## Decision Procedures: An Algorithmic Point of ViewA decision procedure is an algorithm that, given a decision problem, terminates with a correct yes/no answer. Here, the authors focus on theories that are expressive enough to model real problems, but are still decidable. Specifically, the book concentrates on decision procedures for first-order theories that are commonly used in automated verification and reasoning, theorem-proving, compiler optimization and operations research. The techniques described in the book draw from fields such as graph theory and logic, and are routinely used in industry. The authors introduce the basic terminology of satisfiability modulo theories and then, in separate chapters, study decision procedures for each of the following theories: propositional logic; equalities and uninterpreted functions; linear arithmetic; bit vectors; arrays; pointer logic; and quantified formulas. They also study the problem of deciding combined theories and dedicate a chapter to modern techniques based on an interplay between a SAT solver and a decision procedure for the investigated theory. This textbook has been used to teach undergraduate and graduate courses at ETH Zurich, at the Technion, Haifa, and at the University of Oxford. Each chapter includes a detailed bibliography and exercises. Lecturers' slides and a C++ library for rapid prototyping of decision procedures are available from the authors' website. |

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### Contents

Introduction and Basic Concepts | 1 |

11 Two Approaches to Formal Reasoning | 3 |

112 Proof by Enumeration | 4 |

113 Deduction and Enumeration | 5 |

13 Normal Forms and Some of Their Properties | 8 |

14 The Theoretical Point of View | 14 |

141 The Problem We Solve | 17 |

15 Expressiveness vs Decidability | 18 |

62 Deciding BitVector Arithmetic with Flattening | 156 |

622 Arithmetic Operators | 157 |

63 Incremental Bit Flattening | 160 |

632 Enforcing Functional Consistency | 162 |

64 Using Solvers for Linear Arithmetic | 163 |

65 FixedPoint Arithmetic | 165 |

652 Flattening | 167 |

662 BitLevel Encodings of BitVector Arithmetic | 168 |

16 Boolean Structure in Decision Problems | 19 |

17 Problems | 21 |

18 Glossary | 23 |

Decision Procedures for Propositional Logic | 24 |

22 SAT Solvers | 27 |

222 The DPLL Framework | 28 |

223 BCP and the Implication Graph | 30 |

224 Conflict Clauses and Resolution | 35 |

225 Decision Heuristics | 39 |

226 The Resolution Graph and the Unsatisfiable Core | 41 |

Summary | 42 |

23 Binary Decision Diagrams | 43 |

232 Building BDDs from Formulas | 46 |

24 Problems | 50 |

243 Complexity | 51 |

244 DPLL SAT Solving | 52 |

246 Binary Decision Diagrams | 53 |

25 Bibliographic Notes SAT | 54 |

26 Glossary | 57 |

Equality Logic and Uninterpreted Functions | 59 |

312 Boolean Variables | 60 |

321 How Uninterpreted Functions Are Used | 61 |

Proving Equivalence of Programs | 63 |

33 From Uninterpreted Functions to Equality Logic | 64 |

331 Ackermanns Reduction | 66 |

332 Bryants Reduction | 69 |

34 Functional Consistency Is Not Enough | 72 |

35 Two Examples of the Use of Uninterpreted Functions | 74 |

351 Proving Equivalence of Circuits | 75 |

352 Verifying a Compilation Process with Translation Validation | 77 |

36 Problems | 78 |

37 Glossary | 79 |

Decision Procedures for Equality Logic and Uninterpreted Functions | 81 |

42 Basic Concepts | 83 |

43 Simplifications of the Formula | 85 |

44 A GraphBased Reduction to Propositional Logic | 88 |

45 Equalities and SmallDomain Instantiations | 92 |

451 Some Simple Bounds | 93 |

452 GraphBased Domain Allocation | 94 |

453 The Domain Allocation Algorithm | 96 |

454 A Proof of Soundness | 98 |

455 Summary | 101 |

47 Problems | 103 |

472 Reductions | 104 |

473 Complexity | 105 |

474 Domain Allocation | 106 |

49 Glossary | 108 |

Linear Arithmetic | 111 |

511 Solvers for Linear Arithmetic | 112 |

52 The Simplex Algorithm | 113 |

522 Basics of the Simplex Algorithm | 114 |

523 Simplex with Upper and Lower Bounds | 116 |

524 Incremental Problems | 120 |

531 CuttingPlanes | 122 |

54 FourierMotzkin Variable Elimination | 126 |

543 Complexity | 129 |

552 Equality Constraints | 130 |

553 Inequality Constraints | 132 |

56 Preprocessing | 138 |

562 Preprocessing of Integer Linear Systems | 139 |

57 Difference Logic | 140 |

572 A Decision Procedure for Difference Logic | 142 |

582 The Simplex Method | 143 |

584 Omega Test | 144 |

585 Difference Logic | 145 |

510 Glossary | 146 |

Bit Vectors | 148 |

612 Notation | 151 |

613 Semantics | 152 |

663 Using Solvers for Linear Arithmetic | 169 |

68 Glossary | 170 |

Arrays | 171 |

72 Arrays as Uninterpreted Functions | 172 |

73 A Reduction Algorithm for Array Logic | 175 |

732 A Reduction Algorithm | 176 |

74 Problems | 178 |

76 Glossary | 179 |

Pointer Logic | 181 |

812 Dynamic Memory Allocation | 182 |

813 Analysis of Programs with Pointers | 184 |

82 A Simple Pointer Logic | 185 |

822 Semantics | 187 |

823 Axiomatization of the Memory Model | 188 |

824 Adding Structure Types | 189 |

83 Modeling HeapAllocated Data Structures | 190 |

832 Trees | 191 |

84 A Decision Procedure | 193 |

842 Pure Variables | 195 |

843 Partitioning the Memory | 196 |

85 RuleBased Decision Procedures | 197 |

851 A Reachability Predicate for Linked Structures | 198 |

852 Deciding Reachability Predicate Formulas | 199 |

86 Problems | 202 |

862 Reachability Predicates | 203 |

87 Bibliographic Notes | 204 |

88 Glossary | 206 |

Quantified Formulas | 207 |

Quantified Boolean Formulas | 209 |

Quantified Disjunctive Linear Arithmetic | 211 |

922 Quantifier Elimination Algorithms | 213 |

923 Quantifier Elimination for Quantified Boolean Formulas | 214 |

924 Quantifier Elimination for Quantified Disjunctive Linear Arithmetic | 217 |

93 SearchBased Algorithms for Quantified Boolean Formulas | 218 |

94 Problems | 220 |

95 Bibliographic Notes | 223 |

96 Glossary | 224 |

Deciding a Combination of Theories | 225 |

103 The NelsonOppen Combination Procedure | 227 |

1032 Combining Nonconvex Theories | 230 |

1033 Proof of Correctness of the NelsonOppen Procedure | 233 |

104 Problems | 236 |

106 Glossary | 239 |

Propositional Encodings | 240 |

112 Lazy Encodings | 244 |

1122 A Lazy Procedure for Building Propositional Encodings | 245 |

1123 Integration into DPLL | 246 |

1125 Some Implementation Details of DPLLT | 250 |

113 Propositional Encodings with Proofs Advanced | 253 |

1131 Encoding Proofs | 254 |

1132 Complete Proofs | 255 |

1133 Eager Encodings | 257 |

1134 Criteria for Complete Proofs | 258 |

1135 Algorithms for Generating Complete Proofs | 259 |

114 Problems | 263 |

115 Bibliographic Notes | 264 |

116 Glossary | 267 |

The SatisfiabilityModuloTheory Library and Standard SMTLIB | 269 |

A C++ Library for Developing Decision Procedures | 271 |

B2 Graphs and Trees | 272 |

B21 Adding Payload | 274 |

B32 The Problem File Format | 276 |

B33 A Class for Storing Identifiers | 277 |

B4 CNF and SAT | 278 |

B42 Converting the Propositional Skeleton | 281 |

284 | |

299 | |

### Other editions - View all

Decision Procedures: An Algorithmic Point of View Daniel Kroening,Ofer Strichman Limited preview - 2008 |

Decision Procedures: An Algorithmic Point of View Daniel Kroening,Ofer Strichman No preview available - 2009 |

Decision Procedures: An Algorithmic Point of View Daniel Kroening,Ofer Strichman No preview available - 2010 |