, an extensive bibliography is provided for all topics covered.
Mathematics and Computation is useful for undergraduate and graduate students in mathematics, computer science, and related fields, as well as researchers and teachers in these fields. Many parts require little background, and serve as an invitation to newcomers seeking an introduction to the theory of computation.
• Comprehensive coverage of computational complexity theory, and beyond
• High-level, intuitive exposition, which brings conceptual clarity to this central and dynamic scientific discipline
• Historical accounts of the evolution and motivations of central concepts and models
• A broad view of the theory of computation’s influence on science, technology, and society
Avi Wigderson is the Herbert H. Maass Professor in the School of Mathematics at the Institute for Advanced Study in Princeton, New Jersey.
Content :
Acknowledgments xiii
1 Introduction 1 (13)
2 Prelude: Computation, undecidability, and 14 (5)
3 Computational complexity 101: The basics, 19 (28)
P, and NP
4 Problems and classes inside (and around) 47 (14)
NP
5 Lower bounds, Boolean circuits, and 61 (12)
6 Proof complexity 73 (16)
7 Randomness in computation 89 (16)
8 Abstract pseudo-randomness 105 (19)
9 Weak random sources and randomness 124 (8)
extractors
10 Randomness and interaction in proofs 132 (14)
11 Quantum computing 146 (14)
12 Arithmetic complexity 160 (14)
13 Interlude: Concrete interactions between 174 (24)
math and computational complexity
14 Space complexity: Modeling limited memory 198 (9)
15 Communication complexity: Modeling 207 (23)
information bottlenecks
16 On-line algorithms: Coping with an 230 (8)
unknown future
17 Computational learning theory, AI, and 238 (22)
beyond
18 Cryptography: Modeling secrets and lies, 260 (21)
knowledge and trust
19 Distributed computing: Coping with 281 (18)
asynchrony
20 Epilogue: A broader perspective of ToC 299 (50)
References 349
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