And by the induction hypothesis 9 they will also be

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Unformatted text preview: connected in Id (G(')), when connecting the parents of Qt. 2 Theorem 5: Let ' = '(Q1; :::; Qn) be a cnf, G(') its interaction graph, and w3 (d) its 3 induced width along d; then, the size of Ed (') is O(n 1 3w (d) ) and the time complexity of directional resolution along the ordering d is O(n 1 9w3 (d) ). Proof: by O(n 1 3w The result follows from lemmas 1 and 2: the interaction graph of Ed (') is a 3 (d) subgraph of Id (G), and the size of theories having Id (G) as their interaction graph is bounded where jbucketij is the size of the largest bucket (remember, that resolution on a bucket takes time quadratic in the bucket size). From lemma 1 we get jbucketij = O(3w the time complexity is O(n 1 9w 3 (d) 3 (d) ). The time complexity of directional resolution is bounded by O(n 1jbucketi j2), ), therefore ). Note, that this deduction implicitly assumes that the algorithm eliminates duplicate clauses. 2 As follows from the last theorem, theories with the bounded induced width would constitute a tractable class for directional resolution. It is known that the induced width of a graph embedded in a k-tree is bounded by k [1]. Here is a recursive de nition of k -trees. De nition 3: (k-trees) 1. A clique of size k (complete graph with k vertices) is a k -tree. 2. Given a k -tree de ned on Q1; :::; Qi01, a k -tree on Q1; :::; Qi can be generated by selecting a clique of size k and connecting Qi to every node in that clique. Corollary 3: If ' is a formula whose interaction graph can be embedded in a k -tree then there is an ordering d such that the time complexity of directional resolution on that ordering is O(n 1 9k ). 2 Finding an ordering yielding the smallest induced width of a graph is NP-hard [1]. However, any ordering d yields an easily computed bound, w3 (d). Consequently, when given a theory and its interaction graph, we will try to nd an ordering that yields the smallest 10 A1 A3 A5 A7 A2 A4 A6 A8 Figure 3: The interaction graph of '8 in example 3:'8 = f(A1; A2; :A3 ), (:A2; A4), (:A6; A7 ; :A8)g (:A2; A3 ; :A4), (A3; A4; :A5), (:A4 ; A6), (:A4; A5; :A6 ), (A5 ; A6; :A7), (:A6; A8), width possible. Several heuristic orderings are available (see [2]). Important special tractable classes are those having w3 = 1 (namely, the interaction graph is a tree) and those having w3 = 2, called series parallel networks. These classes can be recognized in linear time. In general, given any k , graphs having induced width of k or less can be recognized in O(exp(k )). Example 3: Consider a theory 'n over the alphabet fA1; A2; ; :::; Ang. The theory 'n has a set of clauses indexed by i, where a clause for i odd is given by (Ai; Ai+1 ; :Ai+2 ) and two clauses for i even are given by (:Ai ; Ai+2 ) and (:Ai ; Ai+1 ; :Ai+2 ). The reader can check that the induced width of such theories along the natural order is 2 and thus the size of the the induced graph is identical to the original graph (see gure 3). directional extension will not exceed 18 1 n (see lemma 1). For...
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