Forcing with Random Variables and Proof Complexity.:
A model-theoretic approach to bounded arithmetic and propositional proof complexity.
Gespeichert in:
| 1. Verfasser: | |
|---|---|
| Format: | Elektronisch E-Book |
| Sprache: | English |
| Veröffentlicht: |
Cambridge :
Cambridge University Press,
2010.
|
| Schriftenreihe: | London Mathematical Society Lecture Note Series, 382.
|
| Schlagworte: | |
| Online-Zugang: | Volltext |
| Zusammenfassung: | A model-theoretic approach to bounded arithmetic and propositional proof complexity. |
| Beschreibung: | 19 PHP principle. |
| Beschreibung: | 1 online resource (266 pages) |
| Bibliographie: | Includes bibliographical references (pages 236-242) and indexes. |
| ISBN: | 9781139117333 1139117335 9781139127998 1139127993 9781139115162 1139115162 9781139107211 1139107216 |
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| 490 | 1 | |a London Mathematical Society Lecture Note Series, 382 ; |v v. 382 | |
| 505 | 0 | |a Cover; Title; Copyright; Dedication; Contents; Preface; Acknowledgment; Introduction; Organization of the book; Remarks on the literature; Background; Part I Basics; 1 The definition of the models; 1.1 The ambient model of arithmetic; 1.2 The Boolean algebras; 1.3 The models K(F); 1.4 Valid sentences; 1.5 Possible generalizations; 2 Measure on B; 2.1 A metric on B; 2.2 From Boolean value to probability; 3 Witnessing quantifiers; 3.1 Propositional approximation of truth values; 3.2 Witnessing in definable families; 3.3 Definition by cases by open formulas; 3.4 Compact families. | |
| 505 | 8 | |a 3.5 Propositional computation of truth values4 The truth in N and the validity in K(F); Part II Second-order structures; 5 Structures K(F, G); 5.1 Language L2 and the hierarchy of bounded formulas; 5.2 Cut Mn, languages Ln and L2n; 5.3 Definition of the structures; 5.4 Equality of functions, extensionality and possible generalizations; 5.5 Absoluteness of ASb8-sentences of language Ln; Part III AC0 world; 6 Theories I?0, I?0(R) and V01; 7 Shallow Boolean decision tree model; 7.1 Family Frud; 7.2 Family Grud; 7.3 Properties of Frud and Grud; 8 Open comprehension and open induction. | |
| 505 | 8 | |a 8.1 The ((. . .)) notation8.2 Open comprehension in K(Frud, Grud); 8.3 Open induction in K(Frud, Grud); 8.4 Short open induction; 9 Comprehension and induction via quantifier elimination; 9.1 Bounded quantifier elimination; 9.2 Skolem functions in K(F, G) and quantifierelimination; 9.3 Comprehension and induction for S01,b-formulas; 10 Skolem functions, switching lemma; 10.1 Switching lemma; 10.2 Tree model K(Ftree, Gtree); 11 Quantifier elimination in K(Ftree, Gtree); 11.1 Skolem functions; 11.2 Comprehension and induction for S01,b-formulas. | |
| 505 | 8 | |a 12 Witnessing, independence and definability in V0112.1 Witnessing AX <x EY <x S01,b-formulas; 12.2 Preservation of true sp11,b-sentences; 12.3 Circuit lower bound for parity; Part IV AC0(2) world; 13 Theory Q2V01; 13.1 Q2 quantifier and theory Q2V01; 13.2 Interpreting Q2 in structures; 14 Algebraic model; 14.1 Family Falg; 14.2 Family Galg; 14.3 Open comprehension and open induction; 15 Quantifier elimination and the interpretation of Q2; 15.1 Skolemization and the Razborov -- Smolensky method; 15.2 Interpretation of Q2 in front of an open formula. | |
| 505 | 8 | |a 15.3 Elimination of quantifiers and the interpretation of the Q2 quantifier15.4 Comprehension and induction for Q21,b0-formulas; 16 Witnessing and independence in Q2V01; 16.1 Witnessing AX <xEY <xA Z <xS01,b-formulas; 16.2 Preservation of true sp11,b-sentences; Part V Towards proof complexity; 17 Propositional proof systems; 17.1 Frege and Extended Frege systems; 17.2 Language with connective and constant-depth Frege systems; 18 Lengths-of-proofs lower bounds; 18.1 Formalization of the provability predicate; 18.2 Reflection principles; 18.3 Three conditions for a lower bound. | |
| 500 | |a 19 PHP principle. | ||
| 520 | |a A model-theoretic approach to bounded arithmetic and propositional proof complexity. | ||
| 588 | 0 | |a Print version record. | |
| 504 | |a Includes bibliographical references (pages 236-242) and indexes. | ||
| 650 | 0 | |a Computational complexity. |0 http://id.loc.gov/authorities/subjects/sh85029473 | |
| 650 | 0 | |a Random variables. |0 http://id.loc.gov/authorities/subjects/sh85111355 | |
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| author | Krajícek, Jan |
| author_facet | Krajícek, Jan |
| author_role | |
| author_sort | Krajícek, Jan |
| author_variant | j k jk |
| building | Verbundindex |
| bvnumber | localFWS |
| callnumber-first | Q - Science |
| callnumber-label | QA267 |
| callnumber-raw | QA267.7 .K73 2011 |
| callnumber-search | QA267.7 .K73 2011 |
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| contents | Cover; Title; Copyright; Dedication; Contents; Preface; Acknowledgment; Introduction; Organization of the book; Remarks on the literature; Background; Part I Basics; 1 The definition of the models; 1.1 The ambient model of arithmetic; 1.2 The Boolean algebras; 1.3 The models K(F); 1.4 Valid sentences; 1.5 Possible generalizations; 2 Measure on B; 2.1 A metric on B; 2.2 From Boolean value to probability; 3 Witnessing quantifiers; 3.1 Propositional approximation of truth values; 3.2 Witnessing in definable families; 3.3 Definition by cases by open formulas; 3.4 Compact families. 3.5 Propositional computation of truth values4 The truth in N and the validity in K(F); Part II Second-order structures; 5 Structures K(F, G); 5.1 Language L2 and the hierarchy of bounded formulas; 5.2 Cut Mn, languages Ln and L2n; 5.3 Definition of the structures; 5.4 Equality of functions, extensionality and possible generalizations; 5.5 Absoluteness of ASb8-sentences of language Ln; Part III AC0 world; 6 Theories I?0, I?0(R) and V01; 7 Shallow Boolean decision tree model; 7.1 Family Frud; 7.2 Family Grud; 7.3 Properties of Frud and Grud; 8 Open comprehension and open induction. 8.1 The ((. . .)) notation8.2 Open comprehension in K(Frud, Grud); 8.3 Open induction in K(Frud, Grud); 8.4 Short open induction; 9 Comprehension and induction via quantifier elimination; 9.1 Bounded quantifier elimination; 9.2 Skolem functions in K(F, G) and quantifierelimination; 9.3 Comprehension and induction for S01,b-formulas; 10 Skolem functions, switching lemma; 10.1 Switching lemma; 10.2 Tree model K(Ftree, Gtree); 11 Quantifier elimination in K(Ftree, Gtree); 11.1 Skolem functions; 11.2 Comprehension and induction for S01,b-formulas. 12 Witnessing, independence and definability in V0112.1 Witnessing AX <x EY <x S01,b-formulas; 12.2 Preservation of true sp11,b-sentences; 12.3 Circuit lower bound for parity; Part IV AC0(2) world; 13 Theory Q2V01; 13.1 Q2 quantifier and theory Q2V01; 13.2 Interpreting Q2 in structures; 14 Algebraic model; 14.1 Family Falg; 14.2 Family Galg; 14.3 Open comprehension and open induction; 15 Quantifier elimination and the interpretation of Q2; 15.1 Skolemization and the Razborov -- Smolensky method; 15.2 Interpretation of Q2 in front of an open formula. 15.3 Elimination of quantifiers and the interpretation of the Q2 quantifier15.4 Comprehension and induction for Q21,b0-formulas; 16 Witnessing and independence in Q2V01; 16.1 Witnessing AX <xEY <xA Z <xS01,b-formulas; 16.2 Preservation of true sp11,b-sentences; Part V Towards proof complexity; 17 Propositional proof systems; 17.1 Frege and Extended Frege systems; 17.2 Language with connective and constant-depth Frege systems; 18 Lengths-of-proofs lower bounds; 18.1 Formalization of the provability predicate; 18.2 Reflection principles; 18.3 Three conditions for a lower bound. |
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| dewey-full | 511.36 |
| dewey-hundreds | 500 - Natural sciences and mathematics |
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| discipline | Mathematik |
| format | Electronic eBook |
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| id | ZDB-4-EBA-ocn769341802 |
| illustrated | Not Illustrated |
| indexdate | 2024-11-27T13:18:10Z |
| institution | BVB |
| isbn | 9781139117333 1139117335 9781139127998 1139127993 9781139115162 1139115162 9781139107211 1139107216 |
| language | English |
| oclc_num | 769341802 |
| open_access_boolean | |
| owner | MAIN DE-863 DE-BY-FWS |
| owner_facet | MAIN DE-863 DE-BY-FWS |
| physical | 1 online resource (266 pages) |
| psigel | ZDB-4-EBA |
| publishDate | 2010 |
| publishDateSearch | 2010 |
| publishDateSort | 2010 |
| publisher | Cambridge University Press, |
| record_format | marc |
| series | London Mathematical Society Lecture Note Series, 382. |
| series2 | London Mathematical Society Lecture Note Series, 382 ; |
| spelling | Krajícek, Jan. Forcing with Random Variables and Proof Complexity. Cambridge : Cambridge University Press, 2010. 1 online resource (266 pages) text txt rdacontent computer c rdamedia online resource cr rdacarrier London Mathematical Society Lecture Note Series, 382 ; v. 382 Cover; Title; Copyright; Dedication; Contents; Preface; Acknowledgment; Introduction; Organization of the book; Remarks on the literature; Background; Part I Basics; 1 The definition of the models; 1.1 The ambient model of arithmetic; 1.2 The Boolean algebras; 1.3 The models K(F); 1.4 Valid sentences; 1.5 Possible generalizations; 2 Measure on B; 2.1 A metric on B; 2.2 From Boolean value to probability; 3 Witnessing quantifiers; 3.1 Propositional approximation of truth values; 3.2 Witnessing in definable families; 3.3 Definition by cases by open formulas; 3.4 Compact families. 3.5 Propositional computation of truth values4 The truth in N and the validity in K(F); Part II Second-order structures; 5 Structures K(F, G); 5.1 Language L2 and the hierarchy of bounded formulas; 5.2 Cut Mn, languages Ln and L2n; 5.3 Definition of the structures; 5.4 Equality of functions, extensionality and possible generalizations; 5.5 Absoluteness of ASb8-sentences of language Ln; Part III AC0 world; 6 Theories I?0, I?0(R) and V01; 7 Shallow Boolean decision tree model; 7.1 Family Frud; 7.2 Family Grud; 7.3 Properties of Frud and Grud; 8 Open comprehension and open induction. 8.1 The ((. . .)) notation8.2 Open comprehension in K(Frud, Grud); 8.3 Open induction in K(Frud, Grud); 8.4 Short open induction; 9 Comprehension and induction via quantifier elimination; 9.1 Bounded quantifier elimination; 9.2 Skolem functions in K(F, G) and quantifierelimination; 9.3 Comprehension and induction for S01,b-formulas; 10 Skolem functions, switching lemma; 10.1 Switching lemma; 10.2 Tree model K(Ftree, Gtree); 11 Quantifier elimination in K(Ftree, Gtree); 11.1 Skolem functions; 11.2 Comprehension and induction for S01,b-formulas. 12 Witnessing, independence and definability in V0112.1 Witnessing AX <x EY <x S01,b-formulas; 12.2 Preservation of true sp11,b-sentences; 12.3 Circuit lower bound for parity; Part IV AC0(2) world; 13 Theory Q2V01; 13.1 Q2 quantifier and theory Q2V01; 13.2 Interpreting Q2 in structures; 14 Algebraic model; 14.1 Family Falg; 14.2 Family Galg; 14.3 Open comprehension and open induction; 15 Quantifier elimination and the interpretation of Q2; 15.1 Skolemization and the Razborov -- Smolensky method; 15.2 Interpretation of Q2 in front of an open formula. 15.3 Elimination of quantifiers and the interpretation of the Q2 quantifier15.4 Comprehension and induction for Q21,b0-formulas; 16 Witnessing and independence in Q2V01; 16.1 Witnessing AX <xEY <xA Z <xS01,b-formulas; 16.2 Preservation of true sp11,b-sentences; Part V Towards proof complexity; 17 Propositional proof systems; 17.1 Frege and Extended Frege systems; 17.2 Language with connective and constant-depth Frege systems; 18 Lengths-of-proofs lower bounds; 18.1 Formalization of the provability predicate; 18.2 Reflection principles; 18.3 Three conditions for a lower bound. 19 PHP principle. A model-theoretic approach to bounded arithmetic and propositional proof complexity. Print version record. Includes bibliographical references (pages 236-242) and indexes. Computational complexity. http://id.loc.gov/authorities/subjects/sh85029473 Random variables. http://id.loc.gov/authorities/subjects/sh85111355 Mathematical analysis. http://id.loc.gov/authorities/subjects/sh85082116 Complexité de calcul (Informatique) Variables aléatoires. Analyse mathématique. MATHEMATICS Infinity. bisacsh MATHEMATICS Logic. bisacsh Computational complexity fast Mathematical analysis fast Random variables fast Print version: Krajícek, Jan. Forcing with Random Variables and Proof Complexity. Cambridge : Cambridge University Press, ©2010 9780521154338 London Mathematical Society Lecture Note Series, 382. FWS01 ZDB-4-EBA FWS_PDA_EBA https://search.ebscohost.com/login.aspx?direct=true&scope=site&db=nlebk&AN=399278 Volltext |
| spellingShingle | Krajícek, Jan Forcing with Random Variables and Proof Complexity. London Mathematical Society Lecture Note Series, 382. Cover; Title; Copyright; Dedication; Contents; Preface; Acknowledgment; Introduction; Organization of the book; Remarks on the literature; Background; Part I Basics; 1 The definition of the models; 1.1 The ambient model of arithmetic; 1.2 The Boolean algebras; 1.3 The models K(F); 1.4 Valid sentences; 1.5 Possible generalizations; 2 Measure on B; 2.1 A metric on B; 2.2 From Boolean value to probability; 3 Witnessing quantifiers; 3.1 Propositional approximation of truth values; 3.2 Witnessing in definable families; 3.3 Definition by cases by open formulas; 3.4 Compact families. 3.5 Propositional computation of truth values4 The truth in N and the validity in K(F); Part II Second-order structures; 5 Structures K(F, G); 5.1 Language L2 and the hierarchy of bounded formulas; 5.2 Cut Mn, languages Ln and L2n; 5.3 Definition of the structures; 5.4 Equality of functions, extensionality and possible generalizations; 5.5 Absoluteness of ASb8-sentences of language Ln; Part III AC0 world; 6 Theories I?0, I?0(R) and V01; 7 Shallow Boolean decision tree model; 7.1 Family Frud; 7.2 Family Grud; 7.3 Properties of Frud and Grud; 8 Open comprehension and open induction. 8.1 The ((. . .)) notation8.2 Open comprehension in K(Frud, Grud); 8.3 Open induction in K(Frud, Grud); 8.4 Short open induction; 9 Comprehension and induction via quantifier elimination; 9.1 Bounded quantifier elimination; 9.2 Skolem functions in K(F, G) and quantifierelimination; 9.3 Comprehension and induction for S01,b-formulas; 10 Skolem functions, switching lemma; 10.1 Switching lemma; 10.2 Tree model K(Ftree, Gtree); 11 Quantifier elimination in K(Ftree, Gtree); 11.1 Skolem functions; 11.2 Comprehension and induction for S01,b-formulas. 12 Witnessing, independence and definability in V0112.1 Witnessing AX <x EY <x S01,b-formulas; 12.2 Preservation of true sp11,b-sentences; 12.3 Circuit lower bound for parity; Part IV AC0(2) world; 13 Theory Q2V01; 13.1 Q2 quantifier and theory Q2V01; 13.2 Interpreting Q2 in structures; 14 Algebraic model; 14.1 Family Falg; 14.2 Family Galg; 14.3 Open comprehension and open induction; 15 Quantifier elimination and the interpretation of Q2; 15.1 Skolemization and the Razborov -- Smolensky method; 15.2 Interpretation of Q2 in front of an open formula. 15.3 Elimination of quantifiers and the interpretation of the Q2 quantifier15.4 Comprehension and induction for Q21,b0-formulas; 16 Witnessing and independence in Q2V01; 16.1 Witnessing AX <xEY <xA Z <xS01,b-formulas; 16.2 Preservation of true sp11,b-sentences; Part V Towards proof complexity; 17 Propositional proof systems; 17.1 Frege and Extended Frege systems; 17.2 Language with connective and constant-depth Frege systems; 18 Lengths-of-proofs lower bounds; 18.1 Formalization of the provability predicate; 18.2 Reflection principles; 18.3 Three conditions for a lower bound. Computational complexity. http://id.loc.gov/authorities/subjects/sh85029473 Random variables. http://id.loc.gov/authorities/subjects/sh85111355 Mathematical analysis. http://id.loc.gov/authorities/subjects/sh85082116 Complexité de calcul (Informatique) Variables aléatoires. Analyse mathématique. MATHEMATICS Infinity. bisacsh MATHEMATICS Logic. bisacsh Computational complexity fast Mathematical analysis fast Random variables fast |
| subject_GND | http://id.loc.gov/authorities/subjects/sh85029473 http://id.loc.gov/authorities/subjects/sh85111355 http://id.loc.gov/authorities/subjects/sh85082116 |
| title | Forcing with Random Variables and Proof Complexity. |
| title_auth | Forcing with Random Variables and Proof Complexity. |
| title_exact_search | Forcing with Random Variables and Proof Complexity. |
| title_full | Forcing with Random Variables and Proof Complexity. |
| title_fullStr | Forcing with Random Variables and Proof Complexity. |
| title_full_unstemmed | Forcing with Random Variables and Proof Complexity. |
| title_short | Forcing with Random Variables and Proof Complexity. |
| title_sort | forcing with random variables and proof complexity |
| topic | Computational complexity. http://id.loc.gov/authorities/subjects/sh85029473 Random variables. http://id.loc.gov/authorities/subjects/sh85111355 Mathematical analysis. http://id.loc.gov/authorities/subjects/sh85082116 Complexité de calcul (Informatique) Variables aléatoires. Analyse mathématique. MATHEMATICS Infinity. bisacsh MATHEMATICS Logic. bisacsh Computational complexity fast Mathematical analysis fast Random variables fast |
| topic_facet | Computational complexity. Random variables. Mathematical analysis. Complexité de calcul (Informatique) Variables aléatoires. Analyse mathématique. MATHEMATICS Infinity. MATHEMATICS Logic. Computational complexity Mathematical analysis Random variables |
| url | https://search.ebscohost.com/login.aspx?direct=true&scope=site&db=nlebk&AN=399278 |
| work_keys_str_mv | AT krajicekjan forcingwithrandomvariablesandproofcomplexity |