Presentation on theme: "Www.itu.dk 1 Knowledge Compilation Properties of Tree-of-BDDs Sathiamoorthy Subbarayan AAAI-2007, Vancouver Lucas Bordeaux, Youssef Hamadi IT University."— Presentation transcript:
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
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.,
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
www.itu.dk 4 Binary Decision Diagrams A compressed representation of solutions Answers many queries in polytime
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
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!
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!
www.itu.dk 19 Compilation Time: ToB vs. d-DNNF ToB is much faster than d-DNNF
www.itu.dk 20 Compilation Space: ToB vs. d-DNNF ToB is much smaller than d-DNNF
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
www.itu.dk 23 FormCOVACEIMEQSECTMESize DNNF ToB d-DNNF OBDD Realistic instances often have low treewidth ! ToB exploits it well !! Query Complexity Vs. Size
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
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|)
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 )
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