Bounded variable logics and counting: a study in finite models
Saved in:
| Main Author: | |
|---|---|
| Format: | Thesis Book |
| Language: | English |
| Published: |
Berlin [u.a.]
Springer
1997
|
| Series: | Lecture notes in logic
9 |
| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Physical Description: | IX, 183 S. graph. Darst. |
| ISBN: | 3540620370 |
Staff View
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Record in the Search Index
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| adam_text | Table of Contents
Preface V
0. Introduction 1
0.1 Finite Models, Logic and Complexity 1
0.1.1 Logics for Complexity Classes 1
0.1.2 Semantically Defined Classes 4
0.1.3 Which Logics Are Natural? 7
0.2 Natural Levels of Expressiveness 7
0.2.1 Fixed Point Logics and Their Counting Extensions ... 8
0.2.2 The Framework of Infinitary Logic 9
0.2.3 The Role of Order and Canonization 11
0.3 Guide to the Exposition 12
1. Definitions and Preliminaries 15
1.1 Structures and Types 15
1.1.1 Structures 15
1.1.2 Queries and Global Relations 17
1.1.3 Logics 18
1.1.4 Types 19
1.2 Algorithms on Structures 21
1.2.1 Complexity Classes and Presentations 22
1.2.2 Logics for Complexity Classes 23
1.3 Some Particular Logics 24
1.3.1 First Order Logic and Infinitary Logic 24
1.3.2 Fragments of Infinitary Logic 25
1.3.3 Fixed Point Logics 30
1.3.4 Fixed Point Logics and the L^,w 33
1.4 Types and Definability in the L^w and C*^ 35
1.5 Interpretations 38
1.5.1 Variants of Interpretations 38
1.5.2 Examples 40
1.5.3 Interpretations and Definability 41
VIII Table of Contents
1.6 Lindstrom Quantifiers and Extensions 43
1.6.1 Cardinality Lindstrom Quantifiers 43
1.6.2 Aside on Uniform Families of Quantifiers 44
1.7 Miscellaneous 47
1.7.1 Canonization and Invariants 47
1.7.2 Orderings and Pre Orderings 49
1.7.3 Lexicographic Orderings 49
2. The Games and Their Analysis 51
2.1 The Pebble Games for £ „, and C£,w 51
2.1.1 Examples and Applications 54
2.1.2 Proof of Theorem 2.2 60
2.1.3 Further Analysis of the C* Game 62
2.1.4 The Analogous Treatment for Lk 66
2.2 Colour Refinement and the Stable Colouring 67
2.2.1 The Standard Case: Colourings of Finite Graphs 67
2.2.2 Definability of the Stable Colouring 68
2.2.3 C^ and the Stable Colouring 71
2.2.4 A Variant Without Counting 72
2.3 Order in the Analysis of the Games 73
2.3.1 The Internal View 74
2.3.2 The External View 76
2.3.3 The Analogous Treatment for L* 77
3. The Invariants 79
3.1 Complete Invariants for Lk and Ck 80
3.2 The C* Invariants 81
3.3 The Lfc Invariants 85
3.4 Some Applications 87
3.4.1 Fixed Points and the Invariants 87
3.4.2 The Abiteboul Vianu Theorem 90
3.4.3 Comparison of /c* and ILk 91
3.5 A Partial Reduction to Two Variables 93
4. Fixed Point Logic with Counting 97
4.1 Definition of FP+C and PFP+C 98
4.2 FP+C and the Ck Invariants 106
4.3 The Separation from Ptime 109
4.4 Other Characterizations of FP+C HI
5. Related Lindstrom Extensions 115
5.1 A Structural Padding Technique H7
5.2 Cardinality Lindstrom Quantifiers I24
5.2.1 Proof of Theorem 5.1 125
5.3 Aside on Further Applications 128
Table of Contents IX
6. Canonization Problems 131
6.1 Canonization 131
6.2 Ptime Canonization and Fragments of Ptime 134
6.3 Canonization and Inversion of the Invariants 136
6.4 A Reduction to Three Variables 139
6.4.1 The Proof of Theorems 6.16 and 6.17 141
6.4.2 Remarks on Further Reduction 147
7. Canonization for Two Variables 149
7.1 Game Tableaux and the Inversion Problem 150
7.1.1 Modularity of Realizations 156
7.2 Realizations for 7C2 160
7.2.1 Necessary Conditions 160
7.2.2 Realizations of the Off Diagonal Boxes 162
7.2.3 Magic Squares 163
7.2.4 Realizations of the Diagonal Boxes 166
7.3 Realizations for II* 169
7.3.1 Necessary and Sufficient Conditions 169
7.3.2 On the Special Nature of Two Variables 174
Bibliography 177
Index 181
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| indexdate | 2024-07-09T18:03:28Z |
| institution | BVB |
| isbn | 3540620370 |
| language | English |
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| physical | IX, 183 S. graph. Darst. |
| publishDate | 1997 |
| publishDateSearch | 1997 |
| publishDateSort | 1997 |
| publisher | Springer |
| record_format | marc |
| series | Lecture notes in logic |
| series2 | Lecture notes in logic |
| spelling | Otto, Martin Verfasser aut Bounded variable logics and counting a study in finite models Martin Otto Berlin [u.a.] Springer 1997 IX, 183 S. graph. Darst. txt rdacontent n rdamedia nc rdacarrier Lecture notes in logic 9 Teilw. zugl.: Aachen, Rheinisch-Westfälische Techn. Hochsch., Habil.-Schr., 1995 Logik (DE-588)4036202-4 gnd rswk-swf Modelltheorie (DE-588)4114617-7 gnd rswk-swf (DE-588)4113937-9 Hochschulschrift gnd-content Modelltheorie (DE-588)4114617-7 s DE-604 Logik (DE-588)4036202-4 s Lecture notes in logic 9 (DE-604)BV008909514 9 HBZ Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=007413698&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Otto, Martin Bounded variable logics and counting a study in finite models Lecture notes in logic Logik (DE-588)4036202-4 gnd Modelltheorie (DE-588)4114617-7 gnd |
| subject_GND | (DE-588)4036202-4 (DE-588)4114617-7 (DE-588)4113937-9 |
| 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 | Logik (DE-588)4036202-4 gnd Modelltheorie (DE-588)4114617-7 gnd |
| topic_facet | Logik Modelltheorie Hochschulschrift |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=007413698&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
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