Experimental results on the size of OBDD

Experimental results on the size of OBDDrepresentation of-level redundunciesfunctionsTR-1208-98Macha Nikolska¨ıaLaBRI,Universit Bordeaux-1,351cours de la Libration,33405talence,Francemacha@labri.u-bordeaux.fr1IntroductionIn this report we present experimental results on the size of OBDD rep-resentation of a class of boolean functions that are typical of what we can expect tofind in a fault tree.These functions do not admit BDDs of small size:they either have an exponential size or polynomial size with a coefficient that is unfortunately much too big from the practical point of view.The experiments were carried on a Sparc Ultra-1processor with 512Mb of core memory using a BDD tool Aralia[4].A-level reduncuncies function is is defined as follows.Definition1.A boolean function is a-level reduncuncies function if for some the following holds:Such functions are uniquely defined by a pair of integers which correspond to the number of children a variable has at the level and. For example Figure1represents the graph of the-level reduncuncies function.Note:It is clear that the cardinality of the set of terminal events is .From experimental studies[1]it follows that there are two heuristic methods which can be reasonnably used for such kind of fault trees.They are depthfirst and shuffle[3](inspired by variable interleaving[2]).h2.1h1.1e1.1.2g1e1.2.2e1.3.2h2.3h2.2e2.3.2e2.1.2e2.2.2g2flevel 1level 2level 3h1.3h1.2e1.2.1e1.1.1e1.3.1Fig.1.Example of -level reduncuncies formulaDepth first (DF )consists of a depth-first leftmost traversal of the for-mula graph.The obtained variable ordering is given by the topological order.For example,for the formula on the Figure 1it gives the order,,,,,,,,.Shuffle (SH )tries to assure that variables that are semantically close end up near each other in the variable ordering.Let us note an opera-tion defined over lists of variables in the following way.if xifThe variable ordering corresponding to the variableis defined as where the are children of ,is any boolean operator,and the are the results of the operation for .For the same example it gives the order ,,,,,,,,.2Experimental resultsExperimental results for the -level reduncuncies functions show that:1.depth-first is linear for and exponential for;2.shuffle is linear for and exponential for;3.so,they are both exponential for;4.the number of prime implicants for the case of sum-of-products is;5.the number of prime implicants for product-of-sums is;6.these orderings are optimal for the initial formula wrt.permutationsof arguments;246810357911211572471013408353820095043404191216313061143237623451846894862718576751092456398652027421355683524913997157348031111068036381816410433131783920545043Table1.BDD sizes for depthfirst heuristic246810 357911 21125395367 54280118156 447141235329423 7206442650872 6191701121117212231 9958210832584408 876733255883844110999 11435099801561021240 10307115357276433992952216Table2.BDD sizes for shuffle heuristicTables1and2give the computed size for OBDDs encoding-level reduncuncies functions.The columns make vary the number of childrenof the root,and the lines make vary the number of children of the vari-ables of the level.Figure2shows how evolves the size of the BDD corresponding to -level reduncuncies functions.Thefirst line makes vary for given; and the second linke make vary for given.Two columns correspont to two heuristics.The same results are reported on thefigure3,but the reported data contains theoretically obtained extrapolations.depthfirst shuffledepthfirst shuffle depth-first are added()are added()Size=SizeVW2...WkWk+1figure1).correspond to the subformulas;more pre-cisely they encode the modular part of(i.e.without multiply refer-enced variables).–shuffle In the case of this heuristic corresponds toand to.References1.M.Bouissou,F.Bruy`e re,and A.Rauzy.Bdd based fault-tree processing:A comparison ofvariable ordering heuristics.In Proceedings of European Safety and Reliability Association Conference,ESREL’97,1997.2.H.Fujii,G.Ootomo,and C.Hori.Interleaved Based Variables Ordering Methods for Or-dered Binary Decision Diagrams.In Proceedings of the IEEE International Conference on Computer Aided Design,pages38–41,1993.3.M.Nikolskaia and A.Rauzy.Heuristics for bdd handling of sum-of-products formulae.InBalkema,editor,Proceedings of the European Safety and Reliability Association Conference, ESREL’98,June1998.4. A.Rauzy.Aralia version1.0:the Toolbox Manual.Technical report1093-95,LaBRI–URACNRS1304–Universit´e Bordeaux-I,1995.。