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Unformatted text preview: clauses are initially contained in bucket(A) (highest in the ordering). All other buckets are empty. Following the application of directional resolution along d1 , we get (note that processing is in the reverse order of d): bucket(D) = f(C; D); (D; E )g, bucket(C ) = f(B; C )g, 6 bucket(B ) = f(B; E )g. On the other hand, the directional extension along the ordering d2 = (A; B; C; D; E ) is identical to the input theory, and each bucket contains at most one clause. Example 2: Consider the theory '2 = f(:A; B ); (A; :C ); (:B; D);(C; D; E )g: The directional extensions of ' along the ordering d1 = (A; B; C; D; E ) and d2 = (D; E; C; B; A) are Ed1 (') = ' and Ed2 (') = ' [ f(B; :C ) ; (:C; D); (E; D)g, respectively. In Example 1, A appears in all clauses; hence, it potentially can generate new clauses when resolved upon, unless it is processed last (i.e., appears rst in the ordering), as in d2 . This shows that the interactions among clauses play an important role in the e ectiveness of the algorithm and may suggest orderings that yield smaller extensions. In Example 2, on the other hand, all symbols have the same type of interaction, each (except E ) appearing in two clauses. Nevertheless, D appears positive in both clauses and consequently will not be resolved upon; hence, it can be processed rst. Subsequently, B and C appear only negatively in the remaining theory and can be processed without generating new clauses. In the following, we will provide a connection between the algorithm's complexity and two parameters: a topological parameter, called induced width, and a syntactic parameter, called diversity. Note that directional resolution is tractable for 2-cnf theories in all orderings, since 2cnf are closed under resolution (the resolvents are of size 2 or less) and because the overall number of clauses of size 2 is bounded by O(n2 ). (In this case, unrestricted resolution is also tractable). Clearly, this algorithm is not the most e ective one for satis ability of 2-cnfs. Satis ability for these theories can be decided in linear time [10]. However, as noted earlier, DR achieves more than satis ability, it compiles a theory that allows model generation in linear time. We summarize: Theorem 4: If ' is a 2-cnf theory, then algorithm directional resolution will produce a directional extension of size O(n2 ) in time O(n3 ). 2 7 E E E D D D C C C B B A A (a ) (b ) (c ) B A Figure 2: The interaction graph of '2 Corollary 2: Given a directional extension Ed (') of a 2-cnf theory ', the entailment of any clause involving the rst c symbols in d is O(c3 ). 2 4.1 Induced width Let ' = '(Q1; :::; Qn) be a cnf formula de ned on the variables Q1 ; :::; Qn. The interaction graph of ', denoted G('), is an undirected graph that contains one node for each propositional variable and an arc connecting any two nodes whose associated variables appear in the same clause. The interaction graph of '2 is given in Figure 2a. We can nd a bound on the size of all theories...
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