Bounded arithmetic, propositional logic, and complexity theory:
Saved in:
| Main Author: | |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
Cambridge [u.a.]
Cambridge Univ. Press
1995
|
| Edition: | 1. publ. |
| Series: | Encyclopedia of mathematics and its applications
60 |
| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Physical Description: | XIV, 343 S. |
| ISBN: | 0521452058 |
Staff View
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Titel: Bounded arithmetic, propositional logic and complexity theory
Autor: Krajíček, Jan
Jahr: 1995
CONTENTS Preface page xi Acknowledgments xiv 1 Introduction 1 2 Preliminaries 3 2.1 Logic 3 2.2 Complexity theory 5 3 Basic complexity theory 8 3.1 The P versus NP problem 8 3.2 Bounded arithmetic formulas 17 3.3 Bibliographical and other remarks 22 4 Basic propositional logic 23 4.1 Propositional proof systems 23 4.2 Resolution 25 4.3 Sequent calculus 31 4.4 Frege systems 42 4.5 The extension and the substitution rules 53 4.6 Quantified propositional logic 57 4.7 Bibliographical and other remarks 60 5 Basic bounded arithmetic 62 5.1 Theory/Ao 63 5.2 Theories S 2 and T 2 68 5.3 Theory PV 75 vii
viii Contents 5.4 Coding of sequences 79 5.5 Second order systems 83 5.6 Bibliographical and other remarks 92 6 Definability of computations 93 6.1 Polynomial time with oracles 94 6.2 Bounded number of queries 97 6.3 Interactive computations 97 6.4 Bibliographical and other remarks 101 7 Witnessing theorems 102 7.1 Cut-elimination for bounded arithmetic 102 7.2 E*-definability in S 2 and oracle polynomial time 105 7.3 Ef +2 -and E* +| -definability in S‘ 2 and bounded queries 113 7.4 E, 6 +2 -definability in T 2 and counterexamples 120 7.5 Ef-definability in and polynomial local search 121 7.6 Model-theoretic constructions 126 7.7 Bibliographical and other remarks 131 8 Definability and witnessing in second order theories 132 8.1 Second order computations 132 8.2 Definable functionals 134 8.3 Bibliographical and other remarks 138 9 Translations of arithmetic formulas 139 9.1 Bounded formulas with a predicate 139 9.2 Translation into quantified propositional formulas 144 9.3 Reflection principles and polynomial simulations 158 9.4 Model-theoretic constructions 172 9.5 Witnessing and test trees 180 9.6 Bibliographical and other remarks 183 10 Finite axiomatizability problem 185 10.1 Finite axiomatizability of S 2 and T 2 185 10.2 Tj versus S 2 +1 186 10.3 S 2 versus T 2 193 10.4 Relativized cases 194 10.5 Consistency notions 202 10.6 Bibliographical and other remarks 208 11 Direct independence proofs 210 11.1 Herbrandization of induction axioms 210 11.2 Weak pigeonhole principle 213
Contents ix 11.3 An independence criterion 220 11.4 Lifting independence results 225 11.5 Bibliographical and other remarks 231 12 Bounds for constant-depth Frege systems 232 12.1 Upper bounds 232 12.2 Depth d versus depth d + 1 236 12.3 Complete systems 243 12.4 ¿-evaluations 252 12.5 Lower bounds for the pigeonhole principle and for counting principles 258 12.6 Systems with counting gates 266 12.7 Forcing in nonstandard models 272 12.8 Bibliographical and other remarks 277 13 Bounds for Frege and extended Frege systems 279 13.1 Counting in Frege systems 279 13.2 An approach to lower bounds 286 13.3 Boolean valuations 289 13.4 Bibliographical and other remarks 297 14 Hard tautologies and optimal proof systems 299 14.1 Finitistic consistency statements and optimal proof systems 299 14.2 Hard tautologies 304 14.3 Bibliographical and other remarks 307 15 Strength of bounded arithmetic 308 15.1 Counting 308 15.2 A circuit lower bound 312 15.3 Polynomial hierarchy in models of bounded arithmetic 316 15.4 Bibliographical and other remarks 324 References 327 Subject index 335 Name index 339 Symbol index 341 |
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| series2 | Encyclopedia of mathematics and its applications |
| spelling | Krajiček, Jan Verfasser aut Bounded arithmetic, propositional logic, and complexity theory Jan Krajíček 1. publ. Cambridge [u.a.] Cambridge Univ. Press 1995 XIV, 343 S. txt rdacontent n rdamedia nc rdacarrier Encyclopedia of mathematics and its applications 60 Modelltheorie (DE-588)4114617-7 gnd rswk-swf Komplexitätstheorie (DE-588)4120591-1 gnd rswk-swf Aussagenlogik (DE-588)4136098-9 gnd rswk-swf Modelltheorie (DE-588)4114617-7 s DE-604 Komplexitätstheorie (DE-588)4120591-1 s Aussagenlogik (DE-588)4136098-9 s Encyclopedia of mathematics and its applications 60 (DE-604)BV000903719 60 HBZ Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=007125399&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Krajiček, Jan Bounded arithmetic, propositional logic, and complexity theory Encyclopedia of mathematics and its applications Modelltheorie (DE-588)4114617-7 gnd Komplexitätstheorie (DE-588)4120591-1 gnd Aussagenlogik (DE-588)4136098-9 gnd |
| subject_GND | (DE-588)4114617-7 (DE-588)4120591-1 (DE-588)4136098-9 |
| title | Bounded arithmetic, propositional logic, and complexity theory |
| title_auth | Bounded arithmetic, propositional logic, and complexity theory |
| title_exact_search | Bounded arithmetic, propositional logic, and complexity theory |
| title_full | Bounded arithmetic, propositional logic, and complexity theory Jan Krajíček |
| title_fullStr | Bounded arithmetic, propositional logic, and complexity theory Jan Krajíček |
| title_full_unstemmed | Bounded arithmetic, propositional logic, and complexity theory Jan Krajíček |
| title_short | Bounded arithmetic, propositional logic, and complexity theory |
| title_sort | bounded arithmetic propositional logic and complexity theory |
| topic | Modelltheorie (DE-588)4114617-7 gnd Komplexitätstheorie (DE-588)4120591-1 gnd Aussagenlogik (DE-588)4136098-9 gnd |
| topic_facet | Modelltheorie Komplexitätstheorie Aussagenlogik |
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