ISBN: 3540414924
TITLE: Randomness and Completeness in Computational Complexity
AUTHOR: Melkebeek, Dieter van
TOC:

1. Introduction 1
1.1 Issues in Computational Complexity 1
1.1.1 The Power of Randomness 1
1.1.2 The Power of Guessing 5
1.1.3 The Power of Memory 6
1.2 Contributions in This Dissertation 8
1.2.1 Simulations 8
1.2.2 Separations 9
2. Preliminaries 13
2.1 Computational Problems 13
2.2 Models of Computation and Complexity Classes 18
2.2.1 Turing Machines 19
2.2.2 Uniform Families of Boolean Circuits 21
2.2.3 Nondeterministic Turing Machines 24
2.2.4 Alternating Turing Machines 27
2.3 Randomness 31
2.3.1 Randomized Computations 32
2.3.2 Randomized Proof Checking 35
2.3.3 Pseudo-Random Generators 37
2.4 Reductions and Completeness 42
2.4.1 Relativization 42
2.4.2 Reductions 44
2.4.3 Complete Problems 46
2.5 Resource-Bounded Measure 47
2.5.1 Motivation 47
2.5.2 Martingales 48
2.5.3 Properties 50
3. Derandomizing Arthur-Merlin Games 53
3.1 Introduction 53
3.2 Notation 56
3.3 Derandomizing Arthur-Merlin Games 57
3.3.1 Proof of Lemma 3.3.1 60
3.3.2 Proof of Theorem 3.3.1 61
3.3.3 Proof of Theorem 3.3.2 63
3.4 A General Framework for Derandomization 64
3.5 More Applications 66
3.5.1 Valiant-Vazirani 66
3.5.2 Learning Circuits 69
3.5.3 Rigid Matrices 70
3.5.4 Universal Traversal Sequences 72
3.6 Conclusion and Open Questions 76
4. Sparseness of Complete Languages77
4.1 Introduction 77
4.1.1 The Sparse Hard Language Problem for NP 77
4.1.2 The Sparse Hard Language Problem for P 78
4.1.3 Overview of this Chapter 79
4.2 Deterministic Reductions 81
4.2.1 Previous Work 81
4.2.2 Main Result 85
4.2.3 Generic Theorem for P 92
4.2.4 Extension to Classes Other Than P 95
4.3 Randomized Reductions 98
4.3.1 Previous Work 98
4.3.2 Main Result 99
4.3.3 Randomized Generic Theorem for P 106
4.3.4 Extension to Classes Other Than P 109
4.4 Conclusion and Open Questions 112
5. Autoreducibility of Complete Languages 113
5.1 Introduction 113
5.2 Definitions 115
5.3 Nonautoreducibility Results 115
5.3.1 Adaptive Autoreductions 116
5.3.2 Nonadaptive Autoreductions 121
5.4 Autoreducibility Results 122
5.4.1 Adaptive Autoreductions 123
5.4.2 Nonadaptive Autoreductions 130
5.4.3 Randomized and Nonuniform Autoreductions 134
5.5 Separation Results 137
5.6 Conclusion and Open Questions 139
6. The Size of Randomized Polynomial Time 141
6.1 Introduction 141
6.2 The Zero-One Law for BPP 142
6.3 Generalization 143
6.4 Conclusion and Open Questions 144
7. The Frequency of Complete Languages 145
7.1 Introduction 145
7.2 Small Span Theorem 146
7.3 Complete Languages for EXP under Adaptive Reductions 152
7.4 Complete Languages for EXP under Nonadaptive Reductions 154
7.5 Conclusion and Open Questions 159
8. The Frequency of Autoreducible Languages 161
8.1 Introduction 161
8.2 Betting Games 163
8.3 From Betting Games to Martingales 165
8.4 Sampling Results 168
8.5 Autoreducible Languages 172
8.5.1 Adaptively Autoreducible Languages 172
8.5.2 Nonadaptively Autoreducible Languages 174
8.5.3 Covering Autoreducible Languages by Martingales 175
8.6 Conclusion and Open Questions 181
References 183
Notation Index 191
Subject Index 195
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