1. www.itu.dk 1 Knowledge Compilation Properties of Tree-of-BDDs Sathiamoorthy Subbarayan AAAI-2007, Vancouver Lucas Bordeaux, Youssef Hamadi IT University of Copenhagen Denmark Microsoft Research, Cambridge UK

2. www.itu.dk 2 Knowledge Compilation Compile a system model in to a form –Forms: BDD, DNNF, d-DNNF, Tree-of-BDDs Quickly answer interesting queries Applications: configuration, verification, fault-trees, Bayesian networks, model- based diagnosis, etc.,

3. www.itu.dk 3 Compilation Forms FormSizeCompiler DNNF (JACM 2001) Tree-of-BDDs (CPAIOR 2005) d-DNNF (JAIR 2002) BDD (IEEE Comp. 1986) Tree-of-BDDs even 1000 times smaller than d-DNNF

4. www.itu.dk 4 Binary Decision Diagrams A compressed representation of solutions Answers many queries in polytime

5. www.itu.dk 5 CNF to BDD bdd CNF clauses BDD C1 C4 C2 C3 C5 C7 C6 c1c2c3c4c5c6c7 clause BDDs AND

6. www.itu.dk 6 Tree-of-BDDs (ToB) C1 C2 C3 C4 C5 C6 C7 Tree Decomposition ToB CNF clauses C1 C4 C2 C3 C5 C7 C6 c1c2c3c4c5c6c7 clause BDDs b1 b2 b3 AND+Propagate

7. www.itu.dk 7 Example: CNF to ToB : (¬a+¬e) : (a+b) : (b+¬c) : (c+¬d) : (¬b+¬d) C1 C4 C2 C3 C5 c5c3c1c2c4 C1 C2 C4 C5 C3 Tree Decomposition ToB CNF clauses Group clause- BDDs AND+Propagate

8. www.itu.dk 8 Example Comparison OBDD 14 edges d-DNNF 15 edges 12 edges ToB Exploits tree- width!

9. www.itu.dk 9 Some Interesting Queries Consistency (CO) Validity (VA) Clausal Entailment (CE) Implicant Check (IM) Equality (EQ) Sentential Entailment (SE) Model Counting (CT) Model Enumeration (ME)

10. www.itu.dk 10 Conditioning ToB Condition (ToB, term) //Restrict ToB by term Restrict each BDD with term Propagate Worst case exponential in tree-width!

11. www.itu.dk 11 Clausal Entailment, Implicant Check IsCE (ToB, clause) //Does ToB entails the clause Condition (ToB, ¬clause) if (ToB=false) return true else return false IsIM (ToB, term) //Does the term entails ToB Condition (ToB, term) if ( term entails each BDD ) return true else return false

12. www.itu.dk 12 Complexity of Queries FormCOVACEIMEQSECTME DNNF ToB d-DNNF OBDD Polytime Polytime if P=NP Worst case exponential in w Worst case exponential in (w 1 + w 2 )

13. www.itu.dk 13 Query Complexity Vs. Size FormCOVACEIMEQSECTMESize DNNF ToB d-DNNF OBDD Realistic instances often have low treewidth ! ToB exploits it well !!

14. www.itu.dk 14 Experiments 91 realistic CNF instances –35 Mercedes car configuration problems –7 ISCAS85 circuits –17 ISCAS89 circuits –14 ISCAS93 Addendum circuits –13 ISCAS99 circuits –5 Bounded model checking circuits Compare with c2d, a d-DNNF compiler Simulate CE and IM queries on ToBs

15. www.itu.dk 15 Tools, Instances, Presentation http://www.itu.dk/~sathi/tob/

16. www.itu.dk 16 Compilation Overview Formd-DNNF (hg)d-DNNF (mf)ToB #compiled798391 #failed1280 Total time*1365613176926 Total size*412.7 m602.2 m17.6 m *for successful compilations We could not find any realistic instance where d-DNNF beats ToB in compilation ! Realistic: Industrial instance in SAT competition

17. www.itu.dk 17 Details: Cases hard for d-DNNF Instanced-DNNF (hg)d-DNNF (mf)ToB Name|V||C|wTimeSizeTimeSizeTimeSize c190875120534626926.1TO 0.10.02 c7552318585884437128.9TO 1.020.22 s126962316164011910TO 0.180.05 s32711714426935TO 0.550.15 s48632495643433 FSE 5.091.31 s66693392842326TO 34857.530.440.16 cnt0992073067823TO 1680.011.860.09 cnt10204706856126TO 5.410.21 C168FW19097477110TO 2965.548.050.21 C170FR187410610203TO 3655.9150.58 C202FS1990888388TO 9.580.41 C202FW20381134299TO 650.94 C208FC19227518194TO 2439.56180.93 C210FS1990798284TO 237051.34120.91 C210FW2024970598TO 335271.636633.64 Very low treewidth! ToB even 1000 times smaller than d-DNNF!

18. www.itu.dk 18 CE and IM Simulation InstanceToB - CE TimeToB - IM Time NameMeanMax>1s %MeanMax>1s % c19080.010.0600.010.040 c75520.060.4600.060.490 s12690.020.1300.020.140 s32710.040.1900.040.190 s48630.296.1360.314.495 s66690.030.3100.050.330 cnt090.020.0700.020.070 cnt100.040.1700.040.160 C168FW0.020.2300.040.220 C170FR0.040.7700.110.750 C202FS0.040.7500.10.760 C202FW0.092.6220.212.264 C208FC0.171.4230.171.373 C210FS0.092.1620.222.175 C210FW0.3810.741.0110.4230 Quick responses!

19. www.itu.dk 19 Compilation Time: ToB vs. d-DNNF ToB is much faster than d-DNNF

20. www.itu.dk 20 Compilation Space: ToB vs. d-DNNF ToB is much smaller than d-DNNF

21. www.itu.dk 21 Conclusion ToBs often very small in time and space Even 1000 times smaller than d-DNNF Still quick response to many queries Future work: –EQ Check, usefulness in verification ? –Succinctness of ToB vs d-DNNF –Transformations supported by ToB –Adopting ToBs for multi-core CPUs

22. www.itu.dk 22 Thanks!

23. www.itu.dk 23 FormCOVACEIMEQSECTMESize DNNF ToB d-DNNF OBDD Realistic instances often have low treewidth ! ToB exploits it well !! Query Complexity Vs. Size

24. www.itu.dk 24 Propagation in ToB Use a rooted BFS tree Propagate up –Makes each parent consistent with children Propagate down –Makes each child consistent with parent

25. www.itu.dk 25 Projection in ToB Project (ToB, K) //Project ToB over K variables Conjoin BDDs from bottom to top Early quantification of non-K variables Worst case exponential in (w+|K|)

26. www.itu.dk 26 Equivalence Checking IsEQ (ToB 1, ToB 2 ) Project ToB 1 over vars in each node of ToB 2 -Check whether the two BDDs agree Project ToB 2 over vars in each node of ToB 1 -Check whether the two BDDs agree Worst case exponential in (w 1 +w 2 )