1. Knowledge Compilation Properties of Tree-of-BDDs Sathiamoorthy SubbarayanLucas Bordeaux, Youssef Hamadi IT University of Copenhagen, Denmark Microsoft Research, Cambridge, UK 16 Details: Caseshardford-DNNF 3.6466371.633352TO9897052024C210FW 0.911251.342370TO8479821990C210FS 0.93189.56243TO19475181922C208FC 0.9465TO 99113422038C202FW 0.419.58TO 8888831990C202FS 0.58155.9365TO203106101874C170FR 0.218.055.54296TO11074771909C168FW 0.215.41TO 266856120470cnt10 0.091.860.01168TO23306789207cnt09 0.160.4457.53348TO2684233392s6669 1.315.09FSE 3364342495s4863 0.150.55TO 3542691714s3271 0.050.18TO10119401616623s1269 0.221.02TO28.93714485883185c7552 0.020.1TO26.1269462053751c1908 SizeTimeSizeTimeSizeTimew|C||V|Name ToBd-DNNF (mf)d-DNNF (hg)Instance 1 BinaryDecision Diagrams Acompressedrepresentationofsolutions Answersmanyqueriesinpolytime 7 KnowledgeCompilation Compilea system model in to aform Quicklyanswerinterestingqueries Applications:configuration,fault-trees, Bayesiannetworks,model-based diagnosis,verification, etc., 2 CNF to BDD bdd CNF clauses BDD C1 C4 C2 C3 C5 C7 C6 c1c2 c3 c4 c5 c6 c7 clause BDDs AND 3 Tree-of-BDDs(ToB) C1 C2 C3 C4 C5 C6 C7 Tree Decomposition ToB CNF clauses C1 C4 C2 C3 C5 C7 C6 c1 c2 c3c4 c5 c6 c7 clause BDDs b1 b2 b3 AND+Propagate 4 Propagation inToB UsearootedBFStree Propagateup –Makeseachparent consistentwithchildren Propagatedown –Makeseachchild consistentwithparent 5 Example: CNF toToB : (¬a+¬e) : (a+b) : (b+¬c) : (c+¬d) : (¬b+¬d) C1 C4 C2 C3 C5 c5 c3c1 c2 c4 C1 C2 C4 C5 C3 TreeDecomposition ToB CNF clauses Groupclause- BDDs AND+Propagate 6 ExampleComparison d-DNNF 15edges 12edges ToB Exploits tree-width! 8 SomeInterestingQueries Consistency(CO) Validity(VA) ClausalEntailment(CE) ImplicantCheck (IM) Equality(EQ) SententialEntailment(SE) ModelCounting(CT) ModelEnumeration(ME) 9 ConditioningToB Condition (ToB, term) //RestrictToBby term Restrict each BDD with term Propagate Exponentialintree-width! 10 Clausal Entailment,ImplicantCheck IsCE (ToB,clause) //DoesToBentails the clause Condition (ToB,¬clause) if(ToB=false) returntrue else returnfalse IsIM (ToB, term) //Does the term entailsToB Condition (ToB, term) if( termentailseachBDD ) returntrue else returnfalse 11 Projection inToB Project (ToB, K) //ProjectToBover K variables Conjoin BDDsfrombottomto top Earlyquantification ofnon - Kvariables Exponentialin (w+|K|) 12 Equivalence Checking IsEQ (ToB 1, ToB 2 ) Project ToB 1 overvarsineachnodeofToB 2 -CheckwhetherthetwoBDDsagree Project ToB 2 overvarsineachnodeofToB 1 -CheckwhetherthetwoBDDsagree Exponentialin (w 1 +w 2 ) 13 ComplexityofQueries OBDD d-DNNF ToB DNNF MECTSEEQIMCEVACOForm Polytime Not unless P=NP Exponential inw (w 1 + w 2 ) 14 Experiments 91realisticCNFinstances –35 Mercedescarconfigurationproblems –7 ISCAS85circuits –17 ISCAS89circuits –14 ISCAS93Addendumcircuits –13 ISCAS99circuits –5Boundedmodelcheckingcircuits Comparewithc2d, ad-DNNFcompiler SimulateCE and IMqueriesonToBs 15 CompilationOverview ToBtoolcompilesalltheused instancesin just 926seconds 17.6 m602.2 m412.7 mTotal size* 9261317613656Total time* 0812#failed 918379#compiled ToBd-DNNF (mf)d-DNNF (hg)Form *for successful compilations 17 CE and IM Simulation 18 CompilationTime:ToBvs.d-DNNF 19 CompilationSpace:ToBvs.d-DNNF 20 Conclusion ToBsoftenuseverysmall time and space ToBsstillanswerqueriesformany realisticinstances Futurework: –SuccinctnessofToBvsd-DNNF –TransformationssupportedbyToB –AdoptingToBsformulti-coreCPUs