The strange logic of random graphs: With 13 Figures
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| Main Author: | |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
Berlin, [u.a.]
Springer
[2001]
|
| Series: | Algorithms and Combinatorics
22 |
| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Physical Description: | X, 168 Seiten Illustrationen |
| ISBN: | 3540416544 9783540416548 |
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MARC
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| 100 | 1 | |a Spencer, Joel H. |d 1946- |0 (DE-588)133091546 |4 aut | |
| 245 | 1 | 0 | |a The strange logic of random graphs |b With 13 Figures |c Joel Spencer |
| 264 | 1 | |a Berlin, [u.a.] |b Springer |c [2001] | |
| 264 | 4 | |c © 2001 | |
| 300 | |a X, 168 Seiten |b Illustrationen | ||
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| 490 | 1 | |a Algorithms and Combinatorics |v 22 | |
| 650 | 4 | |a Zufallsgraph - Mathematische Logik | |
| 650 | 4 | |a Logic, Symbolic and mathematical | |
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Record in the Search Index
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| adam_text |
Contents
Part I. Beginnings
0. Two Starting Examples 3
0.1 A Blend of Probability, Logic and Combinatorics 3
0.2 A Random Unary Predicate 8
0.3 Some Comments on References 11
1. Preliminaries 13
1.1 What is the Random Graph G(n,p)7 13
1.1.1 The Erdos Renyi Evolution 14
1.1.2 The Appearance of Small Subgraphs 15
1.2 What is a First Order Theory? 15
1.3 Extension Statements and Rooted Graphs 17
1.4 What is a Zero One Law? 18
1.5 Almost Sure Theories and Complete Theories 20
1.6 Countable Models 20
2. The Ehrenfeucht Game 23
2.1 The Rules of the Game 23
2.2 Equivalence Classes and Ehrenfeucht Value 27
2.3 Connection to First Order Sentences 31
2.4 Inside Outside Strategies 33
2.5 The Bridge to Zero One Laws 37
2.6 Other Structures 39
2.6.1 General First Order Structures 39
2.6.2 The Simple Case of Total Order 40
2.6.3 fc Similar Neighborhoods 42
Part II. Random Graphs
3. Very Sparse Graphs 49
3.1 The Void 50
3.2 On the fc th Day 50
3.3 On Day w 51
VIII Contents
3.3.1 An Excursion into Rooted Trees 51
3.3.2 Two Consequences 55
3.4 Past the Double Jump 56
3.5 Beyond Connectivity 57
3.6 Limiting Probabilities 58
3.6.1 A General Result on Limiting Probabilities 58
3.6.2 In the Beginning 59
3.6.3 On the jfc th Day 60
3.6.4 At the Threshold of Connectivity 61
3.7 The Double Jump in the First Order World 62
3.7.1 Poisson Childbearing 63
3.7.2 Almost Completing the Almost Sure Theory 65
4. The Combinatorics of Rooted Graphs 69
4.1 Sparse, Dense, Rigid, Safe 69
4.2 The i Closure 73
4.3 The Finite Closure Theorem 74
5. The Janson Inequality 79
5.1 Extension Statements 80
5.2 Counting Extensions 82
5.3 Generic Extension 85
6. The Main Theorem 87
6.1 The Look Ahead Strategy 87
6.1.1 The Final Move 88
6.1.2 The Core Argument (Middle Moves) 88
6.1.3 The First Move 89
6.2 The Original Argument 90
7. Countable Models 93
7.1 An Axiomatization for Ta 93
7.1.1 The Schema 93
7.1.2 Completeness Proof 93
7.1.3 The Truth Game 95
7.2 Countable Models 97
7.2.1 Construction 97
7.2.2 Uniqueness of the Model 99
7.2.3 NonUniqueness of the Model 100
7.3 A Continuum of Complete Theories 102
8. Near Rational Powers of n 103
8.1 Infinitely Many Ups and Downs 103
8.1.1 In the Second Order World 103
8.1.2 Replacing Second Order by First Order 105
Contents IX
8.1.3 Are First Order Properties Natural? 107
8.2 Existence of Finite Models 108
8.3 NonSeparability and NonConvergence 109
8.3.1 Representing All Finite Graphs 110
8.3.2 NonSeparability Ill
8.3.3 Arithmetization 112
8.3.4 NonConvergence 113
8.4 The Last Threshold 115
8.4.1 Just Past n a: The Theory T~ 115
8.4.2 Just Past n~a: A Zero One Law 117
Part III. Extras
9. A Dynamic View 121
9.1 More Zero One Laws 121
9.1.1 Near Irrational Powers 121
9.1.2 Dense Random Graphs 122
9.2 The Limit Function 122
9.2.1 Definition 122
9.2.2 Look Ahead Functions 123
9.2.3 Well Ordered Discontinuities 124
9.2.4 Underapproximation sequences 125
9.2.5 Determination in PH 127
10. Strings 131
10.1 Models and Language 131
10.2 Ehrenfeucht Redux 132
10.2.1 The Rules 132
10.2.2 The Semigroup 133
10.2.3 Long Strings 133
10.3 Persistent and Transient 134
10.4 Persistent Strings 136
10.5 Random Strings 137
10.6 Circular Strings 138
10.7 Sparse Unary Predicate 140
11. Stronger Logics 145
11.1 Ordered Graphs 145
11.1.1 Arithmetization 145
11.1.2 Dance Marathon 147
11.1.3 Slow Oscillation 148
11.2 Existential Monadic 149
X Contents
12. Three Final Examples 153
12.1 Random Functions 153
12.2 Distance Random G(n, p) 155
12.2.1 Without Order 155
12.2.2 With Order 157
12.3 Random Lifts 160
Bibliography 165
Index 167 |
| any_adam_object | 1 |
| author | Spencer, Joel H. 1946- |
| author_GND | (DE-588)133091546 |
| author_facet | Spencer, Joel H. 1946- |
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| author_sort | Spencer, Joel H. 1946- |
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| callnumber-label | QA166 |
| callnumber-raw | QA166.17 |
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| callnumber-subject | QA - Mathematics |
| classification_rvk | SK 820 SK 890 SK 170 |
| ctrlnum | (OCoLC)248654608 (DE-599)BVBBV013717459 |
| dewey-full | 511.5 |
| dewey-hundreds | 500 - Natural sciences and mathematics |
| dewey-ones | 511 - General principles of mathematics |
| dewey-raw | 511.5 |
| dewey-search | 511.5 |
| dewey-sort | 3511.5 |
| dewey-tens | 510 - Mathematics |
| discipline | Mathematik |
| format | Book |
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| indexdate | 2025-02-11T07:00:10Z |
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| isbn | 3540416544 9783540416548 |
| language | English |
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| spelling | Spencer, Joel H. 1946- (DE-588)133091546 aut The strange logic of random graphs With 13 Figures Joel Spencer Berlin, [u.a.] Springer [2001] © 2001 X, 168 Seiten Illustrationen txt rdacontent n rdamedia nc rdacarrier Algorithms and Combinatorics 22 Zufallsgraph - Mathematische Logik Logic, Symbolic and mathematical Random graphs Mathematische Logik (DE-588)4037951-6 gnd rswk-swf Zufallsgraph (DE-588)4277661-2 gnd rswk-swf Zufallsgraph (DE-588)4277661-2 s Mathematische Logik (DE-588)4037951-6 s DE-604 Erscheint auch als Online-Ausgabe 978-3-662-04538-1 Algorithms and Combinatorics 22 (DE-604)BV000617357 22 HBZ Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=009373846&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Spencer, Joel H. 1946- The strange logic of random graphs With 13 Figures Algorithms and Combinatorics Zufallsgraph - Mathematische Logik Logic, Symbolic and mathematical Random graphs Mathematische Logik (DE-588)4037951-6 gnd Zufallsgraph (DE-588)4277661-2 gnd |
| subject_GND | (DE-588)4037951-6 (DE-588)4277661-2 |
| title | The strange logic of random graphs With 13 Figures |
| title_auth | The strange logic of random graphs With 13 Figures |
| title_exact_search | The strange logic of random graphs With 13 Figures |
| title_full | The strange logic of random graphs With 13 Figures Joel Spencer |
| title_fullStr | The strange logic of random graphs With 13 Figures Joel Spencer |
| title_full_unstemmed | The strange logic of random graphs With 13 Figures Joel Spencer |
| title_short | The strange logic of random graphs |
| title_sort | the strange logic of random graphs with 13 figures |
| title_sub | With 13 Figures |
| topic | Zufallsgraph - Mathematische Logik Logic, Symbolic and mathematical Random graphs Mathematische Logik (DE-588)4037951-6 gnd Zufallsgraph (DE-588)4277661-2 gnd |
| topic_facet | Zufallsgraph - Mathematische Logik Logic, Symbolic and mathematical Random graphs Mathematische Logik Zufallsgraph |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=009373846&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
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