The foundations of computability theory:
Saved in:
| Main Author: | |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
Heidelberg ; New York ; Dordrecht ; London ; Berlin
Springer
[2015]
|
| Series: | Computer science
|
| Subjects: | |
| Online Access: | Inhaltstext Inhaltsverzeichnis |
| Item Description: | Literaturverzeichnis Seite 311 - 317 |
| Physical Description: | xx, 331 Seiten Illustrationen |
| ISBN: | 9783662448076 9783662448083 |
Staff View
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Record in the Search Index
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|---|---|
| adam_text |
CONTENTS
PART I THE ROOTS OF COMPUTABILITY THEORY
1 INTRODUCTION 3
1.1 ALGORITHMS AND COMPUTATION 3
1.1.1 THE INTUITIVE CONCEPT OF THE ALGORITHM AND COMPUTATION. 3
1.1.2 ALGORITHMS AND COMPUTATIONS BEFORE THE TWENTIETH CENTURY 5
1.2 CHAPTER SUMMARY 7
2 THE FOUNDATIONAL CRISIS OF MATHEMATICS 9
2.1 CRISIS IN SET THEORY 9
2.1.1 AXIOMATIC SYSTEMS 9
2.1.2 CANTOR'S NAIVE SET THEORY 13
2.1.3 LOGICAL PARADOXES 17
2.2 SCHOOLS OF RECOVERY 19
2.2.1 SLOWDOWN AND REVISION 20
2.2.2 LNTUITIONISM 20
2.2.3 LOGICISM 23
2.2.4 FORMALISM 26
2.3 CHAPTER SUMMARY 29
3 FORMALISM 31
3.1 FORMAL AXIOMATIC SYSTEMS AND THEORIES 31
3.1.1 WHAT IS A FORMAL AXIOMATIC SYSTEM? 31
3.1.2 INTERPRETATIONS AND MODELS 35
3.2 FORMALIZATION OF LOGIC, ARITHMETIC, AND SET THEORY 39
3.3 CHAPTER SUMMARY 48
4 HILBERT'S ATTEMPT AT RECOVERY 49
4.1 HILBERT'S PROGRAM 49
4.1.1 FUNDAMENTAL PROBLEMS OF THE FOUNDATIONS OF MATHEMATICS . 49
4.1.2 HILBERT'S PROGRAM 53
XV
HTTP://D-NB.INFO/1054845239
XVI CONTENTS
4.2 THE FATE OF HILBERT'S PROGRAM 54
4.2.1 FORMALIZATION OF MATHEMATICS: FORMAL AXIOMATIC SYSTEM M 54
4.2.2 DECIDABILITY OF M: ENTSCHEIDUNGSPROBLEM 55
4.2.3 COMPLETENESS OF M: GODEL'S FIRST INCOMPLETENESS THEOREM 57
4.2.4 CONSEQUENCES OF THE FIRST INCOMPLETENESS THEOREM 58
4.2.5 CONSISTENCY OF M: GODEL'S SECOND INCOMPLETENESS THEOREM 60
4.2.6 CONSEQUENCES OF THE SECOND INCOMPLETENESS THEOREM 61
4.3 LEGACY OF HILBERT'S PROGRAM 63
4.4 CHAPTER SUMMARY 64
PROBLEMS 64
BIBLIOGRAPHIC NOTES 65
PART II CLASSICAL COMPUTABILITY THEORY
5 THE QUEST FOR A FORMALIZATION 69
5.1 WHAT IS AN ALGORITHM AND WHAT DO WE MEAN BY COMPUTATION? . 69
5.1.1 INTUITION AND DILEMMAS 70
5.1.2 THE NEED FOR FORMALIZATION 71
5.2 MODELS OF COMPUTATION 72
5.2.1 MODELLING AFTER FUNCTIONS 72
5.2.2 MODELLING AFTER HUMANS 80
5.2.3 MODELLING AFTER LANGUAGES 82
5.2.4 REASONABLE MODELS OF COMPUTATION 86
5.3 COMPUTABILITY (CHURCH-TURING) THESIS 87
5.3.1 HISTORY OF THE THESIS 87
5.3.2 THE THESIS 88
5.3.3 DIFFICULTIES WITH TOTAL FUNCTIONS 90
5.3.4 GENERALIZATION TO PARTIAL FUNCTIONS 93
5.3.5 APPLICATIONS OF THE THESIS 97
5.4 CHAPTER SUMMARY 97
PROBLEMS 98
BIBLIOGRAPHIC NOTES 99
6 THE TTARING MACHINE 101
6.1 TURING MACHINE 101
6.1.1 BASIC MODEL 102
6.1.2 GENERALIZED MODELS 107
6.1.3 EQUIVALENCE OF GENERALIZED AND BASIC MODELS 109
6.1.4 REDUCED MODEL 113
6.1.5 EQUIVALENCE OF REDUCED AND BASIC MODELS 114
6.1.6 USE OF DIFFERENT MODELS 114
6.2 UNIVERSAL TURING MACHINE 115
6.2.1 CODING AND ENUMERATION OF TURING MACHINES 115
6.2.2 THE EXISTENCE OF A UNIVERSAL TURING MACHINE 117
6.2.3 THE IMPORTANCE OF THE UNIVERSAL TURING MACHINE 119
CONTENTS XVII
6.2.4 PRACTICAL CONSEQUENCES: DATA VS. INSTRUCTIONS 119
6.2.5 PRACTICAL CONSEQUENCES: GENERAL-PURPOSE COMPUTER 119
6.2.6 PRACTICAL CONSEQUENCES: OPERATING SYSTEM 121
6.2.7 PRACTICAL CONSEQUENCES: RAM MODEL OF COMPUTATION 122
6.3 USE OF A TURING MACHINE 125
6.3.1 FUNCTION COMPUTATION 125
6.3.2 SET GENERATION 127
6.3.3 SET RECOGNITION 130
6.3.4 GENERATION VS. RECOGNITION 133
6.3.5 THE STANDARD UNIVERSES Z* AND N 136
6.3.6 FORMAL LANGUAGES VS. SETS OF NATURAL NUMBERS 137
6.4 CHAPTER SUMMARY 138
PROBLEMS 139
BIBLIOGRAPHIC NOTES 141
7 THE FIRST BASIC RESULTS 143
7.1 SOME BASIC PROPERTIES OF SEMI-DECIDABLE (C.E.) SETS 143
7.2 PADDING LEMMA AND INDEX SETS 145
7.3 PARAMETRIZATION (S-M-N) THEOREM 147
7.3.1 DEDUCTION OF THE THEOREM 148
7.4 RECURSION (FIXED-POINT) THEOREM 149
7.4.1 DEDUCTION OF THE THEOREM 150
7.4.2 INTERPRETATION OF THE THEOREM 151
7.4.3 FIXED POINTS OF FUNCTIONS 152
7.4.4 PRACTICAL CONSEQUENCES: RECURSIVE PROGRAM DEFINITION 153
7.4.5 PRACTICAL CONSEQUENCES: RECURSIVE PROGRAM EXECUTION 154
7.4.6 PRACTICAL CONSEQUENCES: PROCEDURE CALLS IN GENERAL-
PURPOSE COMPUTERS 156
7.5 CHAPTER SUMMARY 158
PROBLEMS 159
BIBLIOGRAPHIC NOTES 160
8 INCOMPUTABLE PROBLEMS 161
8.1 PROBLEM SOLVING 161
8.1.1 DECISION PROBLEMS AND OTHER KINDS OF PROBLEMS 162
8.1.2 LANGUAGE OF A DECISION PROBLEM 163
8.1.3 SUBPROBLEMS OF A DECISION PROBLEM 165
8.2 THERE IS AN INCOMPUTABLE PROBLEM * HALTING PROBLEM 166
8.2.1 CONSEQUENCES: THE BASIC KINDS OF DECISION PROBLEMS 169
8.2.2 CONSEQUENCES: COMPLEMENTARY SETS AND DECISION PROBLEMS 171
8.2.3 CONSEQUENCES: THERE IS AN INCOMPUTABLE FUNCTION 172
8.3 SOME OTHER INCOMPUTABLE PROBLEMS 172
8.3.1 PROBLEMS ABOUT TURING MACHINES 173
8.3.2 POST'S CORRESPONDENCE PROBLEM 175
8.3.3 PROBLEMS ABOUT ALGORITHMS AND COMPUTER PROGRAMS 175
XVIII
CONTENTS
8.3.4 PROBLEMS ABOUT PROGRAMMING LANGUAGES AND GRAMMARS . 177
8.3.5 PROBLEMS ABOUT COMPUTABLE FUNCTIONS 179
8.3.6 PROBLEMS FROM NUMBER THEORY 180
8.3.7 PROBLEMS FROM ALGEBRA 180
8.3.8 PROBLEMS FROM ANALYSIS 182
8.3.9 PROBLEMS FROM TOPOLOGY 183
8.3.10 PROBLEMS FROM MATHEMATICAL LOGIC 184
8.3.11 PROBLEMS ABOUT GAMES 185
8.4 CAN WE OUTWIT INCOMPUTABLE PROBLEMS? 187
8.5 CHAPTER SUMMARY 189
PROBLEMS 189
BIBLIOGRAPHIC NOTES 190
9 METHODS OF PROVING INCOMPUTABILITY 191
9.1 PROVING BY DIAGONALIZATION 191
9.1.1 DIRECT DIAGONALIZATION 191
9.1.2 INDIRECT DIAGONALIZATION 194
9.2 PROVING BY REDUCTION 196
9.2.1 REDUCTIONS IN GENERAL 196
9.2.2 THE M-REDUCTION 197
9.2.3 UNDECIDABILITY AND M-REDUCTION 199
9.2.4 THE 1-REDUCTION 201
9.3 PROVING BY THE RECURSION THEOREM 204
9.4 PROVING BY RICE'S THEOREM 205
9.4.1 RICE'S THEOREM FOR FUNCTIONS 205
9.4.2 RICE'S THEOREM FOR INDEX SETS 206
9.4.3 RICE'S THEOREM FOR SETS 208
9.4.4 CONSEQUENCES: BEHAVIOR OF ABSTRACT COMPUTING MACHINES . 209
9.5 INCOMPUTABILITY OF OTHER KINDS OF PROBLEMS 210
9.6 CHAPTER SUMMARY 213
PROBLEMS 213
BIBLIOGRAPHIC NOTES 215
PART III RELATIVE COMPUTABILITY
10 COMPUTATION WITH EXTERNAL HELP 219
10.1 TURING MACHINES WITH ORACLES 219
10.1.1 TURING'S IDEA OF ORACULAR HELP 220
10.1.2 THE ORACLE TURING MACHINE (O-TM) 223
10.1.3 SOME BASIC PROPERTIES OF O-TMS 225
10.1.4 CODING AND ENUMERATION OF O-TMS 226
10.2 COMPUTATION WITH ORACLES 228
10.2.1 GENERALIZATION OF CLASSICAL DEFINITIONS 228
10.2.2 CONVENTION: THE UNIVERSE N AND SINGLE-ARGUMENT FUNCTIONS231
10.3 OTHER WAYS TO MAKE EXTERNAL HELP AVAILABLE 231
CONTENTS XIX
10.4 RELATIVE COMPUTABILITY THESIS 232
10.5 PRACTICAL CONSEQUENCES: O-TM WITH A DATABASE OR NETWORK 232
10.6 PRACTICAL CONSEQUENCES: ONLINE AND OFFLINE COMPUTATION 233
10.7 CHAPTER SUMMARY 234
BIBLIOGRAPHIC NOTES 234
11 DEGREES OF UNSOLVABILITY 235
11.1 TURING REDUCTION 235
11.1.1 TURING REDUCTION OF A COMPUTATIONAL PROBLEM 236
11.1.2 SOME BASIC PROPERTIES OF THE TURING REDUCTION 237
11.2 TURING DEGREES 240
11.3 CHAPTER SUMMARY 243
PROBLEMS 244
BIBLIOGRAPHIC NOTES 244
12 THE TURING HIERARCHY OF UNSOLVABILITY 245
12.1 THE PERPLEXITIES OF UNSOLVABILITY 245
12.2 THE TURING JUMP 246
12.2.1 PROPERTIES OF THE TURING JUMP OF A SET 247
12.3 HIERARCHIES OF T-DEGREES 249
12.3.1 THE JUMP HIERARCHY 250
12.4 CHAPTER SUMMARY 252
PROBLEMS 252
BIBLIOGRAPHIC NOTES 253
13 THE CLASS 2? OF DEGREES OF UNSOLVABILITY 255
13.1 THE STRUCTURE ( , .') 255
13.2 SOME BASIC PROPERTIES OF (T , ,') 257
13.2.1 CARDINALITY OF DEGREES AND OF THE CLASS X 257
13.2.2 THE CLASS T AS A MATHEMATICAL STRUCTURE 258
13.2.3 INTERMEDIATE 7"-DEGREES 263
13.2.4 CONES 264
13.2.5 MINIMAL T-DEGREES 266
13.3 CHAPTER SUMMARY 267
PROBLEMS 267
BIBLIOGRAPHIC NOTES 267
14 C.E. DEGREES AND THE PRIORITY METHOD 269
14.1 C.E. TURING DEGREES 269
14.2 POST'S PROBLEM 270
14.2.1 POST'S ATTEMPT AT A SOLUTION TO POST'S PROBLEM 271
14.3 THE PRIORITY METHOD AND PRIORITY ARGUMENTS 274
14.3.1 THE PRIORITY METHOD IN GENERAL 274
14.3.2 THE FRIEDBERG-MUCHNIK SOLUTION TO POST'S PROBLEM 278
14.3.3 PRIORITY ARGUMENTS 279
14.4 SOME PROPERTIES OF C.E. DEGREES 279
XX CONTENTS
14.5 CHAPTER SUMMARY 280
PROBLEMS 280
BIBLIOGRAPHIC NOTES 281
15 THE ARITHMETICAL HIERARCHY 283
15.1 DECIDABILITY OF RELATIONS 283
15.2 THE ARITHMETICAL HIERARCHY 284
15.3 THE LINK WITH THE JUMP HIERARCHY 288
15.4 PRACTICAL CONSEQUENCES: PROVING THE INCOMPUTABILITY 290
15.5 CHAPTER SUMMARY 292
PROBLEMS 292
BIBLIOGRAPHIC NOTES 293
16 FURTHER READING 295
A MATHEMATICAL BACKGROUND 297
B NOTATION INDEX 305
REFERENCES 311
INDEX 319 |
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| discipline | Informatik Mathematik |
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| series2 | Computer science |
| spelling | Robič, Borut 1960- Verfasser (DE-588)120990784 aut The foundations of computability theory Borut Robič Heidelberg ; New York ; Dordrecht ; London ; Berlin Springer [2015] © 2015 xx, 331 Seiten Illustrationen txt rdacontent n rdamedia nc rdacarrier Computer science Literaturverzeichnis Seite 311 - 317 Berechnungstheorie (DE-588)4005581-4 gnd rswk-swf Computermathematik (DE-588)4788218-9 gnd rswk-swf Informatik (DE-588)4026894-9 gnd rswk-swf Graduate Computability theory Algorithms Models of computation Turing machine Oracles Relative computability Berechnungstheorie (DE-588)4005581-4 s DE-604 Informatik (DE-588)4026894-9 s Computermathematik (DE-588)4788218-9 s 1\p DE-604 Erscheint auch als Online-Ausgabe 10.1007/978-3-662-44808-3 X:MVB text/html http://deposit.dnb.de/cgi-bin/dokserv?id=4748396&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=028612127&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis 1\p cgwrk 20201028 DE-101 https://d-nb.info/provenance/plan#cgwrk |
| spellingShingle | Robič, Borut 1960- The foundations of computability theory Berechnungstheorie (DE-588)4005581-4 gnd Computermathematik (DE-588)4788218-9 gnd Informatik (DE-588)4026894-9 gnd |
| subject_GND | (DE-588)4005581-4 (DE-588)4788218-9 (DE-588)4026894-9 |
| title | The foundations of computability theory |
| title_auth | The foundations of computability theory |
| title_exact_search | The foundations of computability theory |
| title_full | The foundations of computability theory Borut Robič |
| title_fullStr | The foundations of computability theory Borut Robič |
| title_full_unstemmed | The foundations of computability theory Borut Robič |
| title_short | The foundations of computability theory |
| title_sort | the foundations of computability theory |
| topic | Berechnungstheorie (DE-588)4005581-4 gnd Computermathematik (DE-588)4788218-9 gnd Informatik (DE-588)4026894-9 gnd |
| topic_facet | Berechnungstheorie Computermathematik Informatik |
| url | http://deposit.dnb.de/cgi-bin/dokserv?id=4748396&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=028612127&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| work_keys_str_mv | AT robicborut thefoundationsofcomputabilitytheory |