Bounded variable logics and counting :: a study in finite models /
Since their inception, the 'Perspectives in Logic' and 'Lecture Notes in Logic' series have published seminal works by leading logicians. Many of the original books in the series have been unavailable for years, but they are now in print once again. In this volume, the ninth publ...
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| Format: | Elektronisch E-Book |
| Sprache: | English |
| Veröffentlicht: |
Cambridge ; New York :
Cambridge University Press,
2017.
|
| Schriftenreihe: | Lecture notes in logic ;
9. |
| Schlagworte: | |
| Online-Zugang: | Volltext |
| Zusammenfassung: | Since their inception, the 'Perspectives in Logic' and 'Lecture Notes in Logic' series have published seminal works by leading logicians. Many of the original books in the series have been unavailable for years, but they are now in print once again. In this volume, the ninth publication in the 'Lecture Notes in Logic' series, Martin Otto gives an introduction to finite model theory that indicates the main ideas and lines of inquiry that motivate research in this area. Particular attention is paid to bounded variable infinitary logics, with and without counting quantifiers, related fixed-point logics, and the corresponding fragments of Ptime. The relations with Ptime exhibit the fruitful exchange between ideas from logic and from complexity theory that is characteristic of finite model theory. |
| Beschreibung: | 1 online resource (183 pages) |
| ISBN: | 9781316754719 1316754715 131675278X 9781316752784 |
Internformat
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| 490 | 1 | |a Lecture notes in logic ; |v 9 | |
| 520 | 8 | |a Since their inception, the 'Perspectives in Logic' and 'Lecture Notes in Logic' series have published seminal works by leading logicians. Many of the original books in the series have been unavailable for years, but they are now in print once again. In this volume, the ninth publication in the 'Lecture Notes in Logic' series, Martin Otto gives an introduction to finite model theory that indicates the main ideas and lines of inquiry that motivate research in this area. Particular attention is paid to bounded variable infinitary logics, with and without counting quantifiers, related fixed-point logics, and the corresponding fragments of Ptime. The relations with Ptime exhibit the fruitful exchange between ideas from logic and from complexity theory that is characteristic of finite model theory. | |
| 588 | 0 | |a Print version record. | |
| 505 | 0 | |a Cover; Half-title; Series information; Title page; Copyright information; Preface; Table of contents; 0. Introduction; 0.1 Finite Models, Logic and Complexity; 0.1.1 Logics for Complexity Classes; 0.1.2 Semantically Defined Classes; 0.1.3 Which Logics Are Natural?; 0.2 Natural Levels of Expressiveness; 0.2.1 Fixed-Point Logics and Their Counting Extensions; 0.2.2 The Framework of Infinitary Logic; 0.2.3 The Role of Order and Canonization ; 0.3 Guide to the Exposition; 1. Definitions and Preliminaries; 1.1 Structures and Types; 1.1.1 Structures; 1.1.2 Queries and Global Relations; 1.1.3 Logics | |
| 505 | 8 | |a 1.1.4 Types1.2 Algorithms on Structures; 1.2.1 Complexity Classes and Presentations; 1.2.2 Logics for Complexity Classes; 1.3 Some Particular Logics; 1.3.1 First-Order Logic and Infinitary Logic; 1.3.2 Fragments of Infinitary Logic; 1.3.3 Fixed-Point Logics; 1.3.4 Fixed-Point Logics and the L[sup(k)sub([infty][textomega])] ; 1.4 Types and Definability in the L[sup(k)sub([infty][textomega])] and C[sup(k)sub([infty][textomega])] ; 1.5 Interpretations; 1.5.1 Variants of Interpretations; 1.5.2 Examples; 1.5.3 Interpretations and Definability; 1.6 Lindstrom Quantifiers and Extensions ; 1.6.1 Cardinality Lindstrom Quantifiers | |
| 505 | 8 | |a 1.6.2 Aside on Uniform Families of Quantifiers1.7 Miscellaneous; 1.7.1 Canonization and Invariants; 1.7.2 Orderings and Pre-Orderings; 1.7.3 Lexicographic Orderings; 2. The Games and Their Analysis; 2.1 The Pebble Games for L[sup(k)sub([infty][textomega])] and C[sup(k)sub([infty][textomega])] ; 2.1.1 Examples and Applications; 2.1.2 Proof of Theorem 2.2; 2.1.3 Further Analysis of the C[sup(k)]-Game; 2.1.4 The Analogous Treatment for L[sup(k)]; 2.2 Colour Refinement and the Stable Colouring; 2.2.1 The Standard Case: Colourings of Finite Graphs; 2.2.2 Definability of the Stable Colouring | |
| 505 | 8 | |a 2.2.3 C[sup(2)sub([infty][textomega])] and the Stable Colouring 2.2.4 A Variant Without Counting; 2.3 Order in the Analysis of the Games; 2.3.1 The Internal View; 2.3.2 The External View; 2.3.3 The Analogous Treatment for L[sup(k)]; 3. The Invariants; 3.1 Complete Invariants for L[sup(k)] and C[sup(k)]; 3.2 The C[sup(k)]-Invariants; 3.3 The L[sup(k)]-Invariants; 3.4 Some Applications; 3.4.1 Fixed-Points and the Invariants; 3.4.2 The Abiteboul-Vianu Theorem; 3.4.3 Comparison of I[sub(C[sup(k)])] and I[sub(L[sup(k)])]; 3.5 A Partial Reduction to Two Variables; 4. Fixed-Point Logic with Counting | |
| 505 | 8 | |a 4.1 Definition of FP+C and PFP+C4.2 FP+C and the C[sup(k)]-Invariants; 4.3 The Separation from PTIME; 4.4 Other Characterizations of FP+C; 5. Related Lindstr[ddot(o)]m Extensions; 5.1 A Structural Padding Technique; 5.2 Cardinality Lindstrom Quantifiers ; 5.2.1 Proof of Theorem 5.1; 5.3 Aside on Further Applications; 6. Canonization Problems; 6.1 Canonization; 6.2 PTIME Canonization and Fragments of PTIME; 6.3 Canonization and Inversion of the Invariants; 6.4 A Reduction to Three Variables; 6.4.1 The Proof of Theorems 6.16 and 6.17; 6.4.2 Remarks on Further Reduction | |
| 650 | 0 | |a Model theory. |0 http://id.loc.gov/authorities/subjects/sh85086421 | |
| 650 | 0 | |a Computational complexity. |0 http://id.loc.gov/authorities/subjects/sh85029473 | |
| 650 | 6 | |a Théorie des modèles. | |
| 650 | 6 | |a Complexité de calcul (Informatique) | |
| 650 | 7 | |a MATHEMATICS |x General. |2 bisacsh | |
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| 776 | 0 | 8 | |i Print version: |a Otto, Martin. |t Bounded variable logics and counting. A study in finite models. |d Cambridge : Cambridge University Press 2016 |z 9781107167940 |w (OCoLC)962330962 |
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| author | Otto, Martin, 1961- |
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| contents | Cover; Half-title; Series information; Title page; Copyright information; Preface; Table of contents; 0. Introduction; 0.1 Finite Models, Logic and Complexity; 0.1.1 Logics for Complexity Classes; 0.1.2 Semantically Defined Classes; 0.1.3 Which Logics Are Natural?; 0.2 Natural Levels of Expressiveness; 0.2.1 Fixed-Point Logics and Their Counting Extensions; 0.2.2 The Framework of Infinitary Logic; 0.2.3 The Role of Order and Canonization ; 0.3 Guide to the Exposition; 1. Definitions and Preliminaries; 1.1 Structures and Types; 1.1.1 Structures; 1.1.2 Queries and Global Relations; 1.1.3 Logics 1.1.4 Types1.2 Algorithms on Structures; 1.2.1 Complexity Classes and Presentations; 1.2.2 Logics for Complexity Classes; 1.3 Some Particular Logics; 1.3.1 First-Order Logic and Infinitary Logic; 1.3.2 Fragments of Infinitary Logic; 1.3.3 Fixed-Point Logics; 1.3.4 Fixed-Point Logics and the L[sup(k)sub([infty][textomega])] ; 1.4 Types and Definability in the L[sup(k)sub([infty][textomega])] and C[sup(k)sub([infty][textomega])] ; 1.5 Interpretations; 1.5.1 Variants of Interpretations; 1.5.2 Examples; 1.5.3 Interpretations and Definability; 1.6 Lindstrom Quantifiers and Extensions ; 1.6.1 Cardinality Lindstrom Quantifiers 1.6.2 Aside on Uniform Families of Quantifiers1.7 Miscellaneous; 1.7.1 Canonization and Invariants; 1.7.2 Orderings and Pre-Orderings; 1.7.3 Lexicographic Orderings; 2. The Games and Their Analysis; 2.1 The Pebble Games for L[sup(k)sub([infty][textomega])] and C[sup(k)sub([infty][textomega])] ; 2.1.1 Examples and Applications; 2.1.2 Proof of Theorem 2.2; 2.1.3 Further Analysis of the C[sup(k)]-Game; 2.1.4 The Analogous Treatment for L[sup(k)]; 2.2 Colour Refinement and the Stable Colouring; 2.2.1 The Standard Case: Colourings of Finite Graphs; 2.2.2 Definability of the Stable Colouring 2.2.3 C[sup(2)sub([infty][textomega])] and the Stable Colouring 2.2.4 A Variant Without Counting; 2.3 Order in the Analysis of the Games; 2.3.1 The Internal View; 2.3.2 The External View; 2.3.3 The Analogous Treatment for L[sup(k)]; 3. The Invariants; 3.1 Complete Invariants for L[sup(k)] and C[sup(k)]; 3.2 The C[sup(k)]-Invariants; 3.3 The L[sup(k)]-Invariants; 3.4 Some Applications; 3.4.1 Fixed-Points and the Invariants; 3.4.2 The Abiteboul-Vianu Theorem; 3.4.3 Comparison of I[sub(C[sup(k)])] and I[sub(L[sup(k)])]; 3.5 A Partial Reduction to Two Variables; 4. Fixed-Point Logic with Counting 4.1 Definition of FP+C and PFP+C4.2 FP+C and the C[sup(k)]-Invariants; 4.3 The Separation from PTIME; 4.4 Other Characterizations of FP+C; 5. Related Lindstr[ddot(o)]m Extensions; 5.1 A Structural Padding Technique; 5.2 Cardinality Lindstrom Quantifiers ; 5.2.1 Proof of Theorem 5.1; 5.3 Aside on Further Applications; 6. Canonization Problems; 6.1 Canonization; 6.2 PTIME Canonization and Fragments of PTIME; 6.3 Canonization and Inversion of the Invariants; 6.4 A Reduction to Three Variables; 6.4.1 The Proof of Theorems 6.16 and 6.17; 6.4.2 Remarks on Further Reduction |
| ctrlnum | (OCoLC)982118534 |
| dewey-full | 511.3/3 |
| dewey-hundreds | 500 - Natural sciences and mathematics |
| dewey-ones | 511 - General principles of mathematics |
| dewey-raw | 511.3/3 |
| dewey-search | 511.3/3 |
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| discipline | Mathematik |
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| id | ZDB-4-EBA-ocn982118534 |
| illustrated | Not Illustrated |
| indexdate | 2024-11-27T13:27:46Z |
| institution | BVB |
| isbn | 9781316754719 1316754715 131675278X 9781316752784 |
| language | English |
| oclc_num | 982118534 |
| open_access_boolean | |
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| owner_facet | MAIN DE-863 DE-BY-FWS |
| physical | 1 online resource (183 pages) |
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| publishDate | 2017 |
| publishDateSearch | 2017 |
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| publisher | Cambridge University Press, |
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| series | Lecture notes in logic ; |
| series2 | Lecture notes in logic ; |
| spelling | Otto, Martin, 1961- author. http://id.loc.gov/authorities/names/n96112010 Bounded variable logics and counting : a study in finite models / Martin Otto. Cambridge ; New York : Cambridge University Press, 2017. 1 online resource (183 pages) text txt rdacontent computer c rdamedia online resource cr rdacarrier Lecture notes in logic ; 9 Since their inception, the 'Perspectives in Logic' and 'Lecture Notes in Logic' series have published seminal works by leading logicians. Many of the original books in the series have been unavailable for years, but they are now in print once again. In this volume, the ninth publication in the 'Lecture Notes in Logic' series, Martin Otto gives an introduction to finite model theory that indicates the main ideas and lines of inquiry that motivate research in this area. Particular attention is paid to bounded variable infinitary logics, with and without counting quantifiers, related fixed-point logics, and the corresponding fragments of Ptime. The relations with Ptime exhibit the fruitful exchange between ideas from logic and from complexity theory that is characteristic of finite model theory. Print version record. Cover; Half-title; Series information; Title page; Copyright information; Preface; Table of contents; 0. Introduction; 0.1 Finite Models, Logic and Complexity; 0.1.1 Logics for Complexity Classes; 0.1.2 Semantically Defined Classes; 0.1.3 Which Logics Are Natural?; 0.2 Natural Levels of Expressiveness; 0.2.1 Fixed-Point Logics and Their Counting Extensions; 0.2.2 The Framework of Infinitary Logic; 0.2.3 The Role of Order and Canonization ; 0.3 Guide to the Exposition; 1. Definitions and Preliminaries; 1.1 Structures and Types; 1.1.1 Structures; 1.1.2 Queries and Global Relations; 1.1.3 Logics 1.1.4 Types1.2 Algorithms on Structures; 1.2.1 Complexity Classes and Presentations; 1.2.2 Logics for Complexity Classes; 1.3 Some Particular Logics; 1.3.1 First-Order Logic and Infinitary Logic; 1.3.2 Fragments of Infinitary Logic; 1.3.3 Fixed-Point Logics; 1.3.4 Fixed-Point Logics and the L[sup(k)sub([infty][textomega])] ; 1.4 Types and Definability in the L[sup(k)sub([infty][textomega])] and C[sup(k)sub([infty][textomega])] ; 1.5 Interpretations; 1.5.1 Variants of Interpretations; 1.5.2 Examples; 1.5.3 Interpretations and Definability; 1.6 Lindstrom Quantifiers and Extensions ; 1.6.1 Cardinality Lindstrom Quantifiers 1.6.2 Aside on Uniform Families of Quantifiers1.7 Miscellaneous; 1.7.1 Canonization and Invariants; 1.7.2 Orderings and Pre-Orderings; 1.7.3 Lexicographic Orderings; 2. The Games and Their Analysis; 2.1 The Pebble Games for L[sup(k)sub([infty][textomega])] and C[sup(k)sub([infty][textomega])] ; 2.1.1 Examples and Applications; 2.1.2 Proof of Theorem 2.2; 2.1.3 Further Analysis of the C[sup(k)]-Game; 2.1.4 The Analogous Treatment for L[sup(k)]; 2.2 Colour Refinement and the Stable Colouring; 2.2.1 The Standard Case: Colourings of Finite Graphs; 2.2.2 Definability of the Stable Colouring 2.2.3 C[sup(2)sub([infty][textomega])] and the Stable Colouring 2.2.4 A Variant Without Counting; 2.3 Order in the Analysis of the Games; 2.3.1 The Internal View; 2.3.2 The External View; 2.3.3 The Analogous Treatment for L[sup(k)]; 3. The Invariants; 3.1 Complete Invariants for L[sup(k)] and C[sup(k)]; 3.2 The C[sup(k)]-Invariants; 3.3 The L[sup(k)]-Invariants; 3.4 Some Applications; 3.4.1 Fixed-Points and the Invariants; 3.4.2 The Abiteboul-Vianu Theorem; 3.4.3 Comparison of I[sub(C[sup(k)])] and I[sub(L[sup(k)])]; 3.5 A Partial Reduction to Two Variables; 4. Fixed-Point Logic with Counting 4.1 Definition of FP+C and PFP+C4.2 FP+C and the C[sup(k)]-Invariants; 4.3 The Separation from PTIME; 4.4 Other Characterizations of FP+C; 5. Related Lindstr[ddot(o)]m Extensions; 5.1 A Structural Padding Technique; 5.2 Cardinality Lindstrom Quantifiers ; 5.2.1 Proof of Theorem 5.1; 5.3 Aside on Further Applications; 6. Canonization Problems; 6.1 Canonization; 6.2 PTIME Canonization and Fragments of PTIME; 6.3 Canonization and Inversion of the Invariants; 6.4 A Reduction to Three Variables; 6.4.1 The Proof of Theorems 6.16 and 6.17; 6.4.2 Remarks on Further Reduction Model theory. http://id.loc.gov/authorities/subjects/sh85086421 Computational complexity. http://id.loc.gov/authorities/subjects/sh85029473 Théorie des modèles. Complexité de calcul (Informatique) MATHEMATICS General. bisacsh Modelos matemáticos embne Computational complexity fast Model theory fast has work: Bounded variable logics and counting (Text) https://id.oclc.org/worldcat/entity/E39PCGygyWCX4Q6Y9VgCf673FC https://id.oclc.org/worldcat/ontology/hasWork Print version: Otto, Martin. Bounded variable logics and counting. A study in finite models. Cambridge : Cambridge University Press 2016 9781107167940 (OCoLC)962330962 Lecture notes in logic ; 9. http://id.loc.gov/authorities/names/n93082404 FWS01 ZDB-4-EBA FWS_PDA_EBA https://search.ebscohost.com/login.aspx?direct=true&scope=site&db=nlebk&AN=1475793 Volltext |
| spellingShingle | Otto, Martin, 1961- Bounded variable logics and counting : a study in finite models / Lecture notes in logic ; Cover; Half-title; Series information; Title page; Copyright information; Preface; Table of contents; 0. Introduction; 0.1 Finite Models, Logic and Complexity; 0.1.1 Logics for Complexity Classes; 0.1.2 Semantically Defined Classes; 0.1.3 Which Logics Are Natural?; 0.2 Natural Levels of Expressiveness; 0.2.1 Fixed-Point Logics and Their Counting Extensions; 0.2.2 The Framework of Infinitary Logic; 0.2.3 The Role of Order and Canonization ; 0.3 Guide to the Exposition; 1. Definitions and Preliminaries; 1.1 Structures and Types; 1.1.1 Structures; 1.1.2 Queries and Global Relations; 1.1.3 Logics 1.1.4 Types1.2 Algorithms on Structures; 1.2.1 Complexity Classes and Presentations; 1.2.2 Logics for Complexity Classes; 1.3 Some Particular Logics; 1.3.1 First-Order Logic and Infinitary Logic; 1.3.2 Fragments of Infinitary Logic; 1.3.3 Fixed-Point Logics; 1.3.4 Fixed-Point Logics and the L[sup(k)sub([infty][textomega])] ; 1.4 Types and Definability in the L[sup(k)sub([infty][textomega])] and C[sup(k)sub([infty][textomega])] ; 1.5 Interpretations; 1.5.1 Variants of Interpretations; 1.5.2 Examples; 1.5.3 Interpretations and Definability; 1.6 Lindstrom Quantifiers and Extensions ; 1.6.1 Cardinality Lindstrom Quantifiers 1.6.2 Aside on Uniform Families of Quantifiers1.7 Miscellaneous; 1.7.1 Canonization and Invariants; 1.7.2 Orderings and Pre-Orderings; 1.7.3 Lexicographic Orderings; 2. The Games and Their Analysis; 2.1 The Pebble Games for L[sup(k)sub([infty][textomega])] and C[sup(k)sub([infty][textomega])] ; 2.1.1 Examples and Applications; 2.1.2 Proof of Theorem 2.2; 2.1.3 Further Analysis of the C[sup(k)]-Game; 2.1.4 The Analogous Treatment for L[sup(k)]; 2.2 Colour Refinement and the Stable Colouring; 2.2.1 The Standard Case: Colourings of Finite Graphs; 2.2.2 Definability of the Stable Colouring 2.2.3 C[sup(2)sub([infty][textomega])] and the Stable Colouring 2.2.4 A Variant Without Counting; 2.3 Order in the Analysis of the Games; 2.3.1 The Internal View; 2.3.2 The External View; 2.3.3 The Analogous Treatment for L[sup(k)]; 3. The Invariants; 3.1 Complete Invariants for L[sup(k)] and C[sup(k)]; 3.2 The C[sup(k)]-Invariants; 3.3 The L[sup(k)]-Invariants; 3.4 Some Applications; 3.4.1 Fixed-Points and the Invariants; 3.4.2 The Abiteboul-Vianu Theorem; 3.4.3 Comparison of I[sub(C[sup(k)])] and I[sub(L[sup(k)])]; 3.5 A Partial Reduction to Two Variables; 4. Fixed-Point Logic with Counting 4.1 Definition of FP+C and PFP+C4.2 FP+C and the C[sup(k)]-Invariants; 4.3 The Separation from PTIME; 4.4 Other Characterizations of FP+C; 5. Related Lindstr[ddot(o)]m Extensions; 5.1 A Structural Padding Technique; 5.2 Cardinality Lindstrom Quantifiers ; 5.2.1 Proof of Theorem 5.1; 5.3 Aside on Further Applications; 6. Canonization Problems; 6.1 Canonization; 6.2 PTIME Canonization and Fragments of PTIME; 6.3 Canonization and Inversion of the Invariants; 6.4 A Reduction to Three Variables; 6.4.1 The Proof of Theorems 6.16 and 6.17; 6.4.2 Remarks on Further Reduction Model theory. http://id.loc.gov/authorities/subjects/sh85086421 Computational complexity. http://id.loc.gov/authorities/subjects/sh85029473 Théorie des modèles. Complexité de calcul (Informatique) MATHEMATICS General. bisacsh Modelos matemáticos embne Computational complexity fast Model theory fast |
| subject_GND | http://id.loc.gov/authorities/subjects/sh85086421 http://id.loc.gov/authorities/subjects/sh85029473 |
| title | Bounded variable logics and counting : a study in finite models / |
| title_auth | Bounded variable logics and counting : a study in finite models / |
| title_exact_search | Bounded variable logics and counting : a study in finite models / |
| title_full | Bounded variable logics and counting : a study in finite models / Martin Otto. |
| title_fullStr | Bounded variable logics and counting : a study in finite models / Martin Otto. |
| title_full_unstemmed | Bounded variable logics and counting : a study in finite models / Martin Otto. |
| title_short | Bounded variable logics and counting : |
| title_sort | bounded variable logics and counting a study in finite models |
| title_sub | a study in finite models / |
| topic | Model theory. http://id.loc.gov/authorities/subjects/sh85086421 Computational complexity. http://id.loc.gov/authorities/subjects/sh85029473 Théorie des modèles. Complexité de calcul (Informatique) MATHEMATICS General. bisacsh Modelos matemáticos embne Computational complexity fast Model theory fast |
| topic_facet | Model theory. Computational complexity. Théorie des modèles. Complexité de calcul (Informatique) MATHEMATICS General. Modelos matemáticos Computational complexity Model theory |
| url | https://search.ebscohost.com/login.aspx?direct=true&scope=site&db=nlebk&AN=1475793 |
| work_keys_str_mv | AT ottomartin boundedvariablelogicsandcountingastudyinfinitemodels |