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- Available on iOS & Android
About this book
From the winner of the Turing Award and the Abel Prize, an introduction to computational complexity theory, its connections and interactions with mathematics, and its central role in the natural and social sciences, technology, and philosophy
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
- Extensive bibliography
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Information
Table of contents
- Cover
- Contents
- Acknowledgments
- 1. Introduction
- 2. Prelude: Computation, undecidability, and limits to mathematical knowledge
- 3. Computational complexity 101: The basics, P, and NP
- 4. Problems and classes inside (and around) NP
- 5. Lower bounds, Boolean circuits, and attacks on P vs. NP
- 6. Proof complexity
- 7. Randomness in computation
- 8. Abstract pseudo-randomness
- 9. Weak random sources and randomness extractors
- 10. Randomness and interaction in proofs
- 11. Quantum computing
- 12. Arithmetic complexity
- 13. Interlude: Concrete interactions between math and computational complexity
- 14. Space complexity: Modeling limited memory
- 15. Communication complexity: Modeling information bottlenecks
- 16. On-line algorithms: Coping with an unknown future
- 17. Computational learning theory, AI, and beyond
- 18. Cryptography: Modeling secrets and lies, knowledge and trust
- 19. Distributed computing: Coping with asynchrony
- 20. Epilogue: A broader perspective of ToC
- References