The Calculus of computation: decision procedures with applications to verification
Gespeichert in:
| Hauptverfasser: | , |
|---|---|
| Format: | Buch |
| Sprache: | English |
| Veröffentlicht: |
Berlin [u.a.]
Springer
2007
|
| Schlagworte: | |
| Online-Zugang: | Inhaltsverzeichnis |
| Beschreibung: | Hier auch später erschienene, unveränderte Nachdrucke |
| Beschreibung: | XV, 366 S. graph. Darst. |
| ISBN: | 9783540741121 3540741127 9783642093470 |
Internformat
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| 245 | 1 | 0 | |a The Calculus of computation |b decision procedures with applications to verification |c Aaron R. Bradley ; Zohar Manna |
| 264 | 1 | |a Berlin [u.a.] |b Springer |c 2007 | |
| 300 | |a XV, 366 S. |b graph. Darst. | ||
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Datensatz im Suchindex
| _version_ | 1804137230254473216 |
|---|---|
| adam_text | Contents
Part I Foundations
1 Propositional Logic 3
1.1 Syntax 4
1.2 Semantics 6
1.3 Satisfiability and Validity 8
1.3.1 Truth Tables 9
1.3.2 Semantic Arguments 10
1.4 Equivalence and Implication 14
1.5 Substitution 16
1.6 Normal Forms 18
1.7 Decision Procedures for Satisfiability 21
1.7.1 Simple Decision Procedures 21
1.7.2 Reconsidering the Truth Table Method 22
1.7.3 Conversion to an EquisatLsfiable Formula in CNF 24
1.7.4 The Resolution Procedure 27
1.7.5 DPLL 28
1.8 Summary 31
Bibliographic Remarks 32
Exercises 32
2 First Order Logic 35
2.1 Syntax 35
2.2 Semantics 39
2.3 Satisfiability and Validity 42
2.4 Substitution 45
2.4.1 Safe Substitution 47
2.4.2 Schema Substitution 48
2.5 Normal Forms 51
2.6 Decidability and Complexity 53
2.6.1 Satisfiability as a Formal Language 53
XII Contents
2.6.2 Decidability 54
2.6.3 ^Complexity 54
2.7 *Meta Theorems of First Order Logic 56
2.7.1 Simplifying the Language of FOL 57
2.7.2 Semantic Argument Proof Rules 58
2.7.3 Soundness and Completeness 58
2.7.4 Additional Theorems 61
2.8 Summary 66
Bibliographic Remarks 67
Exercises 67
3 First Order Theories 69
3.1 First Order Theories 69
3.2 Equality 71
3.3 Natural Numbers and Integers 73
3.3.1 Peano Arithmetic 73
3.3.2 Presburger Arithmetic 75
3.3.3 Theory of Integers 76
3.4 Rationals and Reals 79
3.4.1 Theory of Reals 80
3.4.2 Theory of Rationals 82
3.5 Recursive Data Structures 84
3.6 Arrays 87
3.7 *Survey of Decidability and Complexity 90
3.8 Combination Theories 91
3.9 Summary 92
Bibliographic Remarks 93
Exercises 93
4 Induction 95
4.1 Stepwise Induction 95
4.2 Complete Induction 99
4.3 Well Founded Induction 102
4.4 Structural Induction 108
4.5 Summary 110
Bibliographic Remarks Ill
Exercises Ill
5 Program Correctness: Mechanics 113
5.1 pi: A Simple Imperative Language 114
5.1.1 The Language 115
5.1.2 Program Annotations 118
5.2 Partial Correctness 123
5.2.1 Basic Paths: Loops 125
5.2.2 Basic Paths: Function Calls 131
Contents XIII
5.2.3 Program States 135
5.2.4 Verification Conditions 136
5.2.5 P Invariant and P Inductive 142
5.3 Total Correctness 143
5.4 Summary 149
Bibliographic Remarks 150
Exercises 151
6 Program Correctness: Strategies 153
6.1 Developing Inductive Annotations 153
6.1.1 Basic Facts 154
6.1.2 The Precondition Method 156
6.1.3 A Strategy 162
6.2 Extended Example: QuickSort 164
6.2.1 Partial Correctness 167
6.2.2 Total Correctness 171
6.3 Summary 172
Bibliographic Remarks 173
Exercises 173
Part II Algorithmic Reasoning
7 Quantified Linear Arithmetic 183
7.1 Quantifier Elimination 184
7.1.1 Quantifier Elimination 184
7.1.2 A Simplification 185
7.2 Quantifier Elimination over Integers 185
7.2.1 Augmented Theory of Integers 185
7.2.2 Cooper s Method 187
7.2.3 A Symmetric Elimination 194
7.2.4 Eliminating Blocks of Quantifiers 195
7.2.5 ^Solving Divides Constraints 196
7.3 Quantifier Elimination over Rationals 200
7.3.1 Ferrante and Rackoff s Method 200
7.4 * Complexity 204
7.5 Summary 204
Bibliographic Remarks 205
Exercises 205
8 Quantifier Free Linear Arithmetic 207
8.1 Decision Procedures for Quantifier Free Fragments 207
8.2 Preliminary Concepts and Notation 209
8.3 Linear Programs 213
8.4 The Simplex Method 218
XIV Contents
8.4.1 From M to Mo 219
8.4.2 Vertex Traversal 223
8.4.3 *Complexity 237
8.5 Summary 237
Bibliographic Remarks 238
Exercises 238
9 Quantifier Free Equality and Data Structures 241
9.1 Theory of Equality 242
9.2 Congruence Closure Algorithm 244
9.2.1 Relations 245
9.2.2 Congruence Closure Algorithm 247
9.3 Congruence Closure with DAGs 251
9.3.1 Directed Acyclic Graphs 251
9.3.2 Basic Operations 254
9.3.3 Congruence Closure Algorithm 255
9.3.4 Decision Procedure for TE Satisfiability 256
9.3.5 * Complexity 258
9.4 Recursive Data Structures 259
9.5 Arrays 263
9.6 Summary 265
Bibliographic Remarks 266
Exercises 267
10 Combining Decision Procedures 269
10.1 Combining Decision Procedures 269
10.2 Nelson Oppen Method: Nondeterministic Version 271
10.2.1 Phase 1: Variable Abstraction 271
10.2.2 Phase 2: Guess and Check 273
10.2.3 Practical Efficiency 274
10.3 Nelson Oppen Method: Deterministic Version 276
10.3.1 Convex Theories 276
10.3.2 Phase 2: Equality Propagation 278
10.3.3 Equality Propagation: Implementation 282
10.4 Correctness of the Nelson Oppen Method 283
10.5 Complexity 287
10.6 Summary 288
Bibliographic Remarks 288
Exercises 288
11 Arrays 291
11.1 Arrays with Uninterpreted Indices 292
11.1.1 Array Property Fragment 292
11.1.2 Decision Procedure 294
11.2 Integer Indexed Arrays 299
Contents XV
11.2.1 Array Property Fragment 300
11.2.2 Decision Procedure 301
11.3 Hashtables 304
11.3.1 Hashtable Property Fragment 305
11.3.2 Decision Procedure 306
11.4 Larger Fragments 308
11.5 Summary 309
Bibliographic Remarks 310
Exercises 310
12 Invariant Generation 311
12.1 Invariant Generation 311
12.1.1 Weakest Precondition and Strongest Postcondition .... 312
12.1.2 * General Definitions of wp and sp 315
12.1.3 Static Analysis 316
12.1.4 Abstraction 319
12.2 Interval Analysis 325
12.3 Karr s Analysis 333
12.4 ^Standard Notation and Concepts 341
12.5 Summary 344
Bibliographic Remarks 345
Exercises 345
13 Further Reading 347
References 351
Index 357
|
| adam_txt |
Contents
Part I Foundations
1 Propositional Logic 3
1.1 Syntax 4
1.2 Semantics 6
1.3 Satisfiability and Validity 8
1.3.1 Truth Tables 9
1.3.2 Semantic Arguments 10
1.4 Equivalence and Implication 14
1.5 Substitution 16
1.6 Normal Forms 18
1.7 Decision Procedures for Satisfiability 21
1.7.1 Simple Decision Procedures 21
1.7.2 Reconsidering the Truth Table Method 22
1.7.3 Conversion to an EquisatLsfiable Formula in CNF 24
1.7.4 The Resolution Procedure 27
1.7.5 DPLL 28
1.8 Summary 31
Bibliographic Remarks 32
Exercises 32
2 First Order Logic 35
2.1 Syntax 35
2.2 Semantics 39
2.3 Satisfiability and Validity 42
2.4 Substitution 45
2.4.1 Safe Substitution 47
2.4.2 Schema Substitution 48
2.5 Normal Forms 51
2.6 Decidability and Complexity 53
2.6.1 Satisfiability as a Formal Language 53
XII Contents
2.6.2 Decidability 54
2.6.3 ^Complexity 54
2.7 *Meta Theorems of First Order Logic 56
2.7.1 Simplifying the Language of FOL 57
2.7.2 Semantic Argument Proof Rules 58
2.7.3 Soundness and Completeness 58
2.7.4 Additional Theorems 61
2.8 Summary 66
Bibliographic Remarks 67
Exercises 67
3 First Order Theories 69
3.1 First Order Theories 69
3.2 Equality 71
3.3 Natural Numbers and Integers 73
3.3.1 Peano Arithmetic 73
3.3.2 Presburger Arithmetic 75
3.3.3 Theory of Integers 76
3.4 Rationals and Reals 79
3.4.1 Theory of Reals 80
3.4.2 Theory of Rationals 82
3.5 Recursive Data Structures 84
3.6 Arrays 87
3.7 *Survey of Decidability and Complexity 90
3.8 Combination Theories 91
3.9 Summary 92
Bibliographic Remarks 93
Exercises 93
4 Induction 95
4.1 Stepwise Induction 95
4.2 Complete Induction 99
4.3 Well Founded Induction 102
4.4 Structural Induction 108
4.5 Summary 110
Bibliographic Remarks Ill
Exercises Ill
5 Program Correctness: Mechanics 113
5.1 pi: A Simple Imperative Language 114
5.1.1 The Language 115
5.1.2 Program Annotations 118
5.2 Partial Correctness 123
5.2.1 Basic Paths: Loops 125
5.2.2 Basic Paths: Function Calls 131
Contents XIII
5.2.3 Program States 135
5.2.4 Verification Conditions 136
5.2.5 P Invariant and P Inductive 142
5.3 Total Correctness 143
5.4 Summary 149
Bibliographic Remarks 150
Exercises 151
6 Program Correctness: Strategies 153
6.1 Developing Inductive Annotations 153
6.1.1 Basic Facts 154
6.1.2 The Precondition Method 156
6.1.3 A Strategy 162
6.2 Extended Example: QuickSort 164
6.2.1 Partial Correctness 167
6.2.2 Total Correctness 171
6.3 Summary 172
Bibliographic Remarks 173
Exercises 173
Part II Algorithmic Reasoning
7 Quantified Linear Arithmetic 183
7.1 Quantifier Elimination 184
7.1.1 Quantifier Elimination 184
7.1.2 A Simplification 185
7.2 Quantifier Elimination over Integers 185
7.2.1 Augmented Theory of Integers 185
7.2.2 Cooper's Method 187
7.2.3 A Symmetric Elimination 194
7.2.4 Eliminating Blocks of Quantifiers 195
7.2.5 ^Solving Divides Constraints 196
7.3 Quantifier Elimination over Rationals 200
7.3.1 Ferrante and Rackoff's Method 200
7.4 * Complexity 204
7.5 Summary 204
Bibliographic Remarks 205
Exercises 205
8 Quantifier Free Linear Arithmetic 207
8.1 Decision Procedures for Quantifier Free Fragments 207
8.2 Preliminary Concepts and Notation 209
8.3 Linear Programs 213
8.4 The Simplex Method 218
XIV Contents
8.4.1 From M to Mo 219
8.4.2 Vertex Traversal 223
8.4.3 *Complexity 237
8.5 Summary 237
Bibliographic Remarks 238
Exercises 238
9 Quantifier Free Equality and Data Structures 241
9.1 Theory of Equality 242
9.2 Congruence Closure Algorithm 244
9.2.1 Relations 245
9.2.2 Congruence Closure Algorithm 247
9.3 Congruence Closure with DAGs 251
9.3.1 Directed Acyclic Graphs 251
9.3.2 Basic Operations 254
9.3.3 Congruence Closure Algorithm 255
9.3.4 Decision Procedure for TE Satisfiability 256
9.3.5 * Complexity 258
9.4 Recursive Data Structures 259
9.5 Arrays 263
9.6 Summary 265
Bibliographic Remarks 266
Exercises 267
10 Combining Decision Procedures 269
10.1 Combining Decision Procedures 269
10.2 Nelson Oppen Method: Nondeterministic Version 271
10.2.1 Phase 1: Variable Abstraction 271
10.2.2 Phase 2: Guess and Check 273
10.2.3 Practical Efficiency 274
10.3 Nelson Oppen Method: Deterministic Version 276
10.3.1 Convex Theories 276
10.3.2 Phase 2: Equality Propagation 278
10.3.3 Equality Propagation: Implementation 282
10.4 Correctness of the Nelson Oppen Method 283
10.5 Complexity 287
10.6 Summary 288
Bibliographic Remarks 288
Exercises 288
11 Arrays 291
11.1 Arrays with Uninterpreted Indices 292
11.1.1 Array Property Fragment 292
11.1.2 Decision Procedure 294
11.2 Integer Indexed Arrays 299
Contents XV
11.2.1 Array Property Fragment 300
11.2.2 Decision Procedure 301
11.3 Hashtables 304
11.3.1 Hashtable Property Fragment 305
11.3.2 Decision Procedure 306
11.4 Larger Fragments 308
11.5 Summary 309
Bibliographic Remarks 310
Exercises 310
12 Invariant Generation 311
12.1 Invariant Generation 311
12.1.1 Weakest Precondition and Strongest Postcondition . 312
12.1.2 * General Definitions of wp and sp 315
12.1.3 Static Analysis 316
12.1.4 Abstraction 319
12.2 Interval Analysis 325
12.3 Karr's Analysis 333
12.4 ^Standard Notation and Concepts 341
12.5 Summary 344
Bibliographic Remarks 345
Exercises 345
13 Further Reading 347
References 351
Index 357 |
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| index_date | 2024-07-02T19:10:37Z |
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| oclc_num | 255687662 |
| open_access_boolean | |
| owner | DE-703 DE-29T DE-83 DE-11 |
| owner_facet | DE-703 DE-29T DE-83 DE-11 |
| physical | XV, 366 S. graph. Darst. |
| publishDate | 2007 |
| publishDateSearch | 2007 |
| publishDateSort | 2007 |
| publisher | Springer |
| record_format | marc |
| spelling | Bradley, Aaron R. Verfasser aut The Calculus of computation decision procedures with applications to verification Aaron R. Bradley ; Zohar Manna Berlin [u.a.] Springer 2007 XV, 366 S. graph. Darst. txt rdacontent n rdamedia nc rdacarrier Hier auch später erschienene, unveränderte Nachdrucke Programmverifikation - Korrektheit - Computational logic Computer logic Computer programs Verification Programmverifikation (DE-588)4135576-3 gnd rswk-swf Korrektheit (DE-588)4240223-2 gnd rswk-swf Computational logic (DE-588)4255672-7 gnd rswk-swf Programmverifikation (DE-588)4135576-3 s Korrektheit (DE-588)4240223-2 s Computational logic (DE-588)4255672-7 s DE-604 Manna, Zohar 1939-2018 Verfasser (DE-588)128838248 aut HBZ Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=016218429&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Bradley, Aaron R. Manna, Zohar 1939-2018 The Calculus of computation decision procedures with applications to verification Programmverifikation - Korrektheit - Computational logic Computer logic Computer programs Verification Programmverifikation (DE-588)4135576-3 gnd Korrektheit (DE-588)4240223-2 gnd Computational logic (DE-588)4255672-7 gnd |
| subject_GND | (DE-588)4135576-3 (DE-588)4240223-2 (DE-588)4255672-7 |
| title | The Calculus of computation decision procedures with applications to verification |
| title_auth | The Calculus of computation decision procedures with applications to verification |
| title_exact_search | The Calculus of computation decision procedures with applications to verification |
| title_exact_search_txtP | The Calculus of computation decision procedures with applications to verification |
| title_full | The Calculus of computation decision procedures with applications to verification Aaron R. Bradley ; Zohar Manna |
| title_fullStr | The Calculus of computation decision procedures with applications to verification Aaron R. Bradley ; Zohar Manna |
| title_full_unstemmed | The Calculus of computation decision procedures with applications to verification Aaron R. Bradley ; Zohar Manna |
| title_short | The Calculus of computation |
| title_sort | the calculus of computation decision procedures with applications to verification |
| title_sub | decision procedures with applications to verification |
| topic | Programmverifikation - Korrektheit - Computational logic Computer logic Computer programs Verification Programmverifikation (DE-588)4135576-3 gnd Korrektheit (DE-588)4240223-2 gnd Computational logic (DE-588)4255672-7 gnd |
| topic_facet | Programmverifikation - Korrektheit - Computational logic Computer logic Computer programs Verification Programmverifikation Korrektheit Computational logic |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=016218429&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| work_keys_str_mv | AT bradleyaaronr thecalculusofcomputationdecisionprocedureswithapplicationstoverification AT mannazohar thecalculusofcomputationdecisionprocedureswithapplicationstoverification |