Mathematics and Computation Avi Wigderson March 27, 2018 1 Avi Wigderson Mathematics and Computation Draft: March 27, 2018 Dedicated to the memory of my father, Pinchas Wigderson (1921{1988), who loved people, loved puzzles, and inspired me. Ashkhabad, Turkmenistan, 1943 2 Avi Wigderson Mathematics and Computation Draft: March 27, 2018 Acknowledgments In this book I tried to present some of the knowledge and understanding I acquired in my four decades in the field. The main source of this knowledge was the Theory of Computation commu- nity, which has been my academic and social home throughout this period. The members of this wonderful community, especially my teachers, students, postdocs and collaborators, but also the speakers in numerous talks I attended, have been the source of this knowledge and understanding often far more than the books and journals I read. Ours is a generous and interactive community, whose members are happy to share their own knowledge and understanding with others, and are trained by the culture of the field to do so well. These interactions made (and still makes) learning a greatly joyful experience for me! More directly, the content and presentation in this book benefited directly by many. These are friends who carefully read earlier drafts, responded with valuable constructive comments at all levels, which made the book much better. For this I am grateful to Scott Aaronson, Dorit Aharonov, Noga Alon, Sanjeev Arora, Boaz Barak, Zeb Brady, Mark Braverman, Bernard Chazelle, Tom Church, Geoffroy Couteau, Andy Drucker, Ron Fagin, Yuval Filmus, Michael Forbes, Ankit Garg, Sumegha Garg, Oded Goldreich, Renan Gross, Nadia Heninger, Gil Kalai, Vickie Kearn, Pravesh Kothari, James Lee, Alex Lubotzky, Assaf Naor, Ryan O'Donnell, Toni Pitassi, Tim Roughgarden, Sasha Razborov, Mike Saks, Peter Sarnak, Susannah Shoemaker, Amir Shpilka, Alistair Sinclair, Bill Steiger, Arpita Tripathi, Salil Vadhan, Les Valiant, Thomas Vidick, BenLee Volk, Edna Wigderson, Yuval Wigderson, Ronald de Wolf, Amir Yehudayoff, Rich Zemel and David Zuckerman. Special additional thanks are due to Edna and Yuval, who not only read every word (several times), but also helped me overcome many technical problems with the manuscript. Some chapters in this book are revisions and extensions of material taken from my ICM 2006 survey [Wig06], which in turn used parts of a joint survey with Goldreich in this volume [GBGL10]. Last but not least, I am grateful to Tom and Roselyne Nelsen for letting me use their beautiful home in Sun Valley, Idaho - much of this book was written in that serene environment over the past few summers. 3 Avi Wigderson Mathematics and Computation Draft: March 27, 2018 Contents 1 Introduction 9 1.1 On the interactions of math and computation...................... 10 1.2 Computational Complexity Theory............................ 13 1.3 The nature, purpose, style and audience of the book.................. 14 1.4 Organization of the book................................. 15 1.5 Asymptotic Notation.................................... 19 2 Prelude: computation, undecidability and the limits of mathematical knowledge 20 3 Computational complexity 101: the basics 24 3.1 Motivating examples.................................... 24 3.2 Efficient computation and the class P .......................... 26 3.3 Efficient verification and the class NP .......................... 30 3.4 The P versus NP question, its meaning and importance................ 34 3.5 The class coNP, the NP versus coNP question, and efficient characterization... 37 3.6 Reductions: a partial order of computational difficulty................. 40 3.7 Completeness: problems capturing complexity classes.................. 41 3.8 NP-completeness...................................... 42 3.9 Some NP-complete problems............................... 43 3.10 The nature and impact of NP-completeness....................... 45 4 Problems and classes inside (and \around") NP 48 4.1 Other types of computational problems and associated complexity classes...... 48 4.2 Between P and NP .................................... 50 4.3 Constraint Satisfaction Problems (CSPs)......................... 53 4.4 Average-case complexity.................................. 55 4.5 One-way functions, trap-door functions and cryptography............... 56 5 Lower bounds, Boolean Circuits, and attacks on P vs. NP 61 5.1 Diagonalization and relativization............................. 61 5.2 Boolean circuits....................................... 62 5.2.1 Basic results and questions............................ 64 5.2.2 Boolean formulae.................................. 65 5.2.3 Monotone circuits and formulae.......................... 67 5.2.4 Natural Proofs, or, Why is it hard to prove circuit lower bounds?....... 69 6 Proof complexity 71 6.1 The pigeonhole principle|a motivating example.................... 73 6.2 Propositional proof systems and NP vs. coNP ..................... 74 6.3 Concrete proof systems.................................. 76 6.3.1 Algebraic proof systems.............................. 76 6.3.2 Geometric proof systems.............................. 78 6.3.3 Logical proof systems............................... 81 6.4 Proof complexity vs. circuit complexity......................... 83 4 Avi Wigderson Mathematics and Computation Draft: March 27, 2018 7 Randomness in computation 86 7.1 The power of randomness in algorithms......................... 86 7.2 The weakness of randomness in algorithms........................ 89 7.3 Computational pseudo-randomness and pseudo-random generators.......... 92 8 Abstract pseudo-randomness 100 8.1 Motivating examples.................................... 100 8.2 General pseudo-random properties, and finding hay in haystacks........... 101 8.3 The Riemann Hypothesis................................. 103 8.4 P vs. NP .......................................... 104 8.5 Computational pseudo-randomness and de-randomization............... 106 8.6 Quasi-random graphs.................................... 108 8.7 Expanders.......................................... 109 8.8 Structure vs. Pseudo-randomness............................. 113 9 Weak random sources and randomness extractors 117 9.1 Min-entropy and randomness extractors......................... 118 9.2 Explicit constructions of extractors............................ 120 10 Randomness in proofs 123 10.1 Interactive proof systems................................. 124 10.2 Zero-knowledge proof systems............................... 127 10.3 Probabilistically checkable proofs (and hardness of approximation).......... 129 11 Quantum Computing 132 11.1 Building a quantum computer............................... 135 11.2 Quantum proofs and quantum Hamiltonian complexity and dynamics........ 136 11.3 Quantum interactive proofs and testing Quantum Mechanics............. 140 11.4 Quantum randomness: certification and expansion................... 141 12 Arithmetic complexity 144 12.1 Motivation: univariate polynomials............................ 144 12.2 Basic definitions, questions and results.......................... 145 12.3 The complexity of basic polynomials........................... 146 12.4 Reductions and completeness, permanents and determinants.............. 151 12.5 Restricted models...................................... 153 13 Interlude: Concrete interactions between Math and Computational Complexity156 13.1 Number Theory....................................... 156 13.2 Combinatorial geometry.................................. 158 13.3 Operator theory...................................... 159 13.4 Metric Geometry...................................... 161 13.5 Group Theory........................................ 162 13.6 Statistical Physics..................................... 164 13.7 Analysis and Probability.................................. 166 13.8 Lattice Theory....................................... 169 13.9 Invariant Theory...................................... 171 5 Avi Wigderson Mathematics and Computation Draft: March 27, 2018 13.9.1 Geometric Complexity Theory (GCT)...................... 173 13.9.2 Simultaneous Conjugation............................. 174 13.9.3 Left-Right action.................................. 175 14 Space complexity: modeling limited memory 177 14.1 Basic space complexity................................... 177 14.2 Streaming and Sketching.................................. 180 14.3 Finite automata and counting............................... 181 15 Communication complexity: modeling information bottlenecks 185 15.1 Basic definitions and results................................ 185 15.2 Applications......................................... 188 15.2.1 VLSI time-area trade-offs............................. 188 15.2.2 Time-space trade-offs............................... 189 15.2.3 Formula lower bounds............................... 190 15.2.4 Proof complexity.................................. 193 15.2.5 Extension complexity............................... 194 15.2.6 Pseudo-randomness................................ 197 15.3 Interactive information theory and coding theory.................... 198 15.3.1 Information complexity, protocol compression and direct-sum......... 199 15.3.2 Error-correction of interactive communication.................. 202 16 On-line algorithms: coping with an unknown future 206 16.1 Paging, Caching and the k-server problem........................ 208 16.2 Expert advice, portfolio management, repeated games and the multiplicative weights algorithm.........................................