A graphic apology for symmetry and implicitness:
Gespeichert in:
| Beteiligte Personen: | , |
|---|---|
| Format: | Buch |
| Sprache: | Englisch |
| Veröffentlicht: |
Oxford [u.a.]
Oxford Univ. Press
2000
|
| Ausgabe: | 1. publ. |
| Schriftenreihe: | Oxford mathematical monographs
|
| Schlagwörter: | |
| Links: | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=013147030&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| Beschreibung: | Includes bibliographical references (p. [487]-495) and index |
| Umfang: | XVII, 501 S. |
| ISBN: | 0198507291 |
Internformat
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| 245 | 1 | 0 | |a A graphic apology for symmetry and implicitness |c Alessandra Carbone and Stephen Semmes |
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Datensatz im Suchindex
| _version_ | 1819352425484517376 |
|---|---|
| adam_text | Titel: A graphic apology for symmetry and implicitness
Autor: Carbone, Alessandra
Jahr: 2000
CONTENTS
1 Introduction 1
1.1 Examples of implicit descriptions 1
1.2 Formal proofs and cut elimination 4
1.3 Feasibility 5
1.4 Combinatorial models 7
1.5 Formal proofs and algorithmic complexity 8
1.6 The role of symmetry 9
1.7 Partial symmetries 9
1.8 Computational complexity 10
2 Morphisms in logic and complexity 15
2.1 Morphisms and formal proofs 15
2.2 Morphisms and monotonicity 16
2.3 Combinatorial proof systems 17
3 Exponential processes and formal proofs 19
3.1 Preliminaries 19
3.2 A process of branching 22
3.3 A stronger process of branching 24
3.4 Comparisons 28
3.5 The pigeon-hole principle 29
3.6 Proofs, sets, and cells 30
4 Graphs and their visibilities 37
4.1 Optical graphs 37
4.2 The definition of the visibility 38
4.3 Some examples 39
4.4 Visibility and depth 43
4.5 The canonical projection 45
4.6 Basic properties of the visibility 46
4.7 The size of the visibility 48
4.8 Formal proofs and logical flow graphs 50
4.9 Comparison with L-systems 53
4.10 Visibility in Riemannian manifolds 56
4.11 Universal covering spaces 65
4.12 Boolean circuits and expressions 76
4.13 Combinatorial dynamical systems 81
4.14 Exponential expansion 84
xiv
CONTENTS
89
91
104
105
109
109
Asymptotic growth of infinite visibilities
5.1 Introduction
5.2 When loops meet
5.3 When loops do not meet
5.4 Summary and remarks
5.5 Asymptotic geometry
Geometric aspects of cut elimination
6.1 Preliminary remarks
6.2 The process of cut elimination HI
6.3 A first scenario, and the breaking of cycles 114
6.4 A second scenario, and the breaking of focal pairs 115
6.5 A third scenario, and chains of focal pairs 115
6.6 The third scenario, continued 120
6.7 Chains of focal pairs in the second scenario 121
6.8 Recapitulation 122
6.9 Proofs without focal pairs 124
6.10 A fourth scenario, and the creation of focal pairs 125
6.11 Extensions of chains of focal pairs 125
6.12 Steady graphs and cut-free proofs 126
6.13 Steady graphs with oriented cycles 132
6.14 Steady horizons 133
6.15 A simplified model 136
6.16 Comparisons 139
6.17 A brief digression 140
6.18 Proofs with simple cuts 143
Feasibility graphs 155
7.1 Basic concepts 155
7.2 Extensions and comparisons 159
7.3 Some remarks about computability 161
7.4 Feasibility and visibility graphs 162
7.5 Upper bounds 154
7.6 Concrete examples K35
7.7 Measurements of complexity in groups 168
7.8 Trivial words in groups 170
7.9 Examples about numbers 175
7.10 Trees
7.11 Boolean circuits
7.12 Homoinorphisins and comparisons
Bounds for finite visibilities
8.1 The propagator rule
8.2 Visibilities within visibilities
8.3 The Calderon- Zygrnund decomposition
176
177
178
180
180
183
185
CONTENTS xv
8.4 The Corona decomposition 188
8.5 The derived graph 191
8.6 Extensions 192
8.7 A more direct counting argument 195
8.8 Exponential bounds for general graphs 200
8.9 The restrained visibility 207
8.10 Graphs with cycles 210
9 Some related computational questions 212
9.1 The size of the visibility 212
9.2 The visibility recognition problem 216
9.3 An implicit version 220
9.4 The visibility isomorphism problem 221
9.5 Computations with implicit descriptions 228
10 Mappings and graphs 231
10.1 Mappings and weak mappings 231
10.2 Computational questions 235
10.3 Local +-isomorphisms 236
10.4 Some interpretations 242
10.5 The local +-injection problem 243
10.6 A uniqueness result 247
10.7 Minimal representations 248
10.8 Mappings and effective witnesses 252
10.9 The visibility isomorphism problem 253
lO.lOMinimal representations and DP 257
10.11 Minimal folding graphs 258
10.12 Universal constructions 262
10.13The visibility spectrum 266
10.14 The local +-isomorphism problem 271
10.15 Comparisons with /c-provability 275
10.16A partial ordering between graphs 276
10.17Monotonicity properties 278
10.18Possible behavior of mappings 279
10.19PossibIe behavior of mappings, continued 282
H Mappings and comparisons 285
11.1 Locally +-stable mappings 285
11.2 Locally +-uniform mappings 287
11.3 Mappings and symmetry 288
11.4 Labelled graphs 289
11.5 Feasibility graphs 290
12 Adjacency matrices and counting 294
12.1 The adjacency matrix 294
12.2 Counting in the visibility 295
XVI
CONTENTS
12.3 Some concrete examples
12.4 Representation problems
12.5 Mappings and matrices
17 Groups and graphs
299
310
311
313
313
13 Duality and NP-completeness
13.1 The visibility mapping problem
13.2 Monotonicity and stability properties 31
13.3 The visibility surjoction problem 319
13.4 The visibility injection problem 324
14 Finite automata and regular languages 32^
14.1 Definitions and the subset construction 32
14.2 Geometric reformulations 332
14.3 An extended view 33 *
14.4 Markov languages 333
15 Constructions with graphs 333
15.1 Mappings and automata 333
15.2 Cartesian products and concatenation 339
15.3 Free products and positive closure 342
15.4 Unions and intersections 3-13
15.5 Fiber products (in general) 345
15.G Fiber products of graphs 348
15.7 Interpretations for automata 352
16 Stronger forms of recursion 355
1G.1 Feasible numbers 35G
1G.2 Combinatorial interpretations 358
1G.3 Feasibility graphs for feasibility graphs 3G1
1G.4 Correspondence with functions 3G9
1G.5 Implicit representations of functions 372
1G.G Functions and points 373
1G.7 Graphs and numbers 375
1G.8 Graphs and numbers, continued 378
1G.9 Rings and semirings 381
lG.lOFeasibility of sets 384
1G.11 Visual interpretations
lG.12Codings and sets
lG.130ther operations
lG.14Simulatioiis and conversions 395
lG.laSums and visibility graphs 397
1G.16 Back to formal proofs
389
390
392
399
404
17.1 Cay ley graphs and the word metric 404
17.2 Pause for some definitions
CONTENTS xvii
17.3 The Heisenberg groups 408
17.4 Geometry of Heisenberg groups 413
17.5 Automatic groups 418
17.6 Automatic structures for graphs 420
17.7 Between Cayley graphs and graphs in general 430
17.8 Scales and paths 430
17.9 Connections between scales and paths 431
17.10The Mellow traveller property 434
18 Extended notions of automata 436
18.1 Asynchronous automata 437
18.2 Heisenberg groups 440
18.3 Expanding automata 441
18.4 Tapes that cross 445
19 Geometry of scales in metric spaces 449
19.1 Metric spaces and length spaces 449
19.2 Discretizations of metric spaces 451
19.3 The scale-geometry graph 456
19.4 Conditions of bounded geometry 458
19.5 Automatic structures 460
19.6 Making choices 461
19.7 A geometric caveat 463
19.8 The doubling condition 464
20 The Corona decomposition revisited 465
20.1 Interesting paths 465
20.2 Reduced graphs 467
20.3 Crisp paths 470
20.4 A weak mapping between visibilities 473
20.5 Injectivity properties of the weak mapping 476
20.6 Bounds 477
Appendix
A Formal proofs: A brief review 480
A.l Sequent calculus 480
A.2 Cut elimination 483
A.3 The logical flow graph 484
References 4^
Index 496
|
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| dewey-raw | 511.3 |
| dewey-search | 511.3 |
| dewey-sort | 3511.3 |
| dewey-tens | 510 - Mathematics |
| discipline | Mathematik |
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| illustrated | Not Illustrated |
| indexdate | 2024-12-20T12:06:36Z |
| institution | BVB |
| isbn | 0198507291 |
| language | English |
| lccn | 0042739 |
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| physical | XVII, 501 S. |
| publishDate | 2000 |
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| publisher | Oxford Univ. Press |
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| spellingShingle | Carbone, Alessandra Semmes, Stephen A graphic apology for symmetry and implicitness Computational complexity Graph theory Group theory Machine theory Gruppentheorie (DE-588)4072157-7 gnd Graphentheorie (DE-588)4113782-6 gnd |
| subject_GND | (DE-588)4072157-7 (DE-588)4113782-6 |
| title | A graphic apology for symmetry and implicitness |
| title_auth | A graphic apology for symmetry and implicitness |
| title_exact_search | A graphic apology for symmetry and implicitness |
| title_full | A graphic apology for symmetry and implicitness Alessandra Carbone and Stephen Semmes |
| title_fullStr | A graphic apology for symmetry and implicitness Alessandra Carbone and Stephen Semmes |
| title_full_unstemmed | A graphic apology for symmetry and implicitness Alessandra Carbone and Stephen Semmes |
| title_short | A graphic apology for symmetry and implicitness |
| title_sort | a graphic apology for symmetry and implicitness |
| topic | Computational complexity Graph theory Group theory Machine theory Gruppentheorie (DE-588)4072157-7 gnd Graphentheorie (DE-588)4113782-6 gnd |
| topic_facet | Computational complexity Graph theory Group theory Machine theory Gruppentheorie Graphentheorie |
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