A 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.
From the reviews: "This book has two topics as main points: 'decision procedures' and 'algorithms to be applied for these decisions'. ... Such an approach is very useful for graduate students or students in year four or five ... . Each chapter introduces and explains a lot of different concepts and presents good examples, some problems and exercises with algorithm-based solutions, and a glossary at its end, which makes the book very applicable and readable. ... The book is very well written and interesting to read." (Christian Posthoff, Zentralblatt MATH, Vol. 1149, 2008) "There are a number of things that I like about the book. Although the material is often complex, the exposition is extremely clear, precise and meticulous. A definition is almost always followed by a clarifying example. As noted above, Kroening and Strichman go to great efforts to explain the notations used. In fact, I'd be hard pressed to think of another text on an advanced subject that does so much to make the material accessible to the reader." (K. Harrow, ACM Computing Reviews, September 2008) "This book is a developed version of classroom notes for a course on decision procedures, intended for undergraduate and graduate students ... . The developments are intended for programmers who need to know about the algorithms that solve their particular problems, as well as for students and researchers who need to know how to define such algorithms in the framework of appropriate logics. ... Overall, the book is well structured ... ." (Siva Anantharaman, Mathematical Reviews, Issue 2009 m)
Introduction and Basic Concepts | p. 1 |
Two Approaches to Formal Reasoning | p. 3 |
Proof by Deduction | p. 3 |
Proof by Enumeration | p. 4 |
Deduction and Enumeration | p. 5 |
Basic Definitions | p. 5 |
Normal Forms and Some of Their Properties | p. 8 |
The Theoretical Point of View | p. 14 |
The Problem We Solve | p. 17 |
Our Presentation of Theories | p. 17 |
Expressiveness vs. Decidability | p. 18 |
Boolean Structure in Decision Problems | p. 19 |
Problems | p. 21 |
Glossary | p. 23 |
Decision Procedures for Propositional Logic | p. 25 |
Propositional Logic | p. 25 |
Motivation | p. 25 |
SAT Solvers | p. 27 |
The Progress of SAT Solving | p. 27 |
The DPLL Framework | p. 28 |
BCP and the Implication Graph | p. 30 |
Conflict Clauses and Resolution | p. 35 |
Decision Heuristics | p. 39 |
The Resolution Graph and the Unsatisfiable Core | p. 41 |
SAT Solvers: Summary | p. 42 |
Binary Decision Diagrams | p. 43 |
From Binary Decision Trees to ROBDDs | p. 43 |
Building BDDs from Formulas | p. 46 |
Problems | p. 50 |
Warm-up Exercises | p. 50 |
Modeling | p. 50 |
Complexity | p. 51 |
DPLL SAT Solving | p. 52 |
Related Problems | p. 52 |
Binary Decision Diagrams | p. 53 |
Bibliographic Notes | p. 54 |
Glossary | p. 57 |
Equality Logic and Uninterpreted Functions | p. 59 |
Introduction | p. 59 |
Complexity and Expressiveness | p. 59 |
Boolean Variables | p. 60 |
Removing the Constants: A Simplification | p. 60 |
Uninterpreted Functions | p. 60 |
How Uninterpreted Functions Are Used | p. 61 |
An Example: Proving Equivalence of Programs | p. 63 |
From Uninterpreted Functions to Equality Logic | p. 64 |
Ackermann's Reduction | p. 66 |
Bryant's Reduction | p. 69 |
Functional Consistency Is Not Enough | p. 72 |
Two Examples of the Use of Uninterpreted Functions | p. 74 |
Proving Equivalence of Circuits | p. 75 |
Verifying a Compilation Process with Translation Validation | p. 77 |
Problems | p. 78 |
Warm-up Exercises | p. 78 |
Problems | p. 78 |
Glossary | p. 79 |
Decision Procedures for Equality Logic and Uninterpreted Functions | p. 81 |
Congruence Closure | p. 81 |
Basic Concepts | p. 83 |
Simplifications of the Formula | p. 85 |
A Graph-Based Reduction to Propositional Logic | p. 88 |
Equalities and Small-Domain Instantiations | p. 92 |
Some Simple Bounds | p. 93 |
Graph-Based Domain Allocation | p. 94 |
The Domain Allocation Algorithm | p. 96 |
A Proof of Soundness | p. 98 |
Summary | p. 101 |
Ackermann's vs. Bryant's Reduction: Where Does It Matter? | p. 101 |
Problems | p. 103 |
Conjunctions of Equalities and Uninterpreted Functions | p. 103 |
Reductions | p. 104 |
Complexity | p. 105 |
Domain Allocation | p. 106 |
Bibliographic Notes | p. 106 |
Glossary | p. 108 |
Linear Arithmetic | p. 111 |
Introduction | p. 111 |
Solvers for Linear Arithmetic | p. 112 |
The Simplex Algorithm | p. 113 |
Decision Problems and Linear Programs | p. 113 |
Basics of the Simplex Algorithm | p. 114 |
Simplex with Upper and Lower Bounds | p. 116 |
Incremental Problems | p. 120 |
The Branch and Bound Method | p. 120 |
Cutting-Planes | p. 122 |
Fourier-Motzkin Variable Elimination | p. 126 |
Equality Constraints | p. 126 |
Variable Elimination | p. 126 |
Complexity | p. 129 |
The Omega Test | p. 129 |
Problem Description | p. 129 |
Equality Constraints | p. 130 |
Inequality Constraints | p. 132 |
Preprocessing | p. 138 |
Preprocessing of Linear Systems | p. 138 |
Preprocessing of Integer Linear Systems | p. 139 |
Difference Logic | p. 140 |
Introduction | p. 140 |
A Decision Procedure for Difference Logic | p. 142 |
Problems | p. 142 |
Warm-up Exercises | p. 142 |
The Simplex Method | p. 143 |
Integer Linear Systems | p. 143 |
Omega Test | p. 144 |
Difference Logic | p. 145 |
Bibliographic Notes | p. 145 |
Glossary | p. 146 |
Bit Vectors | p. 149 |
Bit-Vector Arithmetic | p. 149 |
Syntax | p. 149 |
Notation | p. 151 |
Semantics | p. 152 |
Deciding Bit-Vector Arithmetic with Flattening | p. 156 |
Converting the Skeleton | p. 156 |
Arithmetic Operators | p. 157 |
Incremental Bit Flattening | p. 160 |
Some Operators Are Hard | p. 160 |
Enforcing Functional Consistency | p. 162 |
Using Solvers for Linear Arithmetic | p. 163 |
Motivation | p. 163 |
Integer Linear Arithmetic for Bit Vectors | p. 163 |
Fixed-Point Arithmetic | p. 165 |
Semantics | p. 165 |
Flattening | p. 167 |
Problems | p. 167 |
Semantics | p. 167 |
Bit-Level Encodings of Bit-Vector Arithmetic | p. 168 |
Using Solvers for Linear Arithmetic | p. 169 |
Bibliographic Notes | p. 169 |
Glossary | p. 170 |
Arrays | p. 171 |
Introduction | p. 171 |
Arrays as Uninterpreted Functions | p. 172 |
A Reduction Algorithm for Array Logic | p. 175 |
Array Properties | p. 175 |
A Reduction Algorithm | p. 176 |
Problems | p. 178 |
Bibliographic Notes | p. 178 |
Glossary | p. 179 |
Pointer Logic | p. 181 |
Introduction | p. 181 |
Pointers and Their Applications | p. 181 |
Dynamic Memory Allocation | p. 182 |
Analysis of Programs with Pointers | p. 184 |
A Simple Pointer Logic | p. 185 |
Syntax | p. 185 |
Semantics | p. 187 |
Axiomatization of the Memory Model | p. 188 |
Adding Structure Types | p. 189 |
Modeling Heap-Allocated Data Structures | p. 190 |
Lists | p. 190 |
Trees | p. 191 |
A Decision Procedure | p. 193 |
Applying the Semantic Translation | p. 193 |
Pure Variables | p. 195 |
Partitioning the Memory | p. 196 |
Rule-Based Decision Procedures | p. 197 |
A Reachability Predicate for Linked Structures | p. 198 |
Deciding Reachability Predicate Formulas | p. 199 |
Problems | p. 202 |
Pointer Formulas | p. 202 |
Reachability Predicates | p. 203 |
Bibliographic Notes | p. 204 |
Glossary | p. 206 |
Quantified Formulas | p. 207 |
Introduction | p. 207 |
Example: Quantified Boolean Formulas | p. 209 |
Example: Quantified Disjunctive Linear Arithmetic | p. 211 |
Quantifier Elimination | p. 211 |
Prenex Normal Form | p. 211 |
Quantifier Elimination Algorithms | p. 213 |
Quantifier Elimination for Quantified Boolean Formulas | p. 214 |
Quantifier Elimination for Quantified Disjunctive Linear Arithmetic | p. 217 |
Search-Based Algorithms for QBF | p. 218 |
Problems | p. 220 |
Warm-up Exercises | p. 220 |
QBF | p. 220 |
Bibliographic Notes | p. 223 |
Glossary | p. 224 |
Deciding a Combination of Theories | p. 225 |
Introduction | p. 225 |
Preliminaries | p. 225 |
The Nelson-Oppen Combination Procedure | p. 227 |
Combining Convex Theories | p. 227 |
Combining Nonconvex Theories | p. 230 |
Proof of Correctness of the Nelson-Oppen Procedure | p. 233 |
Problems | p. 236 |
Bibliographic Notes | p. 236 |
Glossary | p. 239 |
Propositional Encodings | p. 241 |
Overview | p. 241 |
Lazy Encodings | p. 244 |
Definitions and Notations | p. 244 |
Building Propositional Encodings | p. 245 |
Integration into DPLL | p. 246 |
Theory Propagation and the DPLL(T) Framework | p. 246 |
Some Implementation Details of DPLL(T) | p. 250 |
Propositional Encodings with Proofs (Advanced) | p. 253 |
Encoding Proofs | p. 254 |
Complete Proofs | p. 255 |
Eager Encodings | p. 257 |
Criteria for Complete Proofs | p. 258 |
Algorithms for Generating Complete Proofs | p. 259 |
Problems | p. 263 |
Bibliographic Notes | p. 264 |
Glossary | p. 267 |
The SMT-LIB Initiative | p. 269 |
A C++ Library for Developing Decision Procedures | p. 271 |
Introduction | p. 271 |
Graphs and Trees | p. 272 |
Adding "Payload" | p. 274 |
Parsing | p. 274 |
A Grammar for First-Order Logic | p. 274 |
The Problem File Format | p. 276 |
A Class for Storing Identifiers | p. 277 |
The Parse Tree | p. 277 |
CNF and SAT | p. 278 |
Generating CNF | p. 278 |
Converting the Propositional Skeleton | p. 281 |
A Template for a Lazy Decision Procedure | p. 281 |
References | p. 285 |
Index | p. 299 |
Table of Contents provided by Ingram. All Rights Reserved. |
ISBN: 9783540741046
ISBN-10: 3540741046
Series: Texts in Theoretical Computer Science. An EATCS Series
Audience:
Professional
Format:
Hardcover
Language:
English
Number Of Pages: 306
Published: 23rd May 2008
Publisher: Springer-Verlag Berlin and Heidelberg Gmbh & Co. Kg
Country of Publication: DE
Dimensions (cm): 23.5 x 15.5
x 2.29
Weight (kg): 0.68