Proof complexity:
Gespeichert in:
| 1. Verfasser: | |
|---|---|
| Format: | Buch |
| Sprache: | English |
| Veröffentlicht: |
Cambridge
Cambridge University Press
2019
|
| Schriftenreihe: | Encyclopedia of mathematics and its applications
170 |
| Schlagworte: | |
| Online-Zugang: | Inhaltsverzeichnis |
| Beschreibung: | xiv, 516 Seiten |
| ISBN: | 9781108416849 |
Internformat
MARC
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Datensatz im Suchindex
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| adam_text | Contents
Preface page xiii
Acbwwledgements xv
Introduction 1
Parti Basic Concepts 9
1 Concepts and Problems 11
1 1 Propositional Logic 11
1 2 First-Order Logic and Finite Structures 18
1 3 Complexity Classes 22
1 4 Boolean Circuits and Computations 27
1 5 Proof Systems and Fundamental Problems 32
1 6 Bibliographical and Other Remarks 37
2 Frege Systems 39
2 1 Frege Rules and Proofs 39
2 2 Tree-Like Proofs 42
2 3 Reckhow’s Theorem 47
2 4 Extended and Substitution Frege Systems 52
2 5 Overview of Proof Complexity Measures 58
2 6 Bibliographical and Other Remarks 62
3 Sequent Calculus 64
3 1 Sequent Calculus LK 64
3 2 The Strength of LK 66
3 3 Interpolation for Cut-Free Proofs 70
3 4 Constant-Depth Subsystems of LK 73
3 5 Bibliographical and Other Remarks 79
4 Quantified Propositional Calculus 81
4 1 Sequent Calculus G 81
4 2 Relativized Frege systems and Boolean Programs 84
X
Contents
13 6 Games 289
13 7 Bibliographical and Other Remarks 291
14 LKd+i/2 and Combinatorial Restrictions 296
14 1 The Lifting Idea 296
14 2 Restrictions and Switchings 298
14 3 Depth Separation 300
14 4 No-Gap Theorem 302
14 5 DNF Separation 304
14 6 Bibliographical and Other Remarks 304
15 Fa and Logical Restrictions 306
15 1 PHP-Trees and k-Evaluations 307
15 2 The Existence of k-Evaluations 313
15 3 Ajtai’s Theorem 318
15 4 Other Combinatorial Principles 320
15 5 Relations Among Counting Principles 322
15 6 Modular Counting Gates 327
15 7 Bibliographical and Other Remarks 334
16 Algebraic and Geometric Proof Systems 337
16 1 Nullstellensatz 337
16 2 Polynomial Calculus 342
16 3 Positivstellensatz 346
16 4 CP and LS: Rank and Tree-Like Size 348
16 5 Bibliographical and Other Remarks 349
17 Feasible Interpolation: A Framework 353
17 1 The Set-Up 353
17 2 Disjoint NP Pairs Hard to Separate 358
17 3 Win-Win Situation: Non-Automatizability 364
17 4 Communication Protocols 366
17 5 Semantic Feasible Interpolation 370
17 6 Simple Examples 374
17 7 Splitting Proofs 376
17 8 Splitting CNF Formulas 380
17 9 Bibliographical and Other Remarks 380
18 Feasible Interpolation: Applications 384
18 1 CP and Monotone Real Circuits 384
18 2 CP and the Real Game 387
18 3 NS, PC and Span Programs 390
18 4 The Loväsz-Schrijver Proof System 394
18 5 The OBDD Proof System 396
18 6 R(LIN), CP and Randomized Protocols 400
18 7 Limits and Beyond 403
18 8 Bibliographical and Other Remarks 405
Contents xi
Part IV Beyond Bounds 411
19 Hard Tautologies 413
19 1 Levels of Uniformity 414
19 2 Reflection and Consistency 416
19 3 Combinatorics of Consistency 419
19 4 Proof Complexity Generators 426
19 5 Circuit Lower Bounds 432
19 6 Bibliographical and Other Remarks 434
20 Model Theory and Lower Bounds 442
20 1 Short Proofs and Model Extensions 442
20 2 Pseudofinite Structures 447
20 3 Model-Theoretic Forcing 450
20 4 Forcing with Random Variables 452
20 5 Bibliographical and Other Remarks 454
21 Optimality 456
21 1 Hard Sequences 456
21 2 Disjoint NP Pairs 459
21 3 Quantitative Gödel’s Theorem 460
21 4 Diagonalization 463
21 5 The Proof Search Problem 466
21 6 Bibliographical and Other Remarks 468
22 The Nature of Proof Complexity 472
22 1 Algorithmic Level 472
22 2 Combinatorial Level 473
22 3 Logical Level 474
22 4 Mathematical Level 476
22 5 Bibliographical and Other Remarks 477
Bibliography 481
Special Symbols 506
|
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| id | DE-604.BV046411033 |
| illustrated | Not Illustrated |
| indexdate | 2024-07-10T08:43:54Z |
| institution | BVB |
| isbn | 9781108416849 |
| language | English |
| lccn | 018042527 |
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| physical | xiv, 516 Seiten |
| publishDate | 2019 |
| publishDateSearch | 2019 |
| publishDateSort | 2019 |
| publisher | Cambridge University Press |
| record_format | marc |
| series | Encyclopedia of mathematics and its applications |
| series2 | Encyclopedia of mathematics and its applications |
| spelling | Krajíček, Jan 1960- (DE-588)14367630X aut Proof complexity Jan Krajíček, Charles University, Prague Cambridge Cambridge University Press 2019 xiv, 516 Seiten txt rdacontent n rdamedia nc rdacarrier Encyclopedia of mathematics and its applications 170 Proof theory Computational complexity Beweistheorie (DE-588)4145177-6 gnd rswk-swf Komplexitätstheorie (DE-588)4120591-1 gnd rswk-swf Beweistheorie (DE-588)4145177-6 s Komplexitätstheorie (DE-588)4120591-1 s DE-604 Erscheint auch als Online-Ausgabe 978-1-108-24206-6 Encyclopedia of mathematics and its applications 170 (DE-604)BV000903719 170 HEBIS Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=031823654&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Krajíček, Jan 1960- Proof complexity Encyclopedia of mathematics and its applications Proof theory Computational complexity Beweistheorie (DE-588)4145177-6 gnd Komplexitätstheorie (DE-588)4120591-1 gnd |
| subject_GND | (DE-588)4145177-6 (DE-588)4120591-1 |
| title | Proof complexity |
| title_auth | Proof complexity |
| title_exact_search | Proof complexity |
| title_full | Proof complexity Jan Krajíček, Charles University, Prague |
| title_fullStr | Proof complexity Jan Krajíček, Charles University, Prague |
| title_full_unstemmed | Proof complexity Jan Krajíček, Charles University, Prague |
| title_short | Proof complexity |
| title_sort | proof complexity |
| topic | Proof theory Computational complexity Beweistheorie (DE-588)4145177-6 gnd Komplexitätstheorie (DE-588)4120591-1 gnd |
| topic_facet | Proof theory Computational complexity Beweistheorie Komplexitätstheorie |
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