Logical foundations of proof complexity:
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| Main Authors: | , |
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| Format: | Book |
| Language: | English |
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Cambridge
Cambridge Univ. Press
2010
|
| Edition: | 1. publ. |
| Series: | Perspectives in logic
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| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Item Description: | Includes bibliographical references and index |
| Physical Description: | XV, 479 S. |
| ISBN: | 9780521517294 |
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| 245 | 1 | 0 | |a Logical foundations of proof complexity |c Stephen Cook ; Phuong Nguyen |
| 250 | |a 1. publ. | ||
| 264 | 1 | |a Cambridge |b Cambridge Univ. Press |c 2010 | |
| 300 | |a XV, 479 S. | ||
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| 490 | 0 | |a Perspectives in logic | |
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Record in the Search Index
| _version_ | 1804141179713880064 |
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| adam_text | Titel: Logical foundations of proof complexity
Autor: Cook, Stephen
Jahr: 2010
CONTENTS
Preface..........................................................xiii
Chapter I. Introduction...................................... 1
Chapter II. The Predicate Calculus and the System LK....... 9
II.
II.
II.
II.
II.
Propositional Calculus............................... 9
. 1. Gentzen s Propositional Proof System PK............. 10
.2. Soundness and Completeness of PK.................. 12
.3. PK Proofs from Assumptions......................... 13
.4. Propositional Compactness........................... 16
11.2. Predicate Calculus................................... 17
11.2.1. Syntax of the Predicate Calculus...................... 17
11.2.2. Semantics of Predicate Calculus....................... 19
11.2.3. The First-Order Proof System LK.................... 21
11.2.4. Free Variable Normal Form.......................... 23
11.2.5. Completeness of LK without Equality................ 24
11.3. Equality Axioms..................................... 31
11.3.1. Equality Axioms for LK.............................. 32
11.3.2. Revised Soundness and Completeness of LK.......... 33
11.4. Major Corollaries of Completeness................... 34
11.5. The Herbrand Theorem.............................. 35
11.6. Notes................................................ 38
Chapter III. Peano Arithmetic and Its Subsystems............. 39
III. 1. Peano Arithmetic.................................... 39
111.1.1. Minimization........................................ 44
111.1.2. Bounded Induction Scheme.......................... 44
111.1.3. Strong Induction Scheme............................. 44
111.2. Parikh s Theorem.................................... 44
111.3. Conservative Extensions of/Ao....................... 49
111.3.1. Introducing New Function and Predicate Symbols..... 50
111.3.2. /Äo: A Universal Conservative Extension of /Ao....... 54
111.3.3. Defining y = 2x and BIT(i, x) in 7A0................. 59
111.4. /Ao and the Linear Time Hierarchy................... 65
viii Contents
111.4.1. The Polynomial and Linear Time Hierarchies.......... 65
111.4.2. Representability of LTH Relations.................... 66
111.4.3. Characterizing the LTH by /Ao....................... 69
111.5. Buss s S{ Hierarchy: The Road Not Taken............. 70
111.6. Notes................................................ 71
Chapter IV. Two-Sorted Logic and Complexity Classes....... 73
IV. 1. Basic Descriptive Complexity Theory................. 74
IV.2. Two-Sorted First-Order Logic........................ 76
IV.2.1. Syntax............................................... 76
IV.2.2. Semantics............................................ 78
IV.3. Two-Sorted Complexity Classes....................... 80
IV.3.1. Notation for Numbers and Finite Sets................. 80
IV.3.2. Representation Theorems............................. 81
IV.3.3. The LTH Revisited................................... 86
IV.4. The Proof System LK2............................... 87
IV.4.1. Two-Sorted Free Variable Normal Form.............. 90
IV.5. Single-Sorted Logic Interpretation.................... 91
IV.6. Notes................................................ 93
Chapter V. The Theory F° and AC0........................... 95
V. 1. Definition and Basic Properties of V ................. 95
V.2. Two-Sorted Functions................................101
V.3. Parikh s Theorem for Two-Sorted Logic...............104
V.4. Definability in F°....................................106
V.4.1. Ai-Definable Predicates...............................115
V.5. The Witnessing Theorem for F°...................... 117
V.5.1. Independence Follows from the Witnessing Theorem
forF0...........................................118
V.5.2. Proof of the Witnessing Theorem for F°...............119
V.6. V : Universal Conservative Extension of F°...........124
V.6.1. Alternative Proof of the Witnessing Theorem for Vo ... 127
V.7. Finite Axiomatizability...............................129
V.8. Notes................................................130
Chapter VI. The Theory F1 and Polynomial Time..............133
VI. 1. Induction Schemes in V .............................133
VI.2. Characterizing P by F1...............................135
VI.2.1. The If Direction of Theorem VI.2.2.................137
VI.2.2. Application of Cobham s Theorem....................140
VI.3. The Replacement Axiom Scheme.....................142
VI.3.1. Extending F1 by Polytime Functions..................145
VI.4. The Witnessing Theorem for F1...................... 147
VI.4.1. The Sequent System LK2-VX......................... 150
Contents ix
VI.4.2. Proof of the Witnessing Theorem for F1...............154
VI.5. Notes................................................156
Chapter VII. Propositional Translations......................159
VILI. Propositional Proof Systems..........................160
VII.1.1. Treelike vs Daglike Proof Systems.....................162
VII. 1.2. The Pigeonhole Principle and Bounded Depth PK.....163
VII.2. Translating F° to bPK............................... 165
VII.2.1. Translating £¡f Formulas.............................166
VII.2.2. F° and LK2-V°......................................169
VII.2.3. Proof of the Translation Theorem for F°..............170
VII.3. Quantified Propositional Calculus.................... 173
VII.3.1. QPC Proof Systems..................................175
VII.3.2. The System G........................................175
VII.4. The Systems G,and Gf...............................179
VII.4.1. Extended Frege Systems and Witnessing in Gf.........186
VII.5. Propositional Translations for V .....................191
VII.5.1. Translating F° to Bounded Depth G£.................195
VII.6. Notes................................................198
Chapter VIII. Theories for Polynomial Time and Beyond......201
VIII.l. The Theory VP and Aggregate Functions.............201
VIII. 1.1. The Theory VP......................................207
VIII.2. The Theory VPV....................................210
VIII.2.1. Comparing VPV and V1.............................213
VIII.2.2. VPV Is Conservative over VP........................214
VIII.3. TV0 and the TV Hierarchy..........................217
VIII.3.1. TV0 C VPV.........................................220
VIII.3.2. Bit Recursion........................................222
VIII.4. The Theory Vl-HORN...............................223
VIII.5. TV1 and Polynomial Local Search....................228
VIII.6. KPT Witnessing and Replacement....................237
VIII.6.1. Applying KPT Witnessing............................239
VIII.7. More on F and TF .................................243
VIII.7.1. Finite Axiomatizability...............................243
VIII.7.2. Definability in the F°° Hierarchy.....................245
VIII.7.3. Collapse of F°° vs Collapse of PH....................253
VIII.8. RSUV Isomorphism.................................256
VIII.8.1. The Theories S[ and T 2...............................256
VIII.8.2. RSUV Isomorphism.................................258
VIII.8.3. The » Translation.....................................260
VIII.8.4. Theb Translation.....................................262
VIII.8.5. The RSUV Isomorphism between S[ and V..........263
VIII.9. Notes................................................266
x Contents
Chapter IX. Theories for Small Classes.......................267
IX.l. AC0 Reductions......................................269
IX.2. Theories for Subclasses of P..........................272
IX.2.1. The Theories^.....................................273
IX.2.2. The Theory VC_......................................274
IX.2.3. The Theory VC......................................278
IX.2.4. Obtaining Theories for the Classes of Interest..........280
IX.3. Theories for TC°.....................................281
IX.3.1. The Class TC°.......__.............................282
IX.3.2. The Theories VTC°, VTC°, and TTC°................283
IX.3.3. Number Recursion and Number Summation..........287
IX.3.4. The Theory VTC°V..................................289
IX.3.5. Proving the Pigeonhole Principle in VTC°.............291
IX.3.6. Defining String Multiplication in VTC°...............293
LX.3.7. Proving Finite Szpilrajn s Theorem in VTC°...........298
IX.3.8. Proving Bondy s Theorem............................299
IX.4. Theories for AC°(m) and ACC.......................303
IX.4.1. The Classes AC°{m) anàACC........................303
IX.4.2. The Theories F°(2), Vo(2), and Vo(2).................304
IX.4.3. The onto PHP and Parity Principle.................306
IX.4.4. The Theory VAC°(2) V...............................308
IX.4.5. The Jordan Curve Theorem and Related Principles___309
IX.4.6. The Theories for ACf(m) and ACC...................313
IX.4.7. The Modulo m Counting Principles...................316
IX.4.8. The Theory VAC°(6) V...............................318
IX.5. Theories for NC1 and the NC Hierarchy..............319
IX.5.1. Definitions of the Classes.............................320
IX.5.2. BSVPandWC1.......__............................321
IX.5.3. The Theories VNC1, VNC1, and TÑCX...............323
IX.5.4. VTC° C VNC1......................................326
IX.5.5. The Theory VNC1 V.................................333
IX.5.6. Theories for the NC Hierarchy........................335
IX.6. Theories for NL and L^..............................339
IX.6.1. The Theories VNL, VNL, and TÑL...................339
IX.6.2. The Theory V^-KROM..............................343
IX.6.3. The Theories VL, VL,and~VL........................351
IX.6.4. The Theory VLV.....................................356
IX.7. Open Problems......................................358
IX.7.1. Proving Cayley-Hamilton in VNC2...................358
IX.7.2. VSL and VSL =CI.................................358
IX.7.3. Defining [X/Y in VTC°.............................360
IX.7.4. Proving PHP and County in Vo(m)..................360
IX.8. Notes................................................360
Contents xi
Chapter X. Proof Systems and the Reflection Principle......363
X.I. Formalizing Propositional Translations...............364
X.I.I. Verifying Proofs in TC°..............................364
X. 1.2. Computing Propositional Translations in TC°.........373
X.1.3. The Propositional Translation Theorem for TV .......377
X.2. The Reflection Principle..............................382
X.2.1. Truth Definitions.....................................383
X.2.2. Truth Definitions vs Propositional Translations........387
X.2.3. RFN and Consistency for Subsystems of G............396
X.2.4. Axiomatizations Using RFN.........................403
X.2.5. Proving ^-Simulations Using RFN....................407
X.2.6. The Witnessing Problems for G.......................408
X.3. VNC1 and G^........................................410
X.3.1. Propositional Translation for VNC1...................410
X.3.2. The Boolean Sentence Value Problem.................414
X.3.3. Reflection Principle for PK...........................421
X.4. VTC° and Threshold Logic...........................428
X.4.1. The Sequent Calculus PTK...........................428
X.4.2. Reflection Principles for Bounded Depth PTK.........433
X.4.3. Propositional Translation for VTC°...................434
X.4.4. Bounded Depth GTC0...............................441
X.5. Notes................................................442
Appendix A. Computation Models.............................445
A.I. Deterministic Turing Machines.......................445
A.1.1. L, P, PSPACE, and EXP.............................447
A.2. Nondeterministic Turing Machines...................449
A.3. Oracle Turing Machines..............................451
A.4. Alternating Turing Machines.........................452
A.5. Uniform Circuit Families.............................453
Bibliography....................................................457
Index............................................................465
|
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| author_GND | (DE-588)140808973 (DE-588)140809430 |
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| publishDate | 2010 |
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| series2 | Perspectives in logic |
| spelling | Cook, Stephen 1939- Verfasser (DE-588)140808973 aut Logical foundations of proof complexity Stephen Cook ; Phuong Nguyen 1. publ. Cambridge Cambridge Univ. Press 2010 XV, 479 S. txt rdacontent n rdamedia nc rdacarrier Perspectives in logic Includes bibliographical references and index Computational complexity Proof theory Logic, Symbolic and mathematical Konstruktive Mathematik (DE-588)4165105-4 gnd rswk-swf Beweistheorie (DE-588)4145177-6 gnd rswk-swf Beweistheorie (DE-588)4145177-6 s Konstruktive Mathematik (DE-588)4165105-4 s DE-604 Nguyen, Phuong Verfasser (DE-588)140809430 aut HBZ Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=018997048&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Cook, Stephen 1939- Nguyen, Phuong Logical foundations of proof complexity Computational complexity Proof theory Logic, Symbolic and mathematical Konstruktive Mathematik (DE-588)4165105-4 gnd Beweistheorie (DE-588)4145177-6 gnd |
| subject_GND | (DE-588)4165105-4 (DE-588)4145177-6 |
| title | Logical foundations of proof complexity |
| title_auth | Logical foundations of proof complexity |
| title_exact_search | Logical foundations of proof complexity |
| title_full | Logical foundations of proof complexity Stephen Cook ; Phuong Nguyen |
| title_fullStr | Logical foundations of proof complexity Stephen Cook ; Phuong Nguyen |
| title_full_unstemmed | Logical foundations of proof complexity Stephen Cook ; Phuong Nguyen |
| title_short | Logical foundations of proof complexity |
| title_sort | logical foundations of proof complexity |
| topic | Computational complexity Proof theory Logic, Symbolic and mathematical Konstruktive Mathematik (DE-588)4165105-4 gnd Beweistheorie (DE-588)4145177-6 gnd |
| topic_facet | Computational complexity Proof theory Logic, Symbolic and mathematical Konstruktive Mathematik Beweistheorie |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=018997048&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
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