Bounded arithmetic, propositional logic, and complexity theory /:
This book presents an up-to-date, unified treatment of research in bounded arithmetic and complexity of propositional logic, with emphasis on independence proofs and lower bound proofs. The author discusses the deep connections between logic and complexity theory and lists a number of intriguing ope...
Gespeichert in:
| 1. Verfasser: | |
|---|---|
| Körperschaft: | |
| Format: | Elektronisch E-Book |
| Sprache: | English |
| Veröffentlicht: |
Cambridge [England] ; New York, N.Y. :
Cambridge University Press,
1995.
|
| Schriftenreihe: | Encyclopedia of mathematics and its applications ;
v. 60. |
| Schlagworte: | |
| Online-Zugang: | Volltext |
| Zusammenfassung: | This book presents an up-to-date, unified treatment of research in bounded arithmetic and complexity of propositional logic, with emphasis on independence proofs and lower bound proofs. The author discusses the deep connections between logic and complexity theory and lists a number of intriguing open problems. An introduction to the basics of logic and complexity theory is followed by discussion of important results in propositional proof systems and systems of bounded arithmetic. More advanced topics are then treated, including polynomial simulations and conservativity results, various witnessing theorems, the translation of bounded formulas (and their proofs) into propositional ones, the method of random partial restrictions and its applications, direct independence proofs, complete systems of partial relations, lower bounds to the size of constant-depth propositional proofs, the method of Boolean valuations, the issue of hard tautologies and optimal proof systems, combinatorics and complexity theory within bounded arithmetic, and relations to complexity issues of predicate calculus. Students and researchers in mathematical logic and complexity theory will find this comprehensive treatment an excellent guide to this expanding interdisciplinary area. |
| Beschreibung: | 1 online resource (xiv, 343 pages) |
| Bibliographie: | Includes bibliographical references (pages 327-334) and indexes. |
| ISBN: | 9781107088634 1107088631 9780511529948 0511529945 |
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| 245 | 1 | 0 | |a Bounded arithmetic, propositional logic, and complexity theory / |c Jan Krajíček. |
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| 520 | |a This book presents an up-to-date, unified treatment of research in bounded arithmetic and complexity of propositional logic, with emphasis on independence proofs and lower bound proofs. The author discusses the deep connections between logic and complexity theory and lists a number of intriguing open problems. An introduction to the basics of logic and complexity theory is followed by discussion of important results in propositional proof systems and systems of bounded arithmetic. More advanced topics are then treated, including polynomial simulations and conservativity results, various witnessing theorems, the translation of bounded formulas (and their proofs) into propositional ones, the method of random partial restrictions and its applications, direct independence proofs, complete systems of partial relations, lower bounds to the size of constant-depth propositional proofs, the method of Boolean valuations, the issue of hard tautologies and optimal proof systems, combinatorics and complexity theory within bounded arithmetic, and relations to complexity issues of predicate calculus. Students and researchers in mathematical logic and complexity theory will find this comprehensive treatment an excellent guide to this expanding interdisciplinary area. | ||
| 505 | 8 | |a 7.7 Bibliographical and other remarks8 Definability and witnessing in second order theories; 8.1 Second order computations; 8.2 Definable functionals; 8.3 Bibliographical and other remarks; 9 Translations of arithmetic formulas; 9.1 Bounded formulas with a predicate; 9.2 Translation into quantified propositional formulas; 9.3 Reflection principles and polynomial simulations; 9.4 Model-theoretic constructions; 9.5 Witnessing and test trees; 9.6 Bibliographical and other remarks; 10 Finite axiomatizability problem; 10.1 Finite axiomatizability of Si2 and Ti2; 10.2 Ti2 versus Si2+1 | |
| 505 | 8 | |a 10.3 Si2 versus Ti210.4 Relativized cases; 10.5 Consistency notions; 10.6 Bibliographical and other remarks; 11 Direct independence proofs; 11.1 Herbrandization of induction axioms; 11.2 Weak pigeonhole principle; 11.3 An independence criterion; 11.4 Lifting independence results; 11.5 Bibliographical and other remarks; 12 Bounds for constant-depth Frege systems; 12.1 Upper bounds; 12.2 Depth d versus depth d + 1; 12.3 Complete systems; 12.4 k-evaluations; 12.5 Lower bounds for the pigeonhole principle and for counting principles; 12.6 Systems with counting gates | |
| 505 | 8 | |a 12.7 Forcing in nonstandard models12.8 Bibliographical and other remarks; 13 Bounds for Frege and extended Frege systems; 13.1 Counting in Frege systems; 13.2 An approach to lower bounds; 13.3 Boolean valuations; 13.4 Bibliographical and other remarks; 14 Hard tautologies and optimal proof systems; 14.1 Finitistic consistency statements and optimal proof systems; 14.2 Hard tautologies; 14.3 Bibliographical and other remarks; 15 Strength of bounded arithmetic; 15.1 Counting; 15.2 A circuit lower bound; 15.3 Polynomial hierarchy in models of bounded arithmetic | |
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| author | Krajíček, Jan |
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| contents | 7.7 Bibliographical and other remarks8 Definability and witnessing in second order theories; 8.1 Second order computations; 8.2 Definable functionals; 8.3 Bibliographical and other remarks; 9 Translations of arithmetic formulas; 9.1 Bounded formulas with a predicate; 9.2 Translation into quantified propositional formulas; 9.3 Reflection principles and polynomial simulations; 9.4 Model-theoretic constructions; 9.5 Witnessing and test trees; 9.6 Bibliographical and other remarks; 10 Finite axiomatizability problem; 10.1 Finite axiomatizability of Si2 and Ti2; 10.2 Ti2 versus Si2+1 10.3 Si2 versus Ti210.4 Relativized cases; 10.5 Consistency notions; 10.6 Bibliographical and other remarks; 11 Direct independence proofs; 11.1 Herbrandization of induction axioms; 11.2 Weak pigeonhole principle; 11.3 An independence criterion; 11.4 Lifting independence results; 11.5 Bibliographical and other remarks; 12 Bounds for constant-depth Frege systems; 12.1 Upper bounds; 12.2 Depth d versus depth d + 1; 12.3 Complete systems; 12.4 k-evaluations; 12.5 Lower bounds for the pigeonhole principle and for counting principles; 12.6 Systems with counting gates 12.7 Forcing in nonstandard models12.8 Bibliographical and other remarks; 13 Bounds for Frege and extended Frege systems; 13.1 Counting in Frege systems; 13.2 An approach to lower bounds; 13.3 Boolean valuations; 13.4 Bibliographical and other remarks; 14 Hard tautologies and optimal proof systems; 14.1 Finitistic consistency statements and optimal proof systems; 14.2 Hard tautologies; 14.3 Bibliographical and other remarks; 15 Strength of bounded arithmetic; 15.1 Counting; 15.2 A circuit lower bound; 15.3 Polynomial hierarchy in models of bounded arithmetic |
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| discipline | Mathematik |
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| indexdate | 2024-11-27T13:25:23Z |
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| isbn | 9781107088634 1107088631 9780511529948 0511529945 |
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| series | Encyclopedia of mathematics and its applications ; |
| series2 | Encyclopedia of mathematics and its applications ; |
| spelling | Krajíček, Jan. Bounded arithmetic, propositional logic, and complexity theory / Jan Krajíček. Cambridge [England] ; New York, N.Y. : Cambridge University Press, 1995. 1 online resource (xiv, 343 pages) text txt rdacontent computer c rdamedia online resource cr rdacarrier Encyclopedia of mathematics and its applications ; v. 60 Includes bibliographical references (pages 327-334) and indexes. Print version record. This book presents an up-to-date, unified treatment of research in bounded arithmetic and complexity of propositional logic, with emphasis on independence proofs and lower bound proofs. The author discusses the deep connections between logic and complexity theory and lists a number of intriguing open problems. An introduction to the basics of logic and complexity theory is followed by discussion of important results in propositional proof systems and systems of bounded arithmetic. More advanced topics are then treated, including polynomial simulations and conservativity results, various witnessing theorems, the translation of bounded formulas (and their proofs) into propositional ones, the method of random partial restrictions and its applications, direct independence proofs, complete systems of partial relations, lower bounds to the size of constant-depth propositional proofs, the method of Boolean valuations, the issue of hard tautologies and optimal proof systems, combinatorics and complexity theory within bounded arithmetic, and relations to complexity issues of predicate calculus. Students and researchers in mathematical logic and complexity theory will find this comprehensive treatment an excellent guide to this expanding interdisciplinary area. 7.7 Bibliographical and other remarks8 Definability and witnessing in second order theories; 8.1 Second order computations; 8.2 Definable functionals; 8.3 Bibliographical and other remarks; 9 Translations of arithmetic formulas; 9.1 Bounded formulas with a predicate; 9.2 Translation into quantified propositional formulas; 9.3 Reflection principles and polynomial simulations; 9.4 Model-theoretic constructions; 9.5 Witnessing and test trees; 9.6 Bibliographical and other remarks; 10 Finite axiomatizability problem; 10.1 Finite axiomatizability of Si2 and Ti2; 10.2 Ti2 versus Si2+1 10.3 Si2 versus Ti210.4 Relativized cases; 10.5 Consistency notions; 10.6 Bibliographical and other remarks; 11 Direct independence proofs; 11.1 Herbrandization of induction axioms; 11.2 Weak pigeonhole principle; 11.3 An independence criterion; 11.4 Lifting independence results; 11.5 Bibliographical and other remarks; 12 Bounds for constant-depth Frege systems; 12.1 Upper bounds; 12.2 Depth d versus depth d + 1; 12.3 Complete systems; 12.4 k-evaluations; 12.5 Lower bounds for the pigeonhole principle and for counting principles; 12.6 Systems with counting gates 12.7 Forcing in nonstandard models12.8 Bibliographical and other remarks; 13 Bounds for Frege and extended Frege systems; 13.1 Counting in Frege systems; 13.2 An approach to lower bounds; 13.3 Boolean valuations; 13.4 Bibliographical and other remarks; 14 Hard tautologies and optimal proof systems; 14.1 Finitistic consistency statements and optimal proof systems; 14.2 Hard tautologies; 14.3 Bibliographical and other remarks; 15 Strength of bounded arithmetic; 15.1 Counting; 15.2 A circuit lower bound; 15.3 Polynomial hierarchy in models of bounded arithmetic Constructive mathematics. http://id.loc.gov/authorities/subjects/sh85031452 Proposition (Logic) http://id.loc.gov/authorities/subjects/sh85107552 Computational complexity. http://id.loc.gov/authorities/subjects/sh85029473 Mathématiques constructives. Proposition (Logique) Complexité de calcul (Informatique) MATHEMATICS General. bisacsh Computational complexity fast Constructive mathematics fast Proposition (Logic) fast Cambridge University Press. http://id.loc.gov/authorities/names/n50059133 Cambridge books online Mathematics. Print version: Krajíček, Jan. Bounded arithmetic, propositional logic, and complexity theory. Cambridge [England] ; New York, N.Y. : Cambridge University Press, 1995 9780511529948 Encyclopedia of mathematics and its applications ; v. 60. http://id.loc.gov/authorities/names/n42010632 FWS01 ZDB-4-EBA FWS_PDA_EBA https://search.ebscohost.com/login.aspx?direct=true&scope=site&db=nlebk&AN=569329 Volltext |
| spellingShingle | Krajíček, Jan Bounded arithmetic, propositional logic, and complexity theory / Encyclopedia of mathematics and its applications ; 7.7 Bibliographical and other remarks8 Definability and witnessing in second order theories; 8.1 Second order computations; 8.2 Definable functionals; 8.3 Bibliographical and other remarks; 9 Translations of arithmetic formulas; 9.1 Bounded formulas with a predicate; 9.2 Translation into quantified propositional formulas; 9.3 Reflection principles and polynomial simulations; 9.4 Model-theoretic constructions; 9.5 Witnessing and test trees; 9.6 Bibliographical and other remarks; 10 Finite axiomatizability problem; 10.1 Finite axiomatizability of Si2 and Ti2; 10.2 Ti2 versus Si2+1 10.3 Si2 versus Ti210.4 Relativized cases; 10.5 Consistency notions; 10.6 Bibliographical and other remarks; 11 Direct independence proofs; 11.1 Herbrandization of induction axioms; 11.2 Weak pigeonhole principle; 11.3 An independence criterion; 11.4 Lifting independence results; 11.5 Bibliographical and other remarks; 12 Bounds for constant-depth Frege systems; 12.1 Upper bounds; 12.2 Depth d versus depth d + 1; 12.3 Complete systems; 12.4 k-evaluations; 12.5 Lower bounds for the pigeonhole principle and for counting principles; 12.6 Systems with counting gates 12.7 Forcing in nonstandard models12.8 Bibliographical and other remarks; 13 Bounds for Frege and extended Frege systems; 13.1 Counting in Frege systems; 13.2 An approach to lower bounds; 13.3 Boolean valuations; 13.4 Bibliographical and other remarks; 14 Hard tautologies and optimal proof systems; 14.1 Finitistic consistency statements and optimal proof systems; 14.2 Hard tautologies; 14.3 Bibliographical and other remarks; 15 Strength of bounded arithmetic; 15.1 Counting; 15.2 A circuit lower bound; 15.3 Polynomial hierarchy in models of bounded arithmetic Constructive mathematics. http://id.loc.gov/authorities/subjects/sh85031452 Proposition (Logic) http://id.loc.gov/authorities/subjects/sh85107552 Computational complexity. http://id.loc.gov/authorities/subjects/sh85029473 Mathématiques constructives. Proposition (Logique) Complexité de calcul (Informatique) MATHEMATICS General. bisacsh Computational complexity fast Constructive mathematics fast Proposition (Logic) fast |
| subject_GND | http://id.loc.gov/authorities/subjects/sh85031452 http://id.loc.gov/authorities/subjects/sh85107552 http://id.loc.gov/authorities/subjects/sh85029473 |
| title | Bounded arithmetic, propositional logic, and complexity theory / |
| title_alt | Cambridge books online Mathematics. |
| 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 | Constructive mathematics. http://id.loc.gov/authorities/subjects/sh85031452 Proposition (Logic) http://id.loc.gov/authorities/subjects/sh85107552 Computational complexity. http://id.loc.gov/authorities/subjects/sh85029473 Mathématiques constructives. Proposition (Logique) Complexité de calcul (Informatique) MATHEMATICS General. bisacsh Computational complexity fast Constructive mathematics fast Proposition (Logic) fast |
| topic_facet | Constructive mathematics. Proposition (Logic) Computational complexity. Mathématiques constructives. Proposition (Logique) Complexité de calcul (Informatique) MATHEMATICS General. Computational complexity Constructive mathematics |
| url | https://search.ebscohost.com/login.aspx?direct=true&scope=site&db=nlebk&AN=569329 |
| work_keys_str_mv | UT cambridgebooksonlinemathematics AT krajicekjan boundedarithmeticpropositionallogicandcomplexitytheory AT cambridgeuniversitypress boundedarithmeticpropositionallogicandcomplexitytheory |