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Unformatted text preview: given ordering of variables 4.2 Diversity The concept of induced width frequently leads to a loose upper bound on the number of clauses recorded by directional resolution. In example 3, for instance, only 6 clauses were generated by directional-resolution when processed in the given order, even without eliminating subsumption and tautologies in each bucket, while the computed bound is 18 1 8 = 144. for instance, the two clauses (:A; B ); (:C; B ) and the order d = A; C; B . When bucket B The induced graph may not be a tight bound for the interaction graph of Ed ('). Consider, is processed, no clause is added because B is positive in both clauses, nevertheless, nodes A and C will be connected in the induced graph. In this subsection, we introduce a more 11 re ned parameter, called diversity, based on the observation that a propositional letter can be resolved upon only when it appears both positively and negatively in di erent clauses. Extensions to this parameter will later attempt at bounding the number of resolvents in a bucket. De nition 4: (diversity of a theory ) Given a theory ' and an ordering d, let Q+ (Q0 ) denote the number of times Qi appears i i positively (negatively) in bucketi w.r.t. d. The diversity of Qi relative to d, div (Qi), is Q+ 2 Q0 . The diversity of an ordering d, div (d), is the maximum diversity of its variables i i w.r.t. the ordering d and the diversity of a theory, div , is the minimal diversity over all its orderings. Theorem 6: Algorithm min-diversity (Figure 4) generates a minimal diversity ordering of a theory. Proof: Let d be an ordering generated by the algorithm and let Qi be a variable whose diversity equals the diversity of the ordering. If Qi is pushed up, its diversity can only increase and if pushed down, it must be replaced by a variable whose diversity is either equal to or higher than the diversity of Qi . 2 Theorem 7: The complexity of algorithm min-diversity is O(n2 1 c), where c is the number of clauses in the input theory. Proof: Computing the diversity of a variable takes O(c) time, and the algorithm checks at most n variables in order to select one with the smallest diversity at each of n steps. This yields the total O(n2 1 c) complexity. 2 4.2.1 Discovering causal structures The concept of diversity yields new tractable classes. If d is an ordering having a zero diversity, directional resolution will add no clauses to ' along d. Namely, 12 min-diversity (') 1. For i = n to 1 do S 2. Choose symbol Q having the smallest diversity in ' 0 n=i+1 bucketj , and put it in the j i-th position. Figure 4: Algorithm min-diversity Theorem 8: Theories having zero diversity are tractable and can be recognized in linear time. 2 Example 4: Let ' = f(G; E; :F );(G; :E; D); (:A; F ); (A;:E ) (:B;C; :E ) (B; C; D)g. The reader can verify that the ordering d = A; B; C; D; E; F; G is a zero-diversity ordering of '. Note that the diversity of theories in example 3 along the speci ed ordering, is 1. Zero-diversity theo...
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