Introduction to Mathematics of Satisfiability Chapman & Hall/CRC Studies in Informatics Series SERIES EDITOR G. Q. Zhang Case Western Reserve University Department of EECS Cleveland, Ohio, U.S.A. PUBLISHED TITLES Stochastic Relations: Foundations for Markov Transition Systems Ernst-Erich Doberkat Conceptual Structures in Practice Pascal Hitzler and Henrik Schärfe Context-Aware Computing and Self-Managing Systems Waltenegus Dargie Introduction to Mathematics of Satisfiability Victor W. Marek K10101_FM.indd 2 8/7/09 11:32:54 AM Chapman & Hall/CRC Studies in Informatics Series Introduction to Mathematics of Satisfiability Victor W. 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Visit the Taylor & Francis Web site at http://www.taylorandfrancis.com and the CRC Press Web site at http://www.crcpress.com Contents Preface ix 1 Sets, lattices, and Boolean algebras 1 1.1 Setsandset-theoreticnotation . 1 1.2 Posets,lattices,andBooleanalgebras . 3 1.3 Well-orderingsandordinals . 5 1.4 Thefixpointtheorem ......................... 6 1.5 Exercises ............................... 9 2 Introduction to propositional logic 11 2.1 Syntaxofpropositionallogic . 11 2.2 Semanticsofpropositionallogic . 13 2.3 Autarkies ............................... 23 2.4 Tautologiesandsubstitutions . 28 2.5 Lindenbaumalgebra ......................... 32 2.6 Permutations ............................. 34 2.7 Duality ................................ 38 2.8 Semantical consequence, operations Mod and Th .......... 39 2.9 Exercises ............................... 42 3 Normal forms of formulas 45 3.1 Canonicalnegation-normalform . 46 3.2 Occurrencesofvariablesandthree-valuedlogic . 48 3.3 Canonicalforms ........................... 50 3.4 Reducednormalforms ........................ 54 3.5 Completenormalforms .. .... .... .... ... .... .. 56 3.6 Lindenbaumalgebrarevisited . 58 3.7 Othernormalforms.......................... 59 3.8 Exercises ............................... 60 4 The Craig lemma 63 4.1 Craiglemma ............................. 63 4.2 StrongCraiglemma ......................... 66 4.3 Tyinguplooseends ......................... 69 4.4 Exercises ............................... 71 v vi 5 Complete sets of functors 73 5.1 BeyondDeMorganfunctors . 74 5.2 Tables................................. 75 5.3 Field structure in Bool ........................ 78 5.4 Incompletesetsoffunctors,Postclasses . 83 5.5 Postcriterionforcompleteness . 85 5.6 If-then-elsefunctor . .... .... .... .... ... .... .. 88 5.7 Exercises ............................... 90 6 Compactness theorem 93 6.1 K¨oniglemma ............................. 93 6.2 Compactness,denumerablecase . 95 6.3 Continuity of the operator Cn .................... 99 6.4 Exercises ............................... 100 7 Clausal logic and resolution 101 7.1 Clausallogic ............................. 102 7.2 Resolutionrule ............................ 107 7.3 Completenessresults . 110 7.4 Query-answeringwithresolution . 113 7.5 Davis-Putnamlemma . 117 7.6 Semanticresolution .... .... .... .... ... .... .. 119 7.7 Autarkandleansets ......................... 124 7.8 Exercises ............................... 132 8 Testing satisfiability 133 8.1 Tablemethod ............................. 133 8.2 Hintikkasets ............................. 135 8.3 Tableaux ............................... 137 8.4 Davis-Putnamalgorithm . 144 8.5 Booleanconstraintpropagation . 154 8.6 TheDPLLalgorithm ......................... 158 8.7 ImprovementstoDPLL? . 161 8.8 ReductionofthesearchSATtodecisionSAT . 162 8.9 Exercises ............................... 163 9 Polynomial cases of SAT 165 9.1 Positiveandnegativeformulas . 165 9.2 Hornformulas ............................ 167 9.3 AutarkiesforHorntheories . 176 9.4 DualHornformulas ......................... 181 9.5 Kromformulasand2-SAT . 185 9.6 Renameableclassesofformulas . 194 9.7 Affineformulas ............................ 199 9.8 Exercises ............................... 204 vii 10 SAT, integer programming, and matrix algebra 205 10.1 Representingclausesbyinequalities . 206 10.2 Resolutionandotherrulesofproof . 207 10.3 Pigeon-holeprincipleandthe cutting planerule . 209 10.4 Satisfiability and {−1, 1}-integerprogramming . 214 10.5 EmbeddingSATintomatrixalgebra . 216 10.6Exercises ............................... 225 11 Coding runs of Turing machines, NP-completeness 227 11.1 Turingmachines ........................... 228 11.2Thelanguage ............................. 231 11.3 Codingtheruns ............................ 232 11.4 Correctnessofourcoding . 233 11.5 Reductionto3-clauses . 237 11.6 Codingformulasasclausesandcircuits . 239 11.7 Decisionproblemforautarkies . 243 11.8 Searchproblemforautarkies . 245 11.9 Either-orCNFs ............................ 247 11.10Othercases .............................. 249 11.11Exercises ............................... 252 12 Computational knowledge representation with SAT – gettingstarted 253 12.1 EncodingintoSAT,DIMACSformat . 254 12.2 Knowledgerepresentationoverfinitedomains . 261 12.3 Cardinality constraints, the language Lcc .............. 267 12.4 Weightconstraints . 273 12.5 Monotoneconstraints . 276 12.6Exercises ............................... 283 13 Knowledge representation and constraint satisfaction 285 13.1 Extensional and intentional relations, CWA ............. 285 13.2 ConstraintsatisfactionandSAT . 292 13.3 Satisfiability as constraint satisfaction . 297 13.4 PolynomialcasesofBooleanCSP . 300 13.5 Schaeferdichotomytheorem . 305 13.6Exercises ............................... 317 14 Answer set programming 321 14.1 Hornlogicrevisited . 321 14.2 Modelsofprograms .... .... .... .... ... .... .. 322 14.3 Supportedmodels . .... .... .... .... ... .... .. 323 14.4 Stablemodels ............................. 326 14.5 AnswersetprogrammingandSAT . 329 14.6 KnowledgerepresentationandASP . 333 14.6.1 Three-coloringofgraphs . 334 viii 14.6.2 HamiltoniancyclesinASP. 335 14.7 ComplexityissuesforASP . 336 14.8Exercises ............................... 337 15 Conclusions 339 References 343 Index 347 Preface The subject of this book is satisfiability of theories consisting of propositional logic formulas, a topic with over 80 years of history. After an initial push, the subject was abandoned for some time by the mathematicians and analytical philosophers who contributed to its inception. Electrical and computer engineering, and also computer science, picked up the research of this area, starting with the work of Shannon in the 1930s. Several aspects of satisfiability are important for electronic design automa- tion and have been pursued by computer engineers. Yet another reason to look at satisfiability has been a remarkable result of Levin and Cook - the fact that satisfi- ability expresses a huge class of problems of interest to computer science (we will present this fundamental result in Chapter 11). The fundamental algorithms for sat- isfiability testing (the simplest method, that of truth tables, was invented very early, in the 1920s, but is obviously very inefficient) were invented in the 1960s. While there was steady progress in understanding satisfiability, to transform it into useful technology the area had to mature and develop new techniques. Resolution proofs were studied in the decade of the 1970s, but it was not enough to make the area prac- tically important (although the fans of the programming language PROLOG may be of a different opinion). Additional breakthroughs were needed and in the mid-1990s they occurred. The theoretical advances in satisfiability resulted in creation of a class of software, called SAT solvers. Solvers are used to find solutions to search problems encoded by