Higher-order computability:
Saved in:
| Main Authors: | , |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
Berlin ; Heidelberg
Springer
[2015]
|
| Series: | Theory and applications of computability
|
| Subjects: | |
| Online Access: | Inhaltstext Inhaltsverzeichnis |
| Item Description: | Literaturverzeichnis Seite 547 - 559 |
| Physical Description: | xvi, 571 Seiten |
| ISBN: | 3662479915 9783662479919 |
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CONTENTS
PART I BACKGROUND AND GENERAL THEORY
1 INTRODUCTION AND MOTIVATIONS 3
1.1 COMPUTABILITY MODELS 4
1.1.1 DATA AND DATATYPES 4
1.1.2 COMPUTABLE OPERATIONS 7
1.1.3 EXAMPLES OF COMPUTABILITY MODELS 8
1.1.4 SIMULATING ONE MODEL IN ANOTHER 13
1.1.5 EQUIVALENCES OF COMPUTABILITY MODELS 16
1.1.6 HIGHER-ORDER MODELS 18
1.2 MOTIVATIONS AND APPLICATIONS 21
1.2.1 INTERPRETATIONS OF LOGICAL SYSTEMS 21
1.2.2 EXTRACTING COMPUTATIONAL CONTENT FROM PROOFS 24
1.2.3 THEORY OF PROGRAMMING LANGUAGES 25
1.2.4 HIGHER-ORDER PROGRAMS AND PROGRAM CONSTRUCTION 27
1.3 SCOPE AND OUTLINE OF THE BOOK 27
1.3.1 SCOPE OF THE BOOK 28
1.3.2 CHAPTER CONTENTS AND INTERDEPENDENCES 30
1.3.3 STYLE AND PREREQUISITES 32
2 HISTORICAL SURVEY 35
2.1 FORMATIVE INGREDIENTS 35
2.1.1 COMBINATORY LOGIC AND A-CALCULUS 35
2.1.2 RELATIVE COMPUTABILITY AND SECOND-ORDER FUNCTIONALS 37
2.1.3 PROOF THEORY AND SYSTEM T 38
2.2 FULL SET-THEORETIC MODELS 38
2.2.1 KLEENE 39
2.2.2 PLATEK 40
2.2.3 KLEENE AGAIN 41
2.3 CONTINUOUS AND EFFECTIVE MODELS 42
2.3.1 KLEENE-KREISEL 42
XI
HTTP://D-NB.INFO/1072559315
XII CONTENTS
2.3.2 SCOTT-ERSHOV 44
2.3.3 LCF AS A PROGRAMMING LANGUAGE 46
2.3.4 THE QUEST FOR SEQUENTIALITY 47
2.4 THE THIRD MILLENNIUM 48
2.4.1 A UNIFYING FRAMEWORK 48
2.4.2 NESTED SEQUENTIAL PROCEDURES 49
2.5 SUMMARY 50
3 THEORY OF COMPUTABILITY MODELS 51
3.1 HIGHER-ORDER COMPUTABILITY MODELS 52
3.1.1 COMPUTABILITY MODELS 52
3.1.2 EXAMPLES OF COMPUTABILITY MODELS 54
3.1.3 WEAKLY CARTESIAN CLOSED MODELS 56
3.1.4 HIGHER-ORDER MODELS 58
3.1.5 TYPED PARTIAL COMBINATORY ALGEBRAS 61
3.1.6 LAX MODELS 64
3.1.7 TYPE WORLDS 66
3.2 FURTHER EXAMPLES OF HIGHER-ORDER MODELS 68
3.2.1 KLEENE'S SECOND MODEL 68
3.2.2 THE PARTIAL CONTINUOUS FUNCTIONALS 70
3.2.3 TYPED A-CALCULI 73
3.2.4 THE VAN OOSTEN MODEL 76
3.2.5 NESTED SEQUENTIAL PROCEDURES 78
3.3 COMPUTATIONAL STRUCTURE IN HIGHER-ORDER MODELS 82
3.3.1 COMBINATORY COMPLETENESS 82
3.3.2 PAIRING 85
3.3.3 BOOLEANS 86
3.3.4 NUMERALS 87
3.3.5 RECURSION AND MINIMIZATION 89
3.3.6 THE CATEGORY OF ASSEMBLIES 91
3.4 SIMULATIONS BETWEEN COMPUTABILITY MODELS 93
3.4.1 SIMULATIONS AND TRANSFORMATIONS 93
3.4.2 SIMULATIONS AND ASSEMBLIES 97
3.5 EXAMPLES OF SIMULATIONS AND TRANSFORMATIONS 99
3.5.1 EFFECTIVE AND CONTINUOUS MODELS 105
3.6 GENERAL CONSTRUCTIONS ON MODELS 106
3.6.1 QUOTIENTS 106
3.6.2 THE SUBSET CONSTRUCTION 108
3.6.3 PARTIAL EQUIVALENCE RELATIONS 110
3.6.4 TYPE STRUCTURES 113
3.7 PROSPECTUS 114
CONTENTS XIII
4 THEORY OF A-ALGEBRAS 115
4.1 TYPED A-ALGEBRAS 116
4.1.1 INTERPRETATIONS OF A-CALCULUS 116
4.1.2 THE INTERNAL INTERPRETATION 119
4.1.3 CATEGORICAL A-ALGEBRAS 121
4.1.4 THE KAROUBI ENVELOPE 127
4.1.5 RETRACTION SUBALGEBRAS 131
4.2 TYPES AND RETRACTIONS 133
4.2.1 PUREIYPES 135
4.2.2 PRODUCT TYPES 138
4.2.3 FURTHER TYPE STRUCTURE IN TOTAL SETTINGS 142
4.2.4 PARTIALITY, LIFTING AND STRICTNESS 144
4.2.5 UNIVERSAL TYPES 149
4.3 PARTIAL A-ALGEBRAS 150
4.4 PRELOGICAL RELATIONS AND THEIR APPLICATIONS 156
4.4.1 PRELOGICAL AND LOGICAL RELATIONS 157
4.4.2 APPLICATIONS TO SYSTEM T 159
4.4.3 PURE TYPES REVISITED 163
PART II PARTICULAR MODELS
5 KLEENE COMPUTABILITY IN A TOTAL SETTING 167
5.1 DEFINITIONS AND BASIC THEORY 169
5.1.1 KLEENE'S DEFINITION VIA S1-S9 169
5.1.2 BASIC PROPERTIES 174
5.1.3 A A-CALCULUS PRESENTATION 178
5.1.4 MODELS ARISING FROM KLEENE COMPUTABILITY 184
5.1.5 RESTRICTED COMPUTABILITY NOTIONS 186
5.2 COMPUTATIONS, TOTALITY AND LOGICAL COMPLEXITY 188
5.2.1 LOGICAL COMPLEXITY CLASSES 188
5.2.2 TOP-DOWN COMPUTATION TREES 192
5.2.3 TOTALITY AND THE MAXIMAL MODEL 198
5.3 EXISTENTIAL QUANTIFIERS AND EFFECTIVE CONTINUITY 203
5.4 COMPUTABILITY IN NORMAL FUNCTIONALS 208
6 NESTED SEQUENTIAL PROCEDURES 211
6.1 THE NSP MODEL: BASIC THEORY 212
6.1.1 CONSTRUCTION OF THE MODEL 212
6.1.2 THE EVALUATION THEOREM 216
6.1.3 INTERPRETATION OF TYPED A-CALCULUS 225
6.1.4 THE EXTENSIONAL QUOTIENT 229
6.1.5 SIMULATING NSPS IN B 230
6.1.6 CALL-BY-VALUE TYPES 232
6.1.7 AN UNTYPED APPROACH 232
6.2 NSPS AND KLEENE COMPUTABILITY 234
XIV CONTENTS
6.2.1 NSPS AND KLEENE EXPRESSIONS 234
6.2.2 INTERPRETING NSPS IN TOTAL MODELS 235
6.3 SUBSTRUCTURES OF SP 239
6.3.1 FINITE AND WELL-FOUNDED PROCEDURES 239
6.3.2 LEFT-WELL-FOUNDED PROCEDURES 242
6.3.3 LEFT-BOUNDED PROCEDURES 246
6.3.4 EXPRESSIVE POWER OF T + MIN AND KLEX"
110
250
6.3.5 TOTAL MODELS REVISITED 259
6.4 KLEENE COMPUTABILITY IN PARTIAL MODELS 261
6.4.1 THREE INTERPRETATIONS OF KLEENE COMPUTABILITY 262
6.4.2 LEAST FIXED POINTS 267
6.4.3 RELATING PARTIAL AND TOTAL MODELS 272
6.4.4 KLEENE COMPUTABILITY VIA INDUCTIVE DEFINITIONS 276
7 PCF AND ITS MODELS 279
7.1 INTERPRETATIONS OF PCF 281
7.1.1 MODELS OF PCF 282
7.1.2 MATHEMATICAL STRUCTURE IN MODELS OF PCF 285
7.1.3 SOUNDNESS AND ADEQUACY 287
7.1.4 OBSERVATIONAL EQUIVALENCE AND FULL ABSTRACTION 289
7.1.5 COMPLETENESS OF PCF*
2
+ BYVAL FOR SEQUENTIAL PROCEDURES . 293
7.1.6 RELATIONSHIP TO KLEENE EXPRESSIONS 300
7.1.7 INTERPRETATION OF NSPS REVISITED 301
7.2 VARIANTS AND TYPE EXTENSIONS OF PCF 304
7.2.1 FURTHER TYPE STRUCTURE IN MODELS OF PCF 304
7.2.2 CALL-BY-VALUE PCF 306
7.3 EXAMPLES AND NON-EXAMPLES OF PCF-DEFINABLE OPERATIONS 309
7.3.1 COUNTEREXAMPLES TO FULL ABSTRACTION 310
7.3.2 THE POWER OF NESTING 312
7.3.3 BAR RECURSION 313
7.3.4 THE FAN FUNCTIONAL 316
7.3.5 SELECTION FUNCTIONS AND QUANTIFIERS 319
7.4 GAME MODELS 323
7.5 SEQUENTIAL FUNCTIONALS 329
7.5.1 BASIC PROPERTIES 330
7.5.2 SOME PARTIAL CHARACTERIZATIONS 332
7.5.3 EFFECTIVE PRESENTABILITY AND LOADER'S THEOREM 338
7.6 ORDER-THEORETIC STRUCTURE OF THE SEQUENTIAL FUNCTIONALS 342
7.7 ABSENCE OF A UNIVERSAL TYPE 346
8 THE TOTAL CONTINUOUS FUNCTIONALS 349
8.1 DEFINITION AND BASIC THEORY 352
8.1.1 CONSTRUCTION VIA PC 352
8.1.2 THE DENSITY THEOREM 354
8.1.3 TOPOLOGICAL ASPECTS 359
CONTENTS XV
8.1.4 COMPUTABILITY IN CT 363
8.2 ALTERNATIVE CHARACTERIZATIONS 367
8.2.1 KLEENE'S CONSTRUCTION VIA ASSOCIATES 367
8.2.2 THE MODIFIED ASSOCIATE CONSTRUCTION 372
8.2.3 LIMIT SPACES 374
8.2.4 COMPACTLY GENERATED SPACES 381
8.2.5 FILTER SPACES 382
8.2.6 CT AS A REDUCED PRODUCT 383
8.3 EXAMPLES OF CONTINUOUS FUNCTIONALS 384
8.3.1 BAR RECURSION 384
8.3.2 THE FAN FUNCTIONAL 386
8.3.3 THE R FUNCTIONAL 388
8.4 FINE STRUCTURE OF CT 390
8.4.1 CHARACTERIZATIONS OF COMPACT SETS 390
8.4.2 THE KREISEL REPRESENTATION THEOREM 396
8.5 KLEENE COMPUTABILITY AND JI-COMPUTABILITY OVER CT 399
8.5.1 RELATIVE KLEENE COMPUTABILITY 399
8.5.2 RELATIVE/X-COMPUTABILITY 411
8.5.3 COMPUTABLY ENUMERABLE DEGREES OF FUNCTIONALS 417
8.6 COMPUTING WITH REALIZERS 421
8.6.1 SEQUENTIAL COMPUTABILITY AND TOTAL FUNCTIONALS 423
8.6.2 THE UBIQUITY THEOREM 428
9 THE HEREDITARILY EFFECTIVE OPERATIONS 431
9.1 CONSTRUCTIONS BASED ON CONTINUITY 432
9.1.1 HEREDITARILY EFFECTIVE CONTINUOUS FUNCTIONALS 432
9.1.2 THE KLEENE TREE 435
9.2 ALTERNATIVE CHARACTERIZATIONS 440
9.2.1 THE KREISEL-LACOMBE-SHOENFIELD THEOREM 440
9.2.2 THE MODIFIED EXTENSIONAL COLLAPSE OF K\ 446
9.2.3 UBIQUITY OF HEO 450
9.3 SOME NEGATIVE RESULTS 451
9.4 LOGICAL COMPLEXITY 454
9.5 OTHER RELATIVIZATIONS OF THE CONTINUOUS FUNCTIONALS 458
10 THE PARTIAL CONTINUOUS FUNCTIONALS 463
10.1 ORDER-THEORETIC AND TOPOLOGICAL CHARACTERIZATIONS 464
10.1.1 DEFINITION AND ORDER-THEORETIC STRUCTURE 464
10.1.2 TOPOLOGICAL AND LIMIT CHARACTERIZATIONS 467
10.2 A UNIVERSAL DOMAIN 469
10.3 PCF WITH PARALLEL OPERATORS 472
10.3.1 FINITARY PARALLEL OPERATORS 473
10.3.2 INFINITARY PARALLEL OPERATORS 477
10.3.3 DEGREES OF PARALLELISM 480
10.4 REALIZABILITY OVER KI AND K\ 481
XVI CONTENTS
10.5 ENUMERATION AND ITS CONSEQUENCES 485
10.6 SCOTT'S GRAPH MODEL 8?(X) 488
11 THE SEQUENTIALLY REALIZABLE FUNCTIONALS 489
11.1 BASIC STRUCTURE AND EXAMPLES 490
11.2 A UNIVERSAL TYPE 493
11.3 SR AS A MAXIMAL SEQUENTIAL MODEL 498
11.4 THE STRONGLY STABLE MODEL 500
11.5 ORDER-THEORETIC STRUCTURE 503
11.6 THE STABLE MODEL AND ITS GENERALIZATIONS 504
12 SOME INTENSIONAL MODELS 505
12.1 SEQUENTIAL ALGORITHMS AND VAN OOSTEN'S B 506
12.1.1 SEQUENTIAL DATA STRUCTURES 507
12.1.2 A CATEGORY OF SEQUENTIAL ALGORITHMS 509
12.1.3 SIMPLY TYPED MODELS 514
12.1.4 B AS A REFLEXIVE OBJECT 515
12.1.5 A PROGRAMMING LANGUAGE FOR SEQUENTIAL ALGORITHMS 517
12.1.6 AN EXTENSIONAL CHARACTERIZATION 520
12.1.7 RELATION TO OTHER GAME MODELS 525
12.2 KLEENE'S SECOND MODEL KI 526
12.2.1 INTENSIONALITY IN KI 529
12.3 KLEENE'S FIRST MODEL K\ 532
13 RELATED AND FUTURE WORK 535
13.1 DEEPENING AND BROADENING THE PROGRAMME 535
13.1.1 CHALLENGES AND OPEN PROBLEMS 536
13.1.2 OTHER HIGHER-ORDER MODELS 536
13.1.3 FURTHER GENERALIZATIONS 538
13.2 NEIGHBOURING AREAS 539
13.2.1 COMPUTABILITY OVER NON-DISCRETE SPACES 539
13.2.2 COMPUTABILITY IN ANALYSIS AND PHYSICS 541
13.2.3 NORMAL FUNCTIONALS AND SET RECURSION 541
13.2.4 COMPLEXITY AND SUBRECURSION 542
13.2.5 TRANSFINITE TYPES 543
13.2.6 ABSTRACT AND GENERALIZED COMPUTABILITY THEORIES 544
13.3 APPLICATIONS 545
13.3.1 METAMATHEMATICS AND PROOF MINING 545
13.3.2 DESIGN AND LOGIC OF PROGRAMMING LANGUAGES 545
13.3.3 APPLICATIONS TO PROGRAMMING 546
REFERENCES 547
INDEX 561 |
| any_adam_object | 1 |
| author | Longley, John Normann, Dag |
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| indexdate | 2024-08-03T02:48:04Z |
| institution | BVB |
| isbn | 3662479915 9783662479919 |
| language | English |
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| physical | xvi, 571 Seiten |
| publishDate | 2015 |
| publishDateSearch | 2015 |
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| publisher | Springer |
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| series2 | Theory and applications of computability |
| spelling | Longley, John Verfasser (DE-588)1095691686 aut Higher-order computability John Longley, Dag Normann Berlin ; Heidelberg Springer [2015] © 2015 xvi, 571 Seiten txt rdacontent n rdamedia nc rdacarrier Theory and applications of computability Literaturverzeichnis Seite 547 - 559 Berechnungstheorie (DE-588)4005581-4 gnd rswk-swf Research Computability Theory Computability Models Lambda Algebras Kleene Computability Intensional Models Berechnungstheorie (DE-588)4005581-4 s DE-604 Normann, Dag Verfasser (DE-588)1095774662 aut Erscheint auch als Online-Ausgabe 978-3-662-47992-6 X:MVB text/html http://deposit.dnb.de/cgi-bin/dokserv?id=5295573&prov=M&dok_var=1&dok_ext=htm Inhaltstext DNB Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=028600143&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Longley, John Normann, Dag Higher-order computability Berechnungstheorie (DE-588)4005581-4 gnd |
| subject_GND | (DE-588)4005581-4 |
| title | Higher-order computability |
| title_auth | Higher-order computability |
| title_exact_search | Higher-order computability |
| title_full | Higher-order computability John Longley, Dag Normann |
| title_fullStr | Higher-order computability John Longley, Dag Normann |
| title_full_unstemmed | Higher-order computability John Longley, Dag Normann |
| title_short | Higher-order computability |
| title_sort | higher order computability |
| topic | Berechnungstheorie (DE-588)4005581-4 gnd |
| topic_facet | Berechnungstheorie |
| url | http://deposit.dnb.de/cgi-bin/dokserv?id=5295573&prov=M&dok_var=1&dok_ext=htm http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=028600143&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| work_keys_str_mv | AT longleyjohn higherordercomputability AT normanndag higherordercomputability |