Automated theory formation in pure mathematics:
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| Format: | Book |
| Language: | English |
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London ; Berlin ; Heidelberg ; New York ; Barcelona ; Hong Kong ; Milan ; Paris ; Singapore ; Tokyo
Springer
2002
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| Series: | Distinguished dissertations
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| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Item Description: | Literaturverz. S. 365 - 373 |
| Physical Description: | XVI, 380 S. graph Darst. 24 cm |
| ISBN: | 1852336099 9781852336097 |
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| 100 | 1 | |a Colton, Simon |e Verfasser |4 aut | |
| 245 | 1 | 0 | |a Automated theory formation in pure mathematics |c Simon Colton |
| 264 | 1 | |a London ; Berlin ; Heidelberg ; New York ; Barcelona ; Hong Kong ; Milan ; Paris ; Singapore ; Tokyo |b Springer |c 2002 | |
| 300 | |a XVI, 380 S. |b graph Darst. |c 24 cm | ||
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| 337 | |b n |2 rdamedia | ||
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| 490 | 0 | |a Distinguished dissertations | |
| 500 | |a Literaturverz. S. 365 - 373 | ||
| 650 | 4 | |a Datenverarbeitung | |
| 650 | 4 | |a Künstliche Intelligenz | |
| 650 | 4 | |a Mathematik | |
| 650 | 4 | |a Mathematics -- Methodology -- Data processing | |
| 650 | 4 | |a Artificial intelligence | |
| 650 | 4 | |a Automatic theorem proving | |
| 650 | 4 | |a Expert systems (Computer science) | |
| 650 | 0 | 7 | |a Mathematik |0 (DE-588)4037944-9 |2 gnd |9 rswk-swf |
| 650 | 0 | 7 | |a Programm |0 (DE-588)4047394-6 |2 gnd |9 rswk-swf |
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| _version_ | 1804138170081607680 |
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| adam_text | SIMON COLTON AUTOMATED THEORY FORMATION IN PURE MATHEMATICS SPRINGER
CONTENTS PREFACE V ACKNOWLEDGEMENTS VII 1. INTRODUCTION 1 1.1 MOTIVATION
1 1.2 AIMS OF THE PROJECT 3 1.3 CONTRIBUTIONS 5 1.4 ORGANISATION OF THE
BOOK 5 1.5 SUMMARY 8 2. LITERATURE SURVEY 9 2.1 SOME PHILOSOPHICAL
ISSUES 10 2.2 MATHEMATICAL THEORY FORMATION PROGRAMS 13 2.2.1 THE AM
PROGRAM 13 2.2.2 THE GT PROGRAM 14 2.2.3 THE IL PROGRAM 15 2.2.4 THE
BAGAI ET AL. PROGRAM 16 2.3 THE BACON PROGRAMS 17 2.4 CONCEPT INVENTION
18 2.4.1 THE REPRESENTATION OF MATHEMATICAL CONCEPTS 18 2.4.2 INDUCTIVE
LOGIC PROGRAMMING 19 2.5 CONJECTURE MAKING PROGRAMS 21 2.5.1 THE
GRAFFITI PROGRAM 21 2.5.2 THE AUTOGRAPHIX PROGRAM 22 2.5.3 THE PSLQ
ALGORITHM 23 2.6 THE OTTER AND MACE PROGRAMS 24 2.7 THE ENCYCLOPEDIA OF
INTEGER SEQUENCES 25 2.8 SUMMARY 27 3. MATHEMATICAL THEORIES 29 3.1
GROUP THEORY, GRAPH THEORY AND NUMBER THEORY 29 3.1.1 GROUP THEORY 30
3.1.2 GRAPH THEORY 32 X CONTENTS 3.1.3 NUMBER THEORY 33 3.1.4
ISOMORPHISM 35 3.2 MATHEMATICAL DOMAINS 35 3.2.1 REASONS BEHIND THEORY
FORMATION 35 3.2.2 FINITE AND INFINITE DOMAINS 38 3.3 THE CONTENT OF
THEORIES 38 3.3.1 CONCEPTS 38 3.3.2 CONJECTURES, THEOREMS AND PROOFS 41
3.3.3 OTHER ASPECTS OF THEORIES 42 3.4 SUMMARY 43 4. DESIGN
CONSIDERATIONS 45 4.1 ASPECTS OF THEORY FORMATION 45 4.1.1 ASPECTS WHICH
ARE MODELLED 45 4.1.2 SOME ASPECTS WHICH ARE NOT MODELLED 46 4.2 CONCEPT
AND CONJECTURE MAKING DECISIONS 47 4.2.1 THE USE OF EXAMPLES 47 4.2.2
MAKING CONJECTURES 49 4.3 THE DOMAINS HR WORKS IN 49 4.4 REPRESENTATION
ISSUES 50 4.4.1 EXAMPLES OF CONCEPTS 51 4.4.2 DEFINITIONS OF CONCEPTS 53
4.4.3 REPRESENTATION OF CONJECTURES, PROOFS AND COUNTEREXAMPLES 55 4.5
THE HR PROGRAM IN OUTLINE 56 4.6 SUMMARY 57 5. BACKGROUND KNOWLEDGE 59
5.1 OBJECTS OF INTEREST (ENTITIES) 59 5.2 REQUIRED INFORMATION ABOUT
CONCEPTS 60 5.3 INITIAL CONCEPTS FROM THE USER 62 5.3.1 INITIAL CONCEPTS
IN GRAPH THEORY 62 5.3.2 INITIAL CONCEPTS IN NUMBER THEORY 63 5.3.3
INITIAL CONCEPTS IN FINITE ALGEBRAIC SYSTEMS 64 5.4 GENERATING INITIAL
CONCEPTS FROM AXIOMS 66 5.5 SUMMARY 67 6. INVENTING CONCEPTS 69 6.1 AN
OVERVIEW OF THE PRODUCTION RULES 70 6.1.1 SOME COMMON CONSTRUCTION
TECHNIQUES 71 6.2 THE EXISTS PRODUCTION RULE 73 6.2.1 DATA TABLE
CONSTRUCTION AND PARAMETERISATIONS 73 6.2.2 GENERATION OF DEFINITIONS 74
6.3 THE MATCH PRODUCTION RULE 76 6.3.1 DATA TABLE CONSTRUCTION AND
PARAMETERISATIONS 76 CONTENTS XI 6.3.2 GENERATION OF DEFINITIONS 77 6.4
THE NEGATE PRODUCTION RULE 78 6.4.1 DATA TABLE CONSTRUCTION AND
PARAMETERISATIONS 78 6.4.2 GENERATION OF DEFINITIONS 79 6.5 THE SIZE
PRODUCTION RULE 81 6.5.1 DATA TABLE CONSTRUCTION AND PARAMETERISATION 81
6.5.2 GENERATION OF DEFINITIONS 82 6.6 THE SPLIT PRODUCTION RULE 84
6.6.1 DATA TABLE CONSTRUCTION AND PARAMETERISATIONS 84 6.6.2 GENERATION
OF DEFINITIONS 85 6.7 THE COMPOSE PRODUCTION RULE 86 6.7.1 DATA TABLE
CONSTRUCTION AND PARAMETERISATIONS 86 6.7.2 GENERATION OF DEFINITIONS 88
6.7.3 GENERALISATION OF PREVIOUS PRODUCTION RULES 89 6.8 THE FORALL
PRODUCTION RULE 89 6.8.1 DATA TABLE CONSTRUCTION AND PARAMETERISATIONS
89 6.8.2 GENERATION OF DEFINITIONS 92 6.9 EFFICIENCY AND SOUNDNESS
CONSIDERATIONS 93 6.9.1 FORBIDDEN PATHS 93 6.9.2 GENERATED AND STORED
PROPERTIES 97 6.9.3 PROVING CONSISTENCY BETWEEN DATA TABLES AND
DEFINITIONS 97 6.10 EXAMPLE CONSTRUCTIONS 98 6.11 SUMMARY 100 7. MAKING
CONJECTURES 101 7.1 EQUIVALENCE CONJECTURES 102 7.1.1 MAKING EQUIVALENCE
CONJECTURES AUTOMATICALLY 103 7.1.2 IMPLEMENTATION DETAILS 103 7.2
IMPLICATION CONJECTURES 104 7.2.1 MAKING IMPLICATION CONJECTURES
AUTOMATICALLY 105 7.2.2 IMPLEMENTATION DETAILS 106 7.3 NON-EXISTENCE
CONJECTURES 106 7.3.1 MAKING NON-EXISTENCE CONJECTURES AUTOMATICALLY
.... 107 7.4 APPLICABILITY CONJECTURES 108 7.4.1 MAKING APPLICABILITY
CONJECTURES AUTOMATICALLY 108 7.4.2 IMPLEMENTATION DETAILS 109 7.5
CONJECTURE MAKING USING THE ENCYCLOPEDIA OF INTEGER SEQUENCES 110 7.5.1
PRESENTING CONCEPTS AS INTEGER SEQUENCES 110 7.5.2 CONJECTURE TYPES 112
7.5.3 PRUNING METHODS 114 7.5.4 AN EXAMPLE CONJECTURE 116 7.6 ISSUES IN
AUTOMATED CONJECTURE MAKING 117 7.6.1 CHOICE OF CONJECTURE MAKING
TECHNIQUES 117 XII CONTENTS 7.6.2 WHEN TO CHECK FOR CONJECTURES 118
7.6.3 THE USE OF DATA AND PRUNING METHODS 118 7.6.4 OTHER CONJECTURE
FORMATS 119 7.7 SUMMARY 120 8. SETTLING CONJECTURES 121 8.1 REASONS FOR
SETTLING CONJECTURES 121 8.2 PROVING CONJECTURES 122 8.2.1 USING OTTER
TO PROVE CONJECTURES 122 8.2.2 SUB-CONJECTURES AND PRIME IMPLICATES 124
8.2.3 USING HR TO PROVE IMPLICATION CONJECTURES 127 8.2.4 DETAILS OF
HR S THEOREM PROVING 129 8.2.5 ADVANTAGES OF USING HR TO PROVE THEOREMS
131 8.3 DISPROVING CONJECTURES 132 8.3.1 USING MACE TO FIND
COUNTEREXAMPLES 133 8.3.2 USING HR TO FIND COUNTEREXAMPLES 134 8.3.3
FINDING COUNTEREXAMPLES IN NON-ALGEBRAIC DOMAINS .. 136 8.4 RETURNING TO
OPEN CONJECTURES 137 8.5 SUMMARY 139 9. ASSESSING CONCEPTS 141 9.1 THE
AGENDA MECHANISM 142 9.2 THE INTERESTINGNESS OF MATHEMATICAL CONCEPTS
143 9.2.1 WHAT MAKES A CONCEPT INTERESTING? 144 9.2.2 WHAT MAKES A
CONCEPT UNINTERESTING? 145 9.2.3 INTERESTINGNESS GAINED FROM THEORY
FORMATION 146 9.3 INTRINSIC AND RELATIONAL MEASURES OF CONCEPTS 147
9.3.1 COMPREHENSIBILITY 149 9.3.2 PARSIMONY 149 9.3.3 APPLICABILITY 150
9.3.4 NOVELTY 150 9.4 UTILITARIAN PROPERTIES OF CONCEPTS 151 9.4.1
PRODUCTIVITY 152 9.4.2 CLASSIFICATION TASKS 152 9.5 CONJECTURES ABOUT
CONCEPTS 154 9.6 DETAILS OF THE HEURISTIC SEARCHES 155 9.6.1 WHEN AND
HOW TO MEASURE CONCEPTS 155 9.6.2 SORTING THE PRODUCTION RULES 156 9.6.3
RESTRICTING THE SEARCH 156 9.6.4 CHOOSING WEIGHTS 158 9.7 WORKED EXAMPLE
159 9.8 OTHER POSSIBILITIES 161 9.9 SUMMARY 163 CONTENTS XIII 10.
ASSESSING CONJECTURES 165 10.1 GENERIC MEASURES FOR CONJECTURES 165
10.1.1 TYPE OF CONJECTURE 166 10.1.2 SURPRISINGNESS 166 10.1.3 OTHER
GENERIC MEASURES 168 10.2 ADDITIONAL MEASURES FOR THEOREMS 169 10.2.1
DIFFICULTY OF PROOF 169 10.2.2 GENERALITY OF THEOREMS 170 10.3
ADDITIONAL MEASURES FOR NON-THEOREMS 172 10.4 SETTING WEIGHTS FOR
CONJECTURE MEASURES 173 10.5 ASSESSING CONCEPTS USING CONJECTURES 174
10.5.1 INDEPENDENCE OF MEASURES FOR CONJECTURES 174 10.5.2 IDENTIFYING
CONCEPTS DISCUSSED IN CONJECTURES 175 10.5.3 MEASURES FOR CONCEPTS 177
10.6 WORKED EXAMPLE 177 10.7 SUMMARY 179 11. AN EVALUATION OF HR S
THEORIES 181 11.1 ANALYSIS OF TWO THEORIES 182 11.1.1 A THEORY OF
NUMBERS 182 11.1.2 A THEORY OF GROUPS 186 11.2 DESIRABLE QUALITIES OF
THEORIES - CONCEPTS 191 11.2.1 AVERAGE APPLICABILITY OF CONCEPTS 192
11.2.2 AVERAGE COMPREHENSIBILITY OF CONCEPTS 195 11.2.3 NUMBER OF
CATEGORISATIONS 197 11.2.4 NUMBER OF CONCEPTS 199 11.3 DESIRABLE
QUALITIES OF THEORIES - CONJECTURES 201 11.3.1 DIFFICULTY AND
SURPRISINGNESS OF CONJECTURES 202 11.3.2 PROPORTION OF THEOREMS AND OPEN
CONJECTURES 203 11.4 USING THE HEURISTIC SEARCH 204 11.4.1 ROBUSTNESS OF
THE HEURISTIC MEASURES 204 11.4.2 DIFFERENCES BETWEEN DOMAINS 207 11.4.3
PRUNING USING THE HEURISTIC MEASURES 209 11.5 CLASSICALLY INTERESTING
RESULTS 211 11.5.1 GRAPH THEORY 212 11.5.2 GROUP THEORY 214 11.5.3
NUMBER THEORY 216 11.6 CONCLUSIONS 221 12. THE APPLICATION OF HR TO
DISCOVERY TASKS 225 12.1 A CLASSIFICATION PROBLEM 226 12.2 EXPLORATION
OF AN ALGEBRAIC SYSTEM 230 12.3 INVENTION OF INTEGER SEQUENCES 233
12.3.1 ADDITIONS TO THE ENCYCLOPEDIA 233 12.3.2 REFACTORABLE NUMBERS 237
XIV CONTENTS 12.3.3 SEQUENCE A046951 240 12.4 DISCOVERY TASK FAILURES
241 12.5 VALDES-PEREZ S MACHINE DISCOVERY CRITERIA 242 12.5.1 NOVELTY
242 12.5.2 INTERESTINGNESS 243 12.5.3 PLAUSIBILITY 243 12.5.4
INTELLIGIBILITY 244 12.6 CONCLUSIONS 244 13. RELATED WORK 247 13.1 A
COMPARISON OF HR AND THE AM PROGRAM 247 13.1.1 HOW AM FORMED THEORIES
248 13.1.2 MISCONCEPTIONS ABOUT AM 250 13.1.3 PROGRAMS BASED ON AM 253
13.1.4 A QUALITATIVE COMPARISON OF AM AND HR 254 13.1.5 A QUANTITATIVE
COMPARISON OF AM AND HR 259 13.1.6 SUMMARY: THE AM PROGRAM 262 13.2 A
COMPARISON OF HR AND THE GT PROGRAM 264 13.2.1 HOW GT FORMED THEORIES
264 13.2.2 THE SCOT PROGRAM 266 13.2.3 A QUALITATIVE COMPARISON OF GT
AND HR 266 13.2.4 A QUANTITATIVE COMPARISON OF GT AND HR 267 13.3 A
COMPARISON OF HR AND THE IL PROGRAM 269 13.3.1 HOW IL WORKED 269 13.3.2
A QUALITATIVE COMPARISON OF IL AND HR 270 13.4 A COMPARISON OF HR AND
BAGAI ET AL S PROGRAM 271 13.4.1 HOW BAGAI ET AL S PROGRAM WORKED 271
13.4.2 A QUALITATIVE COMPARISON OF HR AND BAGAI ET AL S PROGRAM 272 13.5
A COMPARISON OF HR AND THE GRAFFITI PROGRAM 273 13.5.1 HOW GRAFFITI
WORKS 273 13.5.2 A QUALITATIVE COMPARISON OF GRAFFITI AND HR 275 13.6 A
COMPARISON OF HR AND THE PROGOL PROGRAM 276 13.7 SUMMARY 277 14. FURTHER
WORK 281 14.1 ADDITIONAL THEORY FORMATION ABILITIES 281 14.1.1
ADDITIONAL PRODUCTION RULES 282 14.1.2 IMPROVED PRESENTATIONAL ASPECTS
282 14.1.3 FURTHER POSSIBILITIES FOR MAKING AND SETTLING CONJECTURES 283
14.2 THE APPLICATION OF THEORY FORMATION 284 14.2.1 AUTOMATED CONJECTURE
MAKING IN MATHEMATICS 285 14.2.2 CONSTRAINT SATISFACTION PROBLEMS 286
14.2.3 MACHINE LEARNING 287 CONTENTS XV 14.2.4 AUTOMATED THEOREM PROVING
288 14.2.5 APPLICATION TO OTHER SCIENTIFIC DOMAINS 289 14.3 THEORETICAL
EXPLORATIONS 290 14.3.1 META-THEORY FORMATION 290 14.3.2 CROSS DOMAIN
THEORY FORMATION 291 14.3.3 AGENT BASED COOPERATIVE THEORY FORMATION 292
14.4 SUMMARY 293 15. CONCLUSIONS 295 15.1 HAVE WE ACHIEVED OUR AIMS? 296
15.2 CONTRIBUTIONS 297 15.2.1 FUNCTIONALITY 297 15.2.2 SIMPLICITY OF
ARCHITECTURE 298 15.2.3 CYCLE OF MATHEMATICAL ACTIVITY 298 15.2.4
GENERALITY OF METHODS 299 15.2.5 MATHEMATICAL DISCOVERY 299 15.2.6
EVALUATION TECHNIQUES 300 15.3 AUTOMATED THEORY FORMATION IN PURE MATHS
301 APPENDIX A. USER MANUAL FOR HR1.11 303 A.I INSTALLING HR 1.11 304
A.2 SPECIFYING SETTINGS 305 A.3 INITIALISING THEORIES 307 A.4
CONSTRUCTING THEORIES 309 A.5 INVESTIGATING THEORIES 309 A.5.1 PRINTING
RESULTS TO SCREEN 310 A.5.2 VIEWING GRAPHICAL INFORMATION 310 A.5.3
FINDING CONCEPTS AND CONJECTURES 314 A.5.4 MAKING MORE CONJECTURES 316
A.6 HELP FOR A NEW USER 318 APPENDIX B. EXAMPLE SESSIONS 319 B.I GRAPH
THEORY SHORT SESSION 320 B.I.I SESSION OUTPUT 320 B.I.2 COMMENTARY 323
B.2 THEORY FORMATION SESSION IN GROUP THEORY 326 B.2.1 SESSION OUTPUT
327 B.2.2 COMMENTARY 330 B.3 THEORY FORMATION SESSION IN SEMIGROUP
THEORY 332 B.3.1 SESSION OUTPUT 332 B.3.2 COMMENTARY 337 B.4 INVENTING
AND INVESTIGATING AN INTEGER SEQUENCE 338 B.4.1 SESSION OUTPUT 339 B.4.2
COMMENTARY 341 XVI CONTENTS APPENDIX C. NUMBER THEORY RESULTS 343 C.I
REFACTORABLE NUMBERS 343 C.I.I INITIAL RESULTS 344 C.I.2 RELATION TO
OTHER NUMBER TYPES 345 C.1.3 PAIRS AND TRIPLES OF REFACTORABLES 348
C.1.4 DISTRIBUTION 350 C.2 INTEGERS WITH A PRIME NUMBER OF DIVISORS 352
C3 OTHER RESULTS 354 C.4 DIVISOR GRAPHS 356 GLOSSARY 359 BIBLIOGRAPHY
365 INDEX 375
|
| adam_txt |
SIMON COLTON AUTOMATED THEORY FORMATION IN PURE MATHEMATICS SPRINGER
CONTENTS PREFACE V ACKNOWLEDGEMENTS VII 1. INTRODUCTION 1 1.1 MOTIVATION
1 1.2 AIMS OF THE PROJECT 3 1.3 CONTRIBUTIONS 5 1.4 ORGANISATION OF THE
BOOK 5 1.5 SUMMARY 8 2. LITERATURE SURVEY 9 2.1 SOME PHILOSOPHICAL
ISSUES 10 2.2 MATHEMATICAL THEORY FORMATION PROGRAMS 13 2.2.1 THE AM
PROGRAM 13 2.2.2 THE GT PROGRAM 14 2.2.3 THE IL PROGRAM 15 2.2.4 THE
BAGAI ET AL. PROGRAM 16 2.3 THE BACON PROGRAMS 17 2.4 CONCEPT INVENTION
18 2.4.1 THE REPRESENTATION OF MATHEMATICAL CONCEPTS 18 2.4.2 INDUCTIVE
LOGIC PROGRAMMING 19 2.5 CONJECTURE MAKING PROGRAMS 21 2.5.1 THE
GRAFFITI PROGRAM 21 2.5.2 THE AUTOGRAPHIX PROGRAM 22 2.5.3 THE PSLQ
ALGORITHM 23 2.6 THE OTTER AND MACE PROGRAMS 24 2.7 THE ENCYCLOPEDIA OF
INTEGER SEQUENCES 25 2.8 SUMMARY 27 3. MATHEMATICAL THEORIES 29 3.1
GROUP THEORY, GRAPH THEORY AND NUMBER THEORY 29 3.1.1 GROUP THEORY 30
3.1.2 GRAPH THEORY 32 X CONTENTS 3.1.3 NUMBER THEORY 33 3.1.4
ISOMORPHISM 35 3.2 MATHEMATICAL DOMAINS 35 3.2.1 REASONS BEHIND THEORY
FORMATION 35 3.2.2 FINITE AND INFINITE DOMAINS 38 3.3 THE CONTENT OF
THEORIES 38 3.3.1 CONCEPTS 38 3.3.2 CONJECTURES, THEOREMS AND PROOFS 41
3.3.3 OTHER ASPECTS OF THEORIES 42 3.4 SUMMARY 43 4. DESIGN
CONSIDERATIONS 45 4.1 ASPECTS OF THEORY FORMATION 45 4.1.1 ASPECTS WHICH
ARE MODELLED 45 4.1.2 SOME ASPECTS WHICH ARE NOT MODELLED 46 4.2 CONCEPT
AND CONJECTURE MAKING DECISIONS 47 4.2.1 THE USE OF EXAMPLES 47 4.2.2
MAKING CONJECTURES 49 4.3 THE DOMAINS HR WORKS IN 49 4.4 REPRESENTATION
ISSUES 50 4.4.1 EXAMPLES OF CONCEPTS 51 4.4.2 DEFINITIONS OF CONCEPTS 53
4.4.3 REPRESENTATION OF CONJECTURES, PROOFS AND COUNTEREXAMPLES 55 4.5
THE HR PROGRAM IN OUTLINE 56 4.6 SUMMARY 57 5. BACKGROUND KNOWLEDGE 59
5.1 OBJECTS OF INTEREST (ENTITIES) 59 5.2 REQUIRED INFORMATION ABOUT
CONCEPTS 60 5.3 INITIAL CONCEPTS FROM THE USER 62 5.3.1 INITIAL CONCEPTS
IN GRAPH THEORY 62 5.3.2 INITIAL CONCEPTS IN NUMBER THEORY 63 5.3.3
INITIAL CONCEPTS IN FINITE ALGEBRAIC SYSTEMS 64 5.4 GENERATING INITIAL
CONCEPTS FROM AXIOMS 66 5.5 SUMMARY 67 6. INVENTING CONCEPTS 69 6.1 AN
OVERVIEW OF THE PRODUCTION RULES 70 6.1.1 SOME COMMON CONSTRUCTION
TECHNIQUES 71 6.2 THE EXISTS PRODUCTION RULE 73 6.2.1 DATA TABLE
CONSTRUCTION AND PARAMETERISATIONS 73 6.2.2 GENERATION OF DEFINITIONS 74
6.3 THE MATCH PRODUCTION RULE 76 6.3.1 DATA TABLE CONSTRUCTION AND
PARAMETERISATIONS 76 CONTENTS XI 6.3.2 GENERATION OF DEFINITIONS 77 6.4
THE NEGATE PRODUCTION RULE 78 6.4.1 DATA TABLE CONSTRUCTION AND
PARAMETERISATIONS 78 6.4.2 GENERATION OF DEFINITIONS 79 6.5 THE SIZE
PRODUCTION RULE 81 6.5.1 DATA TABLE CONSTRUCTION AND PARAMETERISATION 81
6.5.2 GENERATION OF DEFINITIONS 82 6.6 THE SPLIT PRODUCTION RULE 84
6.6.1 DATA TABLE CONSTRUCTION AND PARAMETERISATIONS 84 6.6.2 GENERATION
OF DEFINITIONS 85 6.7 THE COMPOSE PRODUCTION RULE 86 6.7.1 DATA TABLE
CONSTRUCTION AND PARAMETERISATIONS 86 6.7.2 GENERATION OF DEFINITIONS 88
6.7.3 GENERALISATION OF PREVIOUS PRODUCTION RULES 89 6.8 THE FORALL
PRODUCTION RULE 89 6.8.1 DATA TABLE CONSTRUCTION AND PARAMETERISATIONS
89 6.8.2 GENERATION OF DEFINITIONS 92 6.9 EFFICIENCY AND SOUNDNESS
CONSIDERATIONS 93 6.9.1 FORBIDDEN PATHS 93 6.9.2 GENERATED AND STORED
PROPERTIES 97 6.9.3 PROVING CONSISTENCY BETWEEN DATA TABLES AND
DEFINITIONS 97 6.10 EXAMPLE CONSTRUCTIONS 98 6.11 SUMMARY 100 7. MAKING
CONJECTURES 101 7.1 EQUIVALENCE CONJECTURES 102 7.1.1 MAKING EQUIVALENCE
CONJECTURES AUTOMATICALLY 103 7.1.2 IMPLEMENTATION DETAILS 103 7.2
IMPLICATION CONJECTURES 104 7.2.1 MAKING IMPLICATION CONJECTURES
AUTOMATICALLY 105 7.2.2 IMPLEMENTATION DETAILS 106 7.3 NON-EXISTENCE
CONJECTURES 106 7.3.1 MAKING NON-EXISTENCE CONJECTURES AUTOMATICALLY
. 107 7.4 APPLICABILITY CONJECTURES 108 7.4.1 MAKING APPLICABILITY
CONJECTURES AUTOMATICALLY 108 7.4.2 IMPLEMENTATION DETAILS 109 7.5
CONJECTURE MAKING USING THE ENCYCLOPEDIA OF INTEGER SEQUENCES 110 7.5.1
PRESENTING CONCEPTS AS INTEGER SEQUENCES 110 7.5.2 CONJECTURE TYPES 112
7.5.3 PRUNING METHODS 114 7.5.4 AN EXAMPLE CONJECTURE 116 7.6 ISSUES IN
AUTOMATED CONJECTURE MAKING 117 7.6.1 CHOICE OF CONJECTURE MAKING
TECHNIQUES 117 XII CONTENTS 7.6.2 WHEN TO CHECK FOR CONJECTURES 118
7.6.3 THE USE OF DATA AND PRUNING METHODS 118 7.6.4 OTHER CONJECTURE
FORMATS 119 7.7 SUMMARY 120 8. SETTLING CONJECTURES 121 8.1 REASONS FOR
SETTLING CONJECTURES 121 8.2 PROVING CONJECTURES 122 8.2.1 USING OTTER
TO PROVE CONJECTURES 122 8.2.2 SUB-CONJECTURES AND PRIME IMPLICATES 124
8.2.3 USING HR TO PROVE IMPLICATION CONJECTURES 127 8.2.4 DETAILS OF
HR'S THEOREM PROVING 129 8.2.5 ADVANTAGES OF USING HR TO PROVE THEOREMS
131 8.3 DISPROVING CONJECTURES 132 8.3.1 USING MACE TO FIND
COUNTEREXAMPLES 133 8.3.2 USING HR TO FIND COUNTEREXAMPLES 134 8.3.3
FINDING COUNTEREXAMPLES IN NON-ALGEBRAIC DOMAINS . 136 8.4 RETURNING TO
OPEN CONJECTURES 137 8.5 SUMMARY 139 9. ASSESSING CONCEPTS 141 9.1 THE
AGENDA MECHANISM 142 9.2 THE INTERESTINGNESS OF MATHEMATICAL CONCEPTS
143 9.2.1 WHAT MAKES A CONCEPT INTERESTING? 144 9.2.2 WHAT MAKES A
CONCEPT UNINTERESTING? 145 9.2.3 INTERESTINGNESS GAINED FROM THEORY
FORMATION 146 9.3 INTRINSIC AND RELATIONAL MEASURES OF CONCEPTS 147
9.3.1 COMPREHENSIBILITY 149 9.3.2 PARSIMONY 149 9.3.3 APPLICABILITY 150
9.3.4 NOVELTY 150 9.4 UTILITARIAN PROPERTIES OF CONCEPTS 151 9.4.1
PRODUCTIVITY 152 9.4.2 CLASSIFICATION TASKS 152 9.5 CONJECTURES ABOUT
CONCEPTS 154 9.6 DETAILS OF THE HEURISTIC SEARCHES 155 9.6.1 WHEN AND
HOW TO MEASURE CONCEPTS 155 9.6.2 SORTING THE PRODUCTION RULES 156 9.6.3
RESTRICTING THE SEARCH 156 9.6.4 CHOOSING WEIGHTS 158 9.7 WORKED EXAMPLE
159 9.8 OTHER POSSIBILITIES 161 9.9 SUMMARY 163 CONTENTS XIII 10.
ASSESSING CONJECTURES 165 10.1 GENERIC MEASURES FOR CONJECTURES 165
10.1.1 TYPE OF CONJECTURE 166 10.1.2 SURPRISINGNESS 166 10.1.3 OTHER
GENERIC MEASURES 168 10.2 ADDITIONAL MEASURES FOR THEOREMS 169 10.2.1
DIFFICULTY OF PROOF 169 10.2.2 GENERALITY OF THEOREMS 170 10.3
ADDITIONAL MEASURES FOR NON-THEOREMS 172 10.4 SETTING WEIGHTS FOR
CONJECTURE MEASURES 173 10.5 ASSESSING CONCEPTS USING CONJECTURES 174
10.5.1 INDEPENDENCE OF MEASURES FOR CONJECTURES 174 10.5.2 IDENTIFYING
CONCEPTS DISCUSSED IN CONJECTURES 175 10.5.3 MEASURES FOR CONCEPTS 177
10.6 WORKED EXAMPLE 177 10.7 SUMMARY 179 11. AN EVALUATION OF HR'S
THEORIES 181 11.1 ANALYSIS OF TWO THEORIES 182 11.1.1 A THEORY OF
NUMBERS 182 11.1.2 A THEORY OF GROUPS 186 11.2 DESIRABLE QUALITIES OF
THEORIES - CONCEPTS 191 11.2.1 AVERAGE APPLICABILITY OF CONCEPTS 192
11.2.2 AVERAGE COMPREHENSIBILITY OF CONCEPTS 195 11.2.3 NUMBER OF
CATEGORISATIONS 197 11.2.4 NUMBER OF CONCEPTS 199 11.3 DESIRABLE
QUALITIES OF THEORIES - CONJECTURES 201 11.3.1 DIFFICULTY AND
SURPRISINGNESS OF CONJECTURES 202 11.3.2 PROPORTION OF THEOREMS AND OPEN
CONJECTURES 203 11.4 USING THE HEURISTIC SEARCH 204 11.4.1 ROBUSTNESS OF
THE HEURISTIC MEASURES 204 11.4.2 DIFFERENCES BETWEEN DOMAINS 207 11.4.3
PRUNING USING THE HEURISTIC MEASURES 209 11.5 CLASSICALLY INTERESTING
RESULTS 211 11.5.1 GRAPH THEORY 212 11.5.2 GROUP THEORY 214 11.5.3
NUMBER THEORY 216 11.6 CONCLUSIONS 221 12. THE APPLICATION OF HR TO
DISCOVERY TASKS 225 12.1 A CLASSIFICATION PROBLEM 226 12.2 EXPLORATION
OF AN ALGEBRAIC SYSTEM 230 12.3 INVENTION OF INTEGER SEQUENCES 233
12.3.1 ADDITIONS TO THE ENCYCLOPEDIA 233 12.3.2 REFACTORABLE NUMBERS 237
XIV CONTENTS 12.3.3 SEQUENCE A046951 240 12.4 DISCOVERY TASK FAILURES
241 12.5 VALDES-PEREZ'S MACHINE DISCOVERY CRITERIA 242 12.5.1 NOVELTY
242 12.5.2 INTERESTINGNESS 243 12.5.3 PLAUSIBILITY 243 12.5.4
INTELLIGIBILITY 244 12.6 CONCLUSIONS 244 13. RELATED WORK 247 13.1 A
COMPARISON OF HR AND THE AM PROGRAM 247 13.1.1 HOW AM FORMED THEORIES
248 13.1.2 MISCONCEPTIONS ABOUT AM 250 13.1.3 PROGRAMS BASED ON AM 253
13.1.4 A QUALITATIVE COMPARISON OF AM AND HR 254 13.1.5 A QUANTITATIVE
COMPARISON OF AM AND HR 259 13.1.6 SUMMARY: THE AM PROGRAM 262 13.2 A
COMPARISON OF HR AND THE GT PROGRAM 264 13.2.1 HOW GT FORMED THEORIES
264 13.2.2 THE SCOT PROGRAM 266 13.2.3 A QUALITATIVE COMPARISON OF GT
AND HR 266 13.2.4 A QUANTITATIVE COMPARISON OF GT AND HR 267 13.3 A
COMPARISON OF HR AND THE IL PROGRAM 269 13.3.1 HOW IL WORKED 269 13.3.2
A QUALITATIVE COMPARISON OF IL AND HR 270 13.4 A COMPARISON OF HR AND
BAGAI ET AL'S PROGRAM 271 13.4.1 HOW BAGAI ET AL'S PROGRAM WORKED 271
13.4.2 A QUALITATIVE COMPARISON OF HR AND BAGAI ET AL'S PROGRAM 272 13.5
A COMPARISON OF HR AND THE GRAFFITI PROGRAM 273 13.5.1 HOW GRAFFITI
WORKS 273 13.5.2 A QUALITATIVE COMPARISON OF GRAFFITI AND HR 275 13.6 A
COMPARISON OF HR AND THE PROGOL PROGRAM 276 13.7 SUMMARY 277 14. FURTHER
WORK 281 14.1 ADDITIONAL THEORY FORMATION ABILITIES 281 14.1.1
ADDITIONAL PRODUCTION RULES 282 14.1.2 IMPROVED PRESENTATIONAL ASPECTS
282 14.1.3 FURTHER POSSIBILITIES FOR MAKING AND SETTLING CONJECTURES 283
14.2 THE APPLICATION OF THEORY FORMATION 284 14.2.1 AUTOMATED CONJECTURE
MAKING IN MATHEMATICS 285 14.2.2 CONSTRAINT SATISFACTION PROBLEMS 286
14.2.3 MACHINE LEARNING 287 CONTENTS XV 14.2.4 AUTOMATED THEOREM PROVING
288 14.2.5 APPLICATION TO OTHER SCIENTIFIC DOMAINS 289 14.3 THEORETICAL
EXPLORATIONS 290 14.3.1 META-THEORY FORMATION 290 14.3.2 CROSS DOMAIN
THEORY FORMATION 291 14.3.3 AGENT BASED COOPERATIVE THEORY FORMATION 292
14.4 SUMMARY 293 15. CONCLUSIONS 295 15.1 HAVE WE ACHIEVED OUR AIMS? 296
15.2 CONTRIBUTIONS 297 15.2.1 FUNCTIONALITY 297 15.2.2 SIMPLICITY OF
ARCHITECTURE 298 15.2.3 CYCLE OF MATHEMATICAL ACTIVITY 298 15.2.4
GENERALITY OF METHODS 299 15.2.5 MATHEMATICAL DISCOVERY 299 15.2.6
EVALUATION TECHNIQUES 300 15.3 AUTOMATED THEORY FORMATION IN PURE MATHS
301 APPENDIX A. USER MANUAL FOR HR1.11 303 A.I INSTALLING HR 1.11 304
A.2 SPECIFYING SETTINGS 305 A.3 INITIALISING THEORIES 307 A.4
CONSTRUCTING THEORIES 309 A.5 INVESTIGATING THEORIES 309 A.5.1 PRINTING
RESULTS TO SCREEN 310 A.5.2 VIEWING GRAPHICAL INFORMATION 310 A.5.3
FINDING CONCEPTS AND CONJECTURES 314 A.5.4 MAKING MORE CONJECTURES 316
A.6 HELP FOR A NEW USER 318 APPENDIX B. EXAMPLE SESSIONS 319 B.I GRAPH
THEORY SHORT SESSION 320 B.I.I SESSION OUTPUT 320 B.I.2 COMMENTARY 323
B.2 THEORY FORMATION SESSION IN GROUP THEORY 326 B.2.1 SESSION OUTPUT
327 B.2.2 COMMENTARY 330 B.3 THEORY FORMATION SESSION IN SEMIGROUP
THEORY 332 B.3.1 SESSION OUTPUT 332 B.3.2 COMMENTARY 337 B.4 INVENTING
AND INVESTIGATING AN INTEGER SEQUENCE 338 B.4.1 SESSION OUTPUT 339 B.4.2
COMMENTARY 341 XVI CONTENTS APPENDIX C. NUMBER THEORY RESULTS 343 C.I
REFACTORABLE NUMBERS 343 C.I.I INITIAL RESULTS 344 C.I.2 RELATION TO
OTHER NUMBER TYPES 345 C.1.3 PAIRS AND TRIPLES OF REFACTORABLES 348
C.1.4 DISTRIBUTION 350 C.2 INTEGERS WITH A PRIME NUMBER OF DIVISORS 352
C3 OTHER RESULTS 354 C.4 DIVISOR GRAPHS 356 GLOSSARY 359 BIBLIOGRAPHY
365 INDEX 375 |
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| author_sort | Colton, Simon |
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| bvnumber | BV023526486 |
| callnumber-first | Q - Science |
| callnumber-label | QA8 |
| callnumber-raw | QA8.4.C64 2002 |
| callnumber-search | QA8.4.C64 2002 |
| callnumber-sort | QA 18.4 C64 42002 |
| callnumber-subject | QA - Mathematics |
| classification_rvk | SK 130 ST 304 |
| ctrlnum | (OCoLC)248805597 (DE-599)BVBBV023526486 |
| dewey-full | 511.3/028521 |
| dewey-hundreds | 500 - Natural sciences and mathematics |
| dewey-ones | 511 - General principles of mathematics |
| dewey-raw | 511.3/0285 21 |
| dewey-search | 511.3/0285 21 |
| dewey-sort | 3511.3 3285 221 |
| dewey-tens | 510 - Mathematics |
| discipline | Informatik Mathematik |
| discipline_str_mv | Informatik Mathematik |
| format | Book |
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| illustrated | Not Illustrated |
| index_date | 2024-07-02T22:34:05Z |
| indexdate | 2024-07-09T21:23:56Z |
| institution | BVB |
| isbn | 1852336099 9781852336097 |
| language | English |
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| physical | XVI, 380 S. graph Darst. 24 cm |
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| spelling | Colton, Simon Verfasser aut Automated theory formation in pure mathematics Simon Colton London ; Berlin ; Heidelberg ; New York ; Barcelona ; Hong Kong ; Milan ; Paris ; Singapore ; Tokyo Springer 2002 XVI, 380 S. graph Darst. 24 cm txt rdacontent n rdamedia nc rdacarrier Distinguished dissertations Literaturverz. S. 365 - 373 Datenverarbeitung Künstliche Intelligenz Mathematik Mathematics -- Methodology -- Data processing Artificial intelligence Automatic theorem proving Expert systems (Computer science) Mathematik (DE-588)4037944-9 gnd rswk-swf Programm (DE-588)4047394-6 gnd rswk-swf Theorie (DE-588)4059787-8 gnd rswk-swf Mathematik (DE-588)4037944-9 s Theorie (DE-588)4059787-8 s Programm (DE-588)4047394-6 s DE-604 GBV Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=016846721&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Colton, Simon Automated theory formation in pure mathematics Datenverarbeitung Künstliche Intelligenz Mathematik Mathematics -- Methodology -- Data processing Artificial intelligence Automatic theorem proving Expert systems (Computer science) Mathematik (DE-588)4037944-9 gnd Programm (DE-588)4047394-6 gnd Theorie (DE-588)4059787-8 gnd |
| subject_GND | (DE-588)4037944-9 (DE-588)4047394-6 (DE-588)4059787-8 |
| title | Automated theory formation in pure mathematics |
| title_auth | Automated theory formation in pure mathematics |
| title_exact_search | Automated theory formation in pure mathematics |
| title_exact_search_txtP | Automated theory formation in pure mathematics |
| title_full | Automated theory formation in pure mathematics Simon Colton |
| title_fullStr | Automated theory formation in pure mathematics Simon Colton |
| title_full_unstemmed | Automated theory formation in pure mathematics Simon Colton |
| title_short | Automated theory formation in pure mathematics |
| title_sort | automated theory formation in pure mathematics |
| topic | Datenverarbeitung Künstliche Intelligenz Mathematik Mathematics -- Methodology -- Data processing Artificial intelligence Automatic theorem proving Expert systems (Computer science) Mathematik (DE-588)4037944-9 gnd Programm (DE-588)4047394-6 gnd Theorie (DE-588)4059787-8 gnd |
| topic_facet | Datenverarbeitung Künstliche Intelligenz Mathematik Mathematics -- Methodology -- Data processing Artificial intelligence Automatic theorem proving Expert systems (Computer science) Programm Theorie |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=016846721&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
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