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Unformatted text preview: backjumping on uniform problems, although not signi cantly, and it recognizes easily unsatis able subproblems having huge amount of deadends if processed by 27 DP-backtracking. 6.4 Directional Resolution as Knowledge Compilation Directional resolution may be viewed as a knowledge compilation algorithm. Our experiments show that answering queries on the directional extension of DR may be signi cantly faster than that on the original theory. In order to determine if a clause is entailed by a theory, we add the negation of each literal in the clause to the theory and run DP-backtracking. In Table 6 we compare the time complexity of query answering for DP-backtracking before and after running DR, on a 3-cnf chain instance with 20 subtheories each containing 5 variables and 13 clauses. We terminate DP-backtracking if the number of deadends exceeds 50000. Deciding on satis ability of that chain problem was also hard for DP-backtracking and took 360.6 seconds when DR solved it in just 0.6 seconds. Answering some queries was easy for DP-backtracking, but most of them required time comparable to that of deciding satis ability. In general, query answering results on di erent random problem generators were similar to those for satis ability checking. We observed that running DR as a preprocssing algorithm on chains is extremely useful, because on chains (and (k; m)-trees in general) the size of the directional extension is bounded, and query answering on the compiled theory is practically backtrack-free. 7 Related work and conclusions Directional resolution belongs to a family of elimination algorithms rst analyzed for optimization tasks in dynamic programming [2] and later used in constraint satisfaction [18, 7] and in belief networks [20]. The complexity of all those elimination algorithms is a function of the induced width w3 of the undirected graph characteristic of each problem instance. Although it is known that determining the w3 of an arbitrary graph is NP-hard, useful heuristics for bounding w3 are available. Since propositional satis ability is a special case of constraint satisfaction, the induced28 width could be obtained by mapping a propositional formula into the relational framework of a constraint satisfaction problem (see [3]), and then applying adaptive consistency, the elimination algorithm tailored for constraint satisfaction problems [7, 18]. We have recently shown, however, that this kind of pairwise elimination operation as performed by directional resolution is more e ective. And, while it can be extended to any row-convex constraint problem [21] or to every 1-tight relations [22] it cannot decide consistency for arbitrary multi-valued networks of relations. The paper makes three main contributions. First, we revive the old Davis-Putnam algorithm (herein called directional resolution) and mitigate the pessimistic analyses of DPelimination by showing that the algorithm admits some known tractable classes for satis ability and constraint...
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