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On Satisfiability Checking for Difference Logic

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On Satisfiability Checking for Difference Logic UNIVERSITE´ JOSEPH FOURIER – GRENOBLE 1 SCIENCES, TECHNOLOGIE ET MEDECINE´ THESE` pour obtenir le grade de DOCTEUR DE L’UNIVERSITE´ JOSEPH FOURIER en Informatique pr´epar´ee au laboratoire VERIMAG pr´esent´ee et soutenue publiquement par Moez MAHFOUDH le 27 Mai 2003 Sur la V´erification de la Satisfaction pour la Logique des Diff´erences Directeur de th`ese : Oded MALER Membres du jury : Anatoli IOUDITSKI, Pr´esident Karem A. SAKALLAH, Rapporteur Henrik Reif ANDERSEN, Rapporteur Oded MALER, Directeur de th`ese Alessandro CIMATTI, Examinateur Eug`ene ASARIN, Examinateur Peter NIEBERT, Examinateur Amam`` ere, `amonp`ere, `amasœur... Remerciements Je tiens `a remercier Anatoli Iouditski, Professeura ` l’Universit´e Joseph Fourier, pour avoir pr´esid´e le jury de cette th`ese. Je remercie aussi messieurs Karem A. Sakallah, Professeura ` l’Universit´e du Michigan, Henrik Reif Andersen, Pro- fesseur `a IT University of Copenhagen, et Alessandro Cimatti, chercheura ` l’Instituto per la Ricerca Scientifica e Tecnologicaa ` Trento, Italie, pour avoir accept´e de juger ce travail avec beaucoup d’int´erˆet et pour avoir particip´eau jury de cette th`ese. Je remercie aussi Eug`ene Asarin, Professeura ` l’Universit´e Joseph Fourier et chercheur au laboratoire VERIMAG, pour avoir toujours ´et´e disponible pour des discussions, souvent longues, concernant mes recherches, et pour avoir fait partie du jury. De mˆeme, je remercie Peter Niebert, Maˆıtre de conf´erence `a l’Universit´ede Provence, pour sa participation au jury. Il a ´et´e la source dinspiration premi`ere de ce travail : Il a jet´e ses bases alors qu’il ´etait encore `aVERIMAGeta continu´e`a suivre son avancement apr`es son d´epart et ce malgr´el’´eloignement g´eographique. Je voudrais aussi exprimer ma gratitudea ` Oded Maler, Directeur de recherche au CNRS et chercheur au laboratoire VERIMAG, qui m’a permis de faire cette th`ese. J’ai appr´eci´e son encadrement lib´eral et l’autonomie qu’il m’a accord´ee pour faire mes recherches. J’aurai toujours le souvenir de nos discus- sions o`u nous confrontions nos visions, souvent diam´etralement oppos´ees, sur l’informatique, les sciences, le monde et la vie. Merci Oded, merci pour tout. Je tiens aussi `a remercier tous les chercheurs de VERIMAG et plus sp´ecia- lement le directeur du laboratoire, Joseph Sifakis, ainsi que Yassine Lakhnech, Marius Bozga, Sergio Yovine, Stavros Tripakis, Saddek Bensalem, et Nicolas Halbwachs, pour leurs aides administratives ou scientifiques. Un grand merci aussi au personnel administratif de VERIMAG et plus sp´ecifiquement `a Jeanne-Marie Colle pour ses conseils et son aide lors de mon installationa ` Grenoble. Je remercie Riadh Robbana, Maˆıtre assistant `a l’Universit´e de Carthage, qui m’a permis d’´etablir le lien initial avec VERIMAG lorsque j’´etais encore ´el`eve-ing´enieur `a l’Ecole Polytechnique de Tunisie. Je remercie Yasmina, Thao, Gordon, Christos, Radu, (ex)membres de VER- IMAG,pourleuramiti´esinc`ere et nos longues discussions dans les caf´es de la ville o`usemˆelaient culture, cin´ema et gamineries. Je remercie Chaker, Ramzi, Cyril, David, Alexandre et Karim, ´etudiantsa ` VERIMAG, avec qui je partageais mes pauses et des discussions sur la vie et la v m´et´eo. Merci aussi `a Marcello pour les pauses caf´e et son aide tr`es appr´eciable lors la pr´eparation de ma soutenance de th`ese. Merci `a mes amis de Grenoble, sp´ecialement Walid Dkhil et Walid Hasni, pour leur soutien et pour nos sorties en ville o`u nous tentions de nous changer les id´ees en oubliant le stress des ´etudes et des th`eses. Merci `a mes amis et anciens de l’Ecole Polytechnique de Tunisie, et plus particuli`erement merci `a Firaz, Olfa, Anis, Amin, Hakim, Fadhel, qui ont su me rendre la vie plus agr´eable durant leurs s´ejours `a Grenoble. Merci aussi `a mes amis de toujours, Badii, Karim, et Helmi, que j’ai connus sur les bancs du coll`ege (et mˆeme bien avant) et qui, bien que vivanta ` Tunis, m’ont toujours fait sentir que je n’´etais pas si ´eloign´e d’eux avec leurs coup de fils, leurs emails et leurs messages. Merci pour votre amiti´e et pour vos encouragements. Finalement, je remercie ma m`ere, Nebiha, mon p`ere, Mohamed, ma sœur, Asma, et tous les membres de ma famille, pour leur amour, leur soutien ind´efectible et leur aide. Ce travail a ´et´e fait en pensant `a vous, il vous est naturellement d´edi´e. Merci pour tout. Que Dieu vous b´enisse. Contents 1 Introduction 1 1.1Motivation.............................. 1 1.2RelatedWork............................. 2 1.3ThesisOutline............................ 3 2 Timed Formalisms 5 2.1 Difference Logic ............................ 5 2.2 Satisfiability . ............................ 6 2.3 Conjunctive Normal Form . .................... 7 2.4TranslatingfromDLtoMX-CNF.................. 8 2.4.1 Trivialtranslation...................... 8 2.4.2 Tseitin’stranslation..................... 9 2.4.3 Wilson’stranslation..................... 12 2.5 Conjunctions of Difference Constraints . ............ 14 2.5.1 Convextimedpolyhedra................... 14 2.5.2 Bounds ............................ 14 2.5.3 RepresentingconvextimedpolyhedrausingDBMs.... 16 2.5.4 CanonicalrepresentationofDBMs............. 17 2.5.5 Conjunction of DBMs .................... 21 3SATSolvers 23 3.1 Overview . ............................ 23 3.2History................................ 24 3.2.1 Before 1960 . ........................ 24 3.2.2 From 1960 to 1990 . .................... 24 3.2.3 From 1990 . ........................ 25 3.3BasicAlgorithm........................... 25 3.3.1 Description.......................... 25 3.3.2 Improvementstothebasicalgorithm............ 26 3.4 Binary constraint propagation .................... 27 3.4.1 Unitresolution........................ 27 3.4.2 Pureliteralsresolution.................... 27 3.4.3 Equivalentliteralsdetection................. 27 3.5Branchingheuristics......................... 27 3.5.1 Maximizingsatisfiedclausesheuristics........... 27 3.5.2 Literalcountheuristics................... 30 vii viii CONTENTS 3.5.3 Variablestateindependentheuristics............ 30 3.6 Conflict Directed Backtracking and Learning . ........ 31 3.6.1 Implicationgraph...................... 32 3.6.2 Conflictanalysis....................... 34 3.6.3 Conflict-freelearning..................... 37 3.6.4 Backtracking......................... 37 3.6.5 Clause recording . .................... 38 3.6.6 Restarts............................38 3.7AlternativeTechniques........................ 38 3.7.1 Resolution `alaDavis-Putnam . ............ 38 3.7.2 Localsearch.......................... 40 3.7.3 St˚almarck’smethod..................... 41 4 A Mixed SAT Solver 45 4.1 Overview . ............................ 45 4.1.1 Solvingcontext........................ 45 4.1.2 Assignments......................... 47 4.1.3 Clauserewriting....................... 47 4.2SimplificationRules......................... 50 4.2.1 Booleanunitresolution................... 50 4.2.2 Booleanliteralsequivalencedetection........... 51 4.2.3 PureBooleanliteralsresolution............... 51 4.2.4 DBMupdating........................ 52 4.2.5 DBManalysis......................... 52 4.3 Davis-Putnam Rules . ........................ 53 4.3.1 Restricted Boolean Davis-Putnam rule application .... 53 4.3.2 Restricted numerical Davis-Putnam rule application . 54 4.4DPLLBranching........................... 55 4.5ConflictAnalysis........................... 61 4.5.1 Extensionofsolvingcontexts................ 61 4.5.2 Changesinoperations.................... 61 4.5.3 Conflictdetectionandanalysis............... 63 4.6MX-SolverImplementation..................... 64 4.6.1 Technical overview . .................... 64 4.6.2 Datastructures........................ 64 4.6.3 Features............................ 66 4.6.4 Designguidelines....................... 67 5 Timed Systems 69 5.1TimedAutomata........................... 69 5.1.1 Flattimedautomata..................... 69 5.1.2 TranslationintoDL..................... 71 5.1.3 Composition......................... 74 5.2AsynchronousDigitalCircuits.................... 77 5.2.1 Usedmodel.......................... 77 5.2.2 Conversiontotimedautomata............... 80 5.3 Non-Preemptive Job-Shops . .................... 81 CONTENTS ix 5.3.1 Introductoryexample.................... 81 5.3.2 Expressionofjob-shopproblems.............. 82 5.3.3 TranslationintoDL..................... 83 6 Experimental Results 85 6.1 Study Methodology . ........................ 85 6.2Job-Shopproblems.......................... 86 6.2.1 Overview . ........................ 86 6.2.2 Instances........................... 86 6.2.3 Results............................ 89 6.3Timed-Automataproblems..................... 90 6.3.1 Overview . ........................ 90 6.3.2 Instances........................... 90 6.3.3 Results............................ 90 6.4Asynchronouscircuitsproblems................... 92 6.4.1 Overview . ........................ 92 6.4.2 Instances........................... 93 6.4.3 Results............................ 93 6.5Conclusions.............................. 94 7 Conclusions 97 7.1 Contribution . ............................ 97 7.2 Future Research Directions . .................... 97 7.2.1 NewAlgorithmsandMethods................ 98 7.2.2 TheMXToolbox....................... 98 AMXToolbox 99 A.1 Overview . ............................ 99 A.2ToDLTranslators.......................... 99 A.2.1TimedautomatatoDL................... 99 A.2.2AsynchronousdigitalcircuitstoDL............101 A.2.3Job-shoptoDL........................102 A.3DLtoMX-CNFTranslator.....................102
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