The backtracking version 15 19 wrongly suggesting

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Unformatted text preview: ion [15, 19], wrongly suggesting that this is the algorithm presented in [6]. In this paper, we wish to \revive" the DP-elimination algorithm by studying its virtues theoretically and by subjecting it to a more extensive empirical testing. First, we show that, in addition to determining satis ability, the algorithm generates an equivalent theory that facilitates model generation and query processing. Consequently, it may be better viewed as a knowledge compilation algorithm. Second, we o set the known worst-case exponential complexities [12, 14] by showing the tractability of DP-elimination for many known tractable classes of satis ability and constraint satisfaction problems (e.g., 2-cnfs, Horn clauses, causal theories and theories having a bounded induced width [7, 8]). Third, we introduce a new parameter, called diversity, that gives rise to new tractable classes. 2 On the empirical side, we qualify prior empirical tests in [5] by showing that for uniform random propositional theories DP-backtracking outperforms DP-elimination by far. However, for a class of instances having a special structure (embeddings in k-trees, for example chains and (k,m)-trees) DP-elimination outperforms DP-backtracking by several orders of magnitude. Also, a restricted version of DP-elimination, called bounded directional resolu- tion, used as a preprocessing for DP-backtracking, improves the performance of the latter both on uniform and structured problems. Empirical results show that the combined algorithm called BDR-DP outperforms both DP-elimination and DP-backtracking. 2 De nition and preliminaries We denote propositional symbols, also called variables, by uppercase letters P; Q; R; :::, propositional literals (e.g., P; :P ) by lowercase letters p; q; r; :::, and disjunctions of literals, or clauses, by ; ; :::. For instance, = (P denote the clause (P _ Q _ R) by fP; Q; Rg. A unit clause is a clause with only one literal. The notation ( _ T ) will be used as a shorthand for the disjunction (P _ Q _ R _ T ), and _ denotes the clause whose literals appear in either or . The resolution operation over two clauses ( _ Q) and ( _ :Q) results in a clause ( _ ), thus eliminating Q. Unit _ Q _ R) is a clause. We will sometime resolution is a resolution operation when one of the clauses is a unit clause. A formula ' in conjunctive normal form (cnf) is represented as a set f 1; :::; tg denoting the conjunction of clauses 1 ; :::; t. The set of models of a formula ' is the set of all truth assignments to its symbols that satisfy '. A clause is entailed by ', ' j= , i is true in all models of '. A Horn formula is a cnf formula whose clauses have at most one positive literal. In a de nite Horn formula, each clause has exactly one positive literal. A clause is positive if it contains only positive literals and is negative if it contains negative literals only. A k -cnf formula is one whose clauses are all of length k or less. 3 3 DP-elimination { Directional Resolution DP-elimination...
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