Computational Complexity : A Conceptual Perspective - Oded Goldreich

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Computational Complexity

A Conceptual Perspective

By: Oded Goldreich

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Hardcover | 28 April 2008

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632 Pages

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25.4 x 17.78 x 3.51

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Complexity theory is a central field of the theoretical foundations of computer science. It is concerned with the general study of the intrinsic complexity of computational tasks; that is, it addresses the question of what can be achieved within limited time (and/or with other limited natural computational resources). This book offers a conceptual perspective on complexity theory. It is intended to serve as an introduction for advanced undergraduate and graduate students, either as a textbook or for self-study. The book will also be useful to experts, since it provides expositions of the various sub-areas of complexity theory such as hardness amplification, pseudorandomness and probabilistic proof systems. In each case, the author starts by posing the intuitive questions that are addressed by the sub-area and then discusses the choices made in the actual formulation of these questions, the approaches that lead to the answers, and the ideas that are embedded in these answers.

Industry Reviews

"This interesting book... is refreshing to read his [Goldreichs'] opinions... The very strong focus on conceptual issues makes the book indispensible as a reference volume for research libraries." M. Bona, University of Florida, CHOICE
"This book provides very well developed material that should interest advanced students either studying or doing new work on computational complexity. It would also be a valuable text for professionals challenged with solving "hard" computing problems of intending to exploit these types of problems when designing of new types computing systems." Brian A. Lawler, Software Engineering Notes
"The book offers a conceptual perspective on several sub-areas of complexity theory and is intended to be used as a textbook for students and educators as well as for experts who seek an overview of of several sub-areas." Gerhard Lischke, Mathematical Reviews

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List of Figures	p. xiii
Preface	p. xv
Organization and Chapter Summaries	p. xvii
Acknowledgments	p. xxiii
Introduction and Preliminaries	p. 1
Introduction	p. 1
A Brief Overview of Complexity Theory	p. 2
Characteristics of Complexity Theory	p. 6
Contents of This Book	p. 8
Approach and Style of This Book	p. 12
Standard Notations and Other Conventions	p. 16
Computational Tasks and Models	p. 17
Representation	p. 18
Computational Tasks	p. 18
Uniform Models (Algorithms)	p. 20
Non-uniform Models (Circuits and Advice)	p. 36
Complexity Classes	p. 42
Chapter Notes	p. 43
P, NP, and NP-Completeness	p. 44
The P Versus NP Question	p. 46
The Search Version: Finding Versus Checking	p. 47
The Decision Version: Proving Versus Verifying	p. 50
Equivalence of the Two Formulations	p. 54
Two Technical Comments Regarding NP	p. 55
The Traditional Definition of NP	p. 55
In Support of P Different from NP	p. 57
Philosophical Meditations	p. 58
Polynomial-Time Reductions	p. 58
The General Notion of a Reduction	p. 59
Reducing Optimization Problems to Search Problems	p. 61
Self-Reducibility of Search Problems	p. 63
Digest and General Perspective	p. 67
NP-Completeness	p. 67
Definitions	p. 68
The Existence of NP-Complete Problems	p. 69
Some Natural NP-Complete Problems	p. 71
NP Sets That Are Neither in P nor NP-Complete	p. 81
Reflections on Complete Problems	p. 85
Three Relatively Advanced Topics	p. 87
Promise Problems	p. 87
Optimal Search Algorithms for NP	p. 92
The Class coNP and Its Intersection with NP	p. 94
Chapter Notes	p. 97
Exercises	p. 99
Variations on P and NP	p. 108
Non-uniform Polynomial Time (P/poly)	p. 108
Boolean Circuits	p. 109
Machines That Take Advice	p. 111
The Polynomial-Time Hierarchy (PH)	p. 113
Alternation of Quantifiers	p. 114
Non-deterministic Oracle Machines	p. 117
The P/poly Versus NP Question and PH	p. 119
Chapter Notes	p. 121
Exercises	p. 122
More Resources, More Power?	p. 127
Non-uniform Complexity Hierarchies	p. 128
Time Hierarchies and Gaps	p. 129
Time Hierarchies	p. 129
Time Gaps and Speedup	p. 136
Space Hierarchies and Gaps	p. 139
Chapter Notes	p. 139
Exercises	p. 140
Space Complexity	p. 143
General Preliminaries and Issues	p. 144
Important Conventions	p. 144
On the Minimal Amount of Useful Computation Space	p. 145
Time Versus Space	p. 146
Circuit Evaluation	p. 153
Logarithmic Space	p. 153
The Class L	p. 154
Log-Space Reductions	p. 154
Log-Space Uniformity and Stronger Notions	p. 155
Undirected Connectivity	p. 155
Non-deterministic Space Complexity	p. 162
Two Models	p. 162
NL and Directed Connectivity	p. 164
A Retrospective Discussion	p. 171
PSPACE and Games	p. 172
Chapter Notes	p. 175
Exercises	p. 175
Randomness and Counting	p. 184
Probabilistic Polynomial Time	p. 185
Basic Modeling Issues	p. 186
Two-Sided Error: The Complexity Class BPP	p. 189
One-Sided Error: The Complexity Classes RP and coRP	p. 193
Zero-Sided Error: The Complexity Class ZPP	p. 199
Randomized Log-Space	p. 199
Counting	p. 202
Exact Counting	p. 202
Approximate Counting	p. 211
Searching for Unique Solutions	p. 217
Uniform Generation of Solutions	p. 220
Chapter Notes	p. 227
Exercises	p. 230
The Bright Side of Hardness	p. 241
One-Way Functions	p. 242
Generating Hard Instances and One-Way Functions	p. 243
Amplification of Weak One-Way Functions	p. 245
Hard-Core Preicates	p. 250
Reflections on Hardness Amplification	p. 255
Hard Problems in E	p. 255
Amplification with Respect to Polynomial-Size Circuits	p. 257
Amplification with Respect to Exponential-Size Circuits	p. 270
Chapter Notes	p. 277
Exercises	p. 278
Pseudorandom Generators	p. 284
Introduction	p. 285
The General Paradigm	p. 288
General-Purpose Pseudorandom Generators	p. 290
The Basic Definition	p. 291
The Archetypical Application	p. 292
Computational Indistinguishability	p. 295
Amplifying the Stretch Function	p. 299
Constructions	p. 301
Non-uniformly Strong Pseudorandom Generators	p. 304
Stronger Notions and Conceptual Reflections	p. 305
Derandomization of Time-Complexity Classes	p. 307
Defining Canonical Derandomizers	p. 308
Constructing Canonical Derandomizers	p. 310
Technical Variations and Conceptual Reflections	p. 313
Space-Bounded Distinguishers	p. 315
Definitional Issues	p. 316
Two Constructions	p. 318
Special-Purpose Generators	p. 325
Pairwise Independence Generators	p. 326
Small-Bias Generators	p. 329
Random Walks on Expanders	p. 332
Chapter Notes	p. 334
Exercises	p. 337
Probabilistic Proof Systems	p. 349
Introduction and Preliminaries	p. 350
Interactive Proof Systems	p. 352
Motivation and Perspective	p. 352
Definition	p. 354
The Power of Interactive Proofs	p. 357
Variants and Finer Structure: An Overview	p. 363
On Computationally Bounded Provers: An Overview	p. 366
Zero-Knowledge Proof Systems	p. 368
Definitional Issues	p. 369
The Power of Zero-Knowledge	p. 372
Proofs of Knowledge - A Parenthetical Subsection	p. 378
Probabilistically Checkable Proof Systems	p. 380
Definition	p. 381
The Power of Probabilistically Checkable Proofs	p. 383
PCP and Approximation	p. 398
More on PCP Itself: An Overview	p. 401
Chapter Notes	p. 404
Exercises	p. 406
Relaxing the Requirements	p. 416
Approximation	p. 417
Search or Optimization	p. 418
Decision or Property Testing	p. 423
Average-Case Complexity	p. 428
The Basic Theory	p. 430
Ramifications	p. 442
Chapter Notes	p. 451
Exercises	p. 453
Epilogue	p. 461
Glossary of Complexity Classes	p. 463
Preliminaries	p. 463
Algorithm-Based Classes	p. 464
Time Complexity Classes	p. 464
Space Complexity Classes	p. 467
Circuit-Based Classes	p. 467
On the Quest for Lower Bounds	p. 469
Preliminaries	p. 469
Boolean Circuit Complexity	p. 471
Basic Results and Questions	p. 472
Monotone Circuits	p. 473
Bounded-Depth Circuits	p. 473
Formula Size	p. 474
Arithmetic Circuits	p. 475
Univariate Polynomials	p. 476
Multivariate Polynomials	p. 476
Proof Complexity	p. 478
Logical Proof Systems	p. 480
Algebraic Proof Systems	p. 480
Geometric Proof Systems	p. 481
On the Foundations of Modern Cryptography	p. 482
Introduction and Preliminaries	p. 482
The Underlying Principles	p. 483
The Computational Model	p. 485
Organization and Beyond	p. 486
Computational Difficulty	p. 487
One-Way Functions	p. 487
Hard-Core Predicates	p. 489
Pseudorandomness	p. 490
Computational Indistinguishability	p. 490
Pseudorandom Generators	p. 491
Pseudorandom Functions	p. 492
Zero-Knowledge	p. 494
The Simulation Paradigm	p. 494
The Actual Definition	p. 494
A General Result and a Generic Application	p. 495
Definitional Variations and Related Notions	p. 497
Encryption Schemes	p. 500
Definitions	p. 502
Constructions	p. 504
Beyond Eavesdropping Security	p. 505
Signatures and Message Authentication	p. 507
Definitions	p. 508
Constructions	p. 509
General Cryptographic Protocols	p. 511
The Definitional Approach and Some Models	p. 512
Some Known Results	p. 517
Construction Paradigms and Two Simple Protocols	p. 517
Concluding Remarks	p. 522
Probabilistic Preliminaries and Advanced Topics in Randomization	p. 523
Probabilistic Preliminaries	p. 523
Notational Conventions	p. 523
Three Inequalities	p. 524
Hashing	p. 528
Definitions	p. 528
Constructions	p. 529
The Leftover Hash Lemma	p. 529
Sampling	p. 533
Formal Setting	p. 533
Known Results	p. 534
Hitters	p. 535
Randomnes Extractors	p. 536
Definitions and Various Perspectives	p. 537
Constructions	p. 541
Explicit Constructions	p. 545
Error-Correcting Codes	p. 546
Basic Notions	p. 546
A Few Popular Codes	p. 547
Two Additional Computational Problems	p. 551
A List-Decoding Bound	p. 553
Expander Graphs	p. 554
Definitions and Properties	p. 555
Constructions	p. 561
Some Omitted Proofs	p. 566
Proving That PH Reduces to #P	p. 566
Proving That IP(f) [characters not reproducible] AM(O(f)) [characters not reproducible] AM(f)	p. 572
Emulating General Interactive Proofs by AM-Games	p. 572
Linear Speedup for AM	p. 578
Some Computational Problems	p. 583
Graphs	p. 583
Boolean Formulae	p. 585
Finite Fields, Polynomials, and Vector Spaces	p. 586
The Determinant and the Permanent	p. 587
Primes and Composite Numbers	p. 587
Bibliography	p. 589
Index	p. 601
Table of Contents provided by Ingram. All Rights Reserved.

Computational Complexity

A Conceptual Perspective

At a Glance

Hardcover

Industry Reviews

Shipping

How to return your order

Defective items

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