Preface |
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vii | |
Invitation |
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vii | |
Usage |
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viii | |
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1 The Self-Reducibility Technique |
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1 | (30) |
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1.1 GEM: There Are No Sparse NP-Complete Sets Unless P=NP |
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2 | (16) |
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18 | (4) |
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1.3 The Case of Merely Putting Sparse Sets in NP - P: The Hartmanis-Immerman-Sewelson Encoding |
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22 | (4) |
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1.4 OPEN ISSUE: Does the Disjunctive Case Hold? |
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26 | (1) |
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26 | (5) |
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2 The One-Way Function Technique |
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31 | (14) |
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2.1 GEM: Characterizing the Existence of One-Way Functions |
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32 | (3) |
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2.2 Unambiguous One-Way Functions Exist If and Only If Bounded-Ambiguity One-Way Functions Exist |
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35 | (1) |
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2.3 Strong, Total, Commutative, Associative One-Way Functions Exist If and Only If One-Way Functions Exist |
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36 | (6) |
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2.4 OPEN ISSUE: Low-Ambiguity, Commutative, Associative One-Way Functions? |
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42 | (1) |
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43 | (2) |
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3 The Tournament Divide and Conquer Technique |
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45 | (22) |
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3.1 GEM: The Semi-feasible Sets Have Small Circuits |
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45 | (3) |
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3.2 Optimal Advice for the Semi-feasible Sets |
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48 | (8) |
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3.3 Unique Solutions Collapse the Polynomial Hierarchy |
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56 | (7) |
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3.4 OPEN ISSUE: Are the Semi-feasible Sets in P/linear? |
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63 | (1) |
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63 | (4) |
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4 The Isolation Technique |
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67 | (24) |
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4.1 GEM: Isolating a Unique Solution |
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68 | (4) |
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4.2 Toda's Theorem: PH PPP |
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72 | (10) |
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82 | (5) |
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4.4 OPEN ISSUE: Do Ambiguous and Unambiguous Nondeterminism Coincide? |
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87 | (1) |
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87 | (4) |
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5 The Witness Reduction Technique |
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91 | (18) |
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5.1 Framing the Question: Is #P Closed Under Proper Subtraction? |
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91 | (2) |
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5.2 GEM: A Complexity Theory for Feasible Closure Properties of #P |
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93 | (6) |
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5.3 Intermediate Potential Closure Properties |
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99 | (4) |
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5.4 A Complexity Theory for Feasible Closure Properties of OptP |
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103 | (2) |
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5.5 OPEN ISSUE: Characterizing Closure Under Proper Decrement |
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105 | (1) |
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106 | (3) |
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6 The Polynomial Interpolation Technique |
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109 | (58) |
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6.1 GEM: Interactive Protocols for the Permanent |
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110 | (9) |
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6.2 Enumerators for the Permanent |
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119 | (3) |
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122 | (11) |
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133 | (30) |
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6.5 OPEN ISSUE: The Power of the Provers |
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163 | (1) |
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163 | (4) |
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7 The Nonsolvable Group Technique |
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167 | (30) |
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7.1 GEM: Width-5 Branching Programs Capture Nonuniform-NC1 |
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168 | (8) |
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7.2 Width-5 Bottleneck Machines Capture PSPACE |
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176 | (5) |
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7.3 Width-2 Bottleneck Computation |
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181 | (11) |
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7.4 OPEN ISSUE: How Complex Is Majority-Based Probabilistic Symmetric Bottleneck Computation? |
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192 | (1) |
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192 | (5) |
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8 The Random Restriction Technique |
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197 | (38) |
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8.1 GEM: The Random Restriction Technique and a Polynomial-Size Lower Bound for Parity |
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197 | (10) |
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8.2 An Exponential-Size Lower Bound for Parity |
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207 | (11) |
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8.3 PH and PSPACE Differ with Probability One |
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218 | (4) |
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8.4 Oracles That Make the Polynomial Hierarchy Infinite |
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222 | (9) |
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8.5 OPEN ISSUE: Is the Polynomial Hierarchy Infinite with Probability One? |
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231 | (1) |
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231 | (4) |
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9 The Polynomial Technique |
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235 | (28) |
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9.1 GEM: The Polynomial Technique |
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236 | (5) |
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9.2 Closure Properties of PP |
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241 | (11) |
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9.3 The Probabilistic Logspace Hierarchy Collapses |
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252 | (7) |
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9.4 OPEN ISSUE: Is PP Closed Under Polynomial-Time Turing Reductions? |
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259 | (1) |
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260 | (3) |
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A A Rogues' Gallery of Complexity Classes |
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263 | (42) |
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264 | (2) |
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266 | (2) |
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A.3 Oracles and Relativized Worlds |
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268 | (2) |
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A.4 The Polynomial Hierarchy and Polynomial Space: The Power of Quantifiers |
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270 | (4) |
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274 | (2) |
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A.6 P/Poly: Small Circuits |
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276 | (1) |
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A.7 L, NL, etc.: Logspace Classes |
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277 | (2) |
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A.8 NC, AC, LOGCFL: Circuit Classes |
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279 | (2) |
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A.9 UP, FewP, and US: Ambiguity-Bounded Computation and Unique Computation |
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281 | (5) |
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A.10 #P: Counting Solutions |
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286 | (2) |
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A.11 ZPP, RP, coRP, and BPP: Error-Bounded Probabilism |
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288 | (2) |
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A.12 PP, C = P, and SPP: Counting Classes |
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290 | (1) |
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A.13 FP, NPSV, and NPMV: Deterministic and Nondeterministic Functions |
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291 | (3) |
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A.14 P-Sel: Semi-feasible Computation |
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294 | (3) |
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A.15 P, ModkP: Modulo-Based Computation |
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297 | (1) |
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A.16 SpanP, OptP: Output-Cardinality and Optimization Function Classes |
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297 | (2) |
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A.17 IP and MIP: Interactive Proof Classes |
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299 | (1) |
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A.18 PBP, SF, SSF: Branching Programs and Bottleneck Computation |
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300 | (5) |
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B A Rogues' Gallery of Reductions |
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305 | (4) |
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B.1 Reduction Definitions: ≤pm, ≤pT |
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305 | (2) |
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307 | (1) |
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B.3 Facts about Reductions |
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307 | (1) |
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B.4 Circuit-Based Reductions: NCk and ACk |
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308 | (1) |
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308 | (1) |
References |
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309 | (26) |
Index |
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335 | |