Fundamental proof methods in computer science: a computer-based approach
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| Main Authors: | , |
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| Format: | Book |
| Language: | English |
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Cambridge, MA
MIT Press
[2017]
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| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Physical Description: | XXVII, 941 Seiten Diagramme |
| ISBN: | 9780262035538 |
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| adam_text | Contents
Preface xvii
I INTRODUCTION 1
1 An Overview of Fundamental Proof Methods 3
1.1 Equality chaining 3
1.2 Induction 8
1.3 Case analysis 10
1.4 Proof by contradiction 12
1.5 Abstraction/specialization 13
1.6 The usual case: Proof methods in combination 15
1.7 Automated proof 15
1.8 Structure of the book 16
2 Introduction to Athena 19
2.1 Interacting with Athena 22
2.2 Domains and function symbols 23
2.3 Terms 27
2.4 Sentences 34
2.5 Definitions 40
2.6 Assumption bases 42
2.7 Datatypes 44
2.8 Polymorphism 50
2.8.1 Polymorphic domains and sort identity 50
2.8.2 Polymorphic function symbols 52
2.8.3 Polymorphic datatypes 57
2.8.4 Integers and reals 59
2.9 Meta-identifiers 61
2.10 Expressions and deductions 62
i 2.10.1 Compositions 65
2.10.2 Nested method calls 68
2.10.3 Let expressions and deductions 69
2.10.4 Conclusion-annotated deductions 70
2.10.5 Conditional expressions and deductions 71
2.10.6 Pattern-matching expressions and deductions 72
2.10.7 Backtracking expressions and deductions 73
2.10.8 Defining procedures and methods 74
2.11 More on pattern matching 78
2.12 Directives 85
2.13 Overloading 86
2.14 Programming 89
2.14.1 Characters 90
2.14.2 Strings 90
2.14.3 Cells and vectors 90
2.14.4 Tables and maps 91
2.14.5 While loops 93
2.14.6 Expression sequences 93
2.14.7 Recursion 93
2.14.8 Substitutions 94
2.15 A consequence of static scoping 98
2.16 Miscellanea 99
2.17 Summary and notational conventions 103
2.18 Exercises 105
II FUNDAMENTAL PROOF METHODS 111
3 Proving Equalities 113
3.1 Numeric equations 113
3.2 Equality chaining preview 116
3.3 Terms and sentences as trees 117
3.4 The logic behind equality chaining 120
3.5 More examples of equality chaining 128
3.6 A more substantial proof example 130
3.7 A better proof 134
3.8 The principle of mathematical induction 135
3.8.1 Different ways of understanding mathematical induction 139
3.9 List equations 140
3,9.1 Polymorphic datatypes 147
3.10 Evaluation of ground terms 150
3.11 Top-down proof development 152
3.12 * Input expahsion and output transformation 158
3.12.1 Converters 158
3.12.2 Input expansion 162
3.12.3 Output transformation 165
3.12.4 Combining input expansion and output transformation with overloading 166
3.12.5 Using declare with auxiliary information 167
3.13 * Conjecture falsification 168
3.14 * Conditional rewriting and additional chaining features 172
3.15 ■*· Proper function definitions 179
3.16 Summary 185
3.17 Additional exercises 186
3.18 Chapter notes 189
4 Sentential Logic 191
4.1 Working with the Boolean constants 191
4.2 Working with conjunctions 192
4.2.1 Using conjunctions 192
4.2.2 Deriving conjunctions 193
4.3 Working with conditionals 194
4.3.1 Using conditionals 194
4.3.2 Deriving conditionals: hypothetical reasoning 195
4.4 Working with disjunctions 199
4.4.1 Using disjunctions: reasoning by cases 199
4.4.2 Deriving disjunctions 201
4.5 Working with negations 202
4.5.1 Using negations 202
4.5.2 Deriving negations: reasoning by contradiction 202
4.6 Working with biconditionals 206
4.6.1 Using biconditionals 206
4.6.2 Deriving biconditionals 207
4.7 Forcing a proof 207
4.8 Putting it all together 209
4.9 A library of useful methods for sentential reasoning 211
4.10 Recursive proof methods 226
4.11 Dealing with large conjunctions and disjunctions 236
4.12 Sentential logic semantics 238
4.13 SAT solving 246
4.14 Proof heuristics for sentential logic 272
4.14.1 Backward tactics 274
4.14.2 Forward tactics 276
4.14.3 Replacement tactics 282
4.14.4 Strategies for deploying the tactics 282
4.15 * A theorem prover for sentential logic 294
4.16 Additional exercises 304
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4.17 Chapter notes
First-Order Logic
5.1 Working with universal quantifications
5.1.1 Using universal quantifications
5.1.2 Deriving universal quantifications
5.2 Working with existential quantifications
5.2.1 Deriving existential quantifications
5.2.2 Using existential quantifications
5.3 Some examples
5.4 Methods for quantifier reasoning
5.5 Proof heuristics for first-order logic
5.5.1 Backward tactics for quantifiers
5.5.2 Forward tactics for quantifiers
5.5.3 Proof strategy for first-order logic
5.6 First-order logic semantics
5.7 Additional exercises
5.8 Chapter notes
Implication Chaining
6.1 Implication chains
6.2 Using sentences as justifiers
6.2.1 Nested rules
6.3 Implication chaining through sentential structure
6.4 Using chains with chain-last
6.5 Backward chains and chain-first
6.6 Equivalence chains
6.7 Mixing equational, implication, and equivalence steps
6.8 Chain nesting
6.9 Exercises
PROOFS ABOUT FUNDAMENTAL DATATYPES
I
Organizing Theory Development with Athena Modules
7.1 Introducing a module
7.2 Natural numbers using modules
7.3 Extending a module
7.4 Modules for function symbols
7.5 Additional module features
7.6 Additional module procedures 436
7.7 A note on indentation 437
8 Natural Number Orderings 439
8.1 Properties of natural number ordering functions 439
8.1.1 Trichotomy properties 443
8.1.2 Transitive and asymmetric properties 444
8.1.3 Less-equal properties 447
8.1.4 Combining ordering and arithmetic 450
8.2 Natural number subtraction 451
8.3 Ordered lists 459
8.4 Binary search trees 463
8.5 Summary and a connecting theorem 469
8.6 Additional exercises 472
8.7 Chapter notes 473
9 Integer Representations and Proof Mappings 475
9.1 Declarations and axioms 475
9.2 First proofs of integer properties 477
9.3 Another integer representation 478
9.4 Mappings between the signed and pair representations 480
9.5 Additive homomorphism property 481
9.6 Associativity and commutativity of integer addition 483
9.7 Power series 484
9.8 Summary and looking ahead 487
9.9 Additional exercises 488
10 Fundamental Discrete Structures 491
10.1 Ordered pairs 493
10.1.1 Representation and notation 493
10.1.2 Results and methods 494
10.2 Options 497
10.2.1 Representation and notation 497
10.2.2 Some useful results 498
10.3 Sets, relations, and functions 499
10.3.1 Representation and notation 499
10.3.2 Set membership, the subset relation, and set identity 501
10.3.3 Set operations 508
10.3.4 Cartesian products 518
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10.3.5 Relations
10.3.6 Set cardinality
10.3.7 Powersets
10.4 Maps
10.4.1 Representation and notation
10.4.2 Map operations and theorems
10.4.3 Default maps
10.5 Chapter notes
PROOFS ABOUT ALGORITHMS
A Binary Search Algorithm
11.1 Defining the algorithm
11.1.1 Efficiency considerations
11.1.2 Correspondence to definitions in other languages
11.1.3 Interface design
11.1.4 Testing with evaluation
11.2 First correctness properties
11.3 Specifying requirements on a function to be defined
11.4 Correctness of an optimized binary search algorithm
11.5 Summary and looking ahead
11.6 Additional exercises
A Fast Exponentiation Algorithm
12.1 Mathematical background
12.2 Strong induction
12.3 Properties of half
12.4 Properties of odd and even
12.5 Properties of power
12.6 Properties of fast-power
12.7 Tail recursion, a potential optimization
12.8 Transforming strong induction into ordinary induction
12.9 Measure induction
12.10 Summary and looking ahead
12.11 Additional exercises
Euclid’s Algorithm for Greatest Common Divisors
13.1 Quotient and remainder
13.2 The division algorithm
13.3 Divisibility 608
13.3.1 A cancellation lemma 610
13.3.2 Proof of the characterization theorem 610
13.3.3 Additional properties of divisibility 611
13.4 Euclid’s algorithm 616
13.5 Summary 621
13.6 Additional exercises 621
13.7 Chapter notes 623
PROOFS AT AN ABSTRACT LEVEL 625
Abstract Structures 627
14.1 Group properties 627
14.2 Theory refinement 631
14.3 Writing proofs at the level of a theory 635
14.4 Abstract proof method conventions 638
14.5 Dynamic evolution of theories 641
14.6 Testing abstract proofs 642
14.7 Group theory refinements 644
14.7.1 Abelian group theory 644
14.7.2 Multiplicative theories 646
14.7.3 Ring theory 649
14.7.4 Integral domain 652
14.7.5 Algebraic theory diagram 652
14.8 * Permutations as a group 653
14.8.1 Function theory 654
14.8.2 Permutation theory 658
14.9 Ordering properties at an abstract level 664
14.9.1 Binary-Relation 664
14.9.2 Irreflexive 665
14.9.3 Transitive 665
14.9.4 Strict partial order 666
14.9.5 Nonstrict partial orders 668
14.9.6 Strict weak order 671
14.9,7 A preorder 673
14.9.8 Strict total order 674
14.9.9 Lists over a strict weak order 674
14.9.10 Relational theory diagram 678
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14.10 Additional exercises
Abstract Algorithms
15.1 An abstract binary search algorithm
15.1.1 Abstract-level binary search trees
15.1.2 Abstract-level binary search correctness theorems
15.2 An abstract fast-power algorithm
15.2.1 Raising to a power in a monoid
15.2.2 A monoid version of fast-power
15.2.3 Multiplicative version of fast power
15.2.4 A nonnumeric application
Algorithms on Memory Abstractions
16.1 Axioms and theorems for individual memory locations
16.2 Iterators and ranges
16.2.1 Iterator and range axioms and theorems
16.2.2 Trivial iterator: The base of a hierarchy of iterator theories
16.2.3 Forward iterators
16.3 Range count algorithm
16.4 Range replace algorithm
16.5 Range copy algorithm
16.6 Range copy-backward algorithm
16.7 Adapters: Reverse-iterator and reverse-range
16.8 Implementing copy-backward
16.9 Random-access iterators
16.9.1 Relationships among iterator functions
16.9.2 New properties of the length function
16.9.3 Theorems about collecting locations
16.9.4 Ordered range
16.10 A binary Search algorithm
(%A
16.11 Summary and suggestions for continued study
PROOFS ABOUT PROGRAMMING LANGUAGES
A Correctness Proof for a Toy Compiler
17.1 Interpreting and compiling numeric expressions
17.1.1 Representation and notation
17.1.2 Defining the interpreter
17.1.3 An instruction set and a virtual machine
17.1.4 Compiling numeric expressions 774
17.1.5 Correctness 775
17.2 Handling errors explicitly 783
17.2.1 Extending the compiler with error handling 784
17.3 Chapter notes 797
18 A Simple Imperative Programming Language 799
18.1 A simple imperative language 799
18.2 Semantics of expressions 808
18.3 Semantics of commands 817
18.4 * Testing the semantics 827
18.5 Some lemmas 833
18.6 Reasoning about the language 844
18.7 Chapter notes 852
A Athena Reference 857
A.l Syntax 857
A.2 Values 858
A.3 Operational semantics 861
A.4 Pattern matching 875
A.5 Selectors 881
A,6 Prefix syntax 882
B Logic Programming and Prolog 885
B.l Basics of logic programming 885
B.2 Examples 888
B.3 Implementing a Prolog interpreter 893
B.4 Integration with external Prolog systems 898
B.5 Automating the handling of equations 902
Bibliography 905
Glossary 911
Index 921
|
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| spelling | Arkoudas, Konstantine aut Fundamental proof methods in computer science a computer-based approach Konstantine Arkoudas and David Musser Cambridge, MA MIT Press [2017] XXVII, 941 Seiten Diagramme txt rdacontent n rdamedia nc rdacarrier Informatik Mathematik Computer science Mathematics Proof theory Beweistheorie (DE-588)4145177-6 gnd rswk-swf Mathematische Logik (DE-588)4037951-6 gnd rswk-swf Datenverarbeitung (DE-588)4011152-0 gnd rswk-swf Beweistheorie (DE-588)4145177-6 s Mathematische Logik (DE-588)4037951-6 s Datenverarbeitung (DE-588)4011152-0 s DE-604 Musser, David R. aut Digitalisierung UB Passau - ADAM Catalogue Enrichment application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=029642335&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Arkoudas, Konstantine Musser, David R. Fundamental proof methods in computer science a computer-based approach Informatik Mathematik Computer science Mathematics Proof theory Beweistheorie (DE-588)4145177-6 gnd Mathematische Logik (DE-588)4037951-6 gnd Datenverarbeitung (DE-588)4011152-0 gnd |
| subject_GND | (DE-588)4145177-6 (DE-588)4037951-6 (DE-588)4011152-0 |
| title | Fundamental proof methods in computer science a computer-based approach |
| title_auth | Fundamental proof methods in computer science a computer-based approach |
| title_exact_search | Fundamental proof methods in computer science a computer-based approach |
| title_full | Fundamental proof methods in computer science a computer-based approach Konstantine Arkoudas and David Musser |
| title_fullStr | Fundamental proof methods in computer science a computer-based approach Konstantine Arkoudas and David Musser |
| title_full_unstemmed | Fundamental proof methods in computer science a computer-based approach Konstantine Arkoudas and David Musser |
| title_short | Fundamental proof methods in computer science |
| title_sort | fundamental proof methods in computer science a computer based approach |
| title_sub | a computer-based approach |
| topic | Informatik Mathematik Computer science Mathematics Proof theory Beweistheorie (DE-588)4145177-6 gnd Mathematische Logik (DE-588)4037951-6 gnd Datenverarbeitung (DE-588)4011152-0 gnd |
| topic_facet | Informatik Mathematik Computer science Mathematics Proof theory Beweistheorie Mathematische Logik Datenverarbeitung |
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