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Clause the interaction graph of 2 is given in figure

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Unformatted text preview: having the same interaction graph using some properties of the graph. De nition 1: Given a graph G and an ordering of its nodes d, the parent set of a node A relative to d is the set of nodes connected to A that precede A in the ordering d. The size of this parent set is the width of A relative to d. The width w(d) of an ordering d is the maximum width of nodes along the ordering, and the width w of a graph is the minimal width of all its orderings [11, 7]. Lemma 1: Given an interaction graph G(') and an ordering d, if A is a node having k 0 1 parents, then there are no more than 3k clauses in the bucket of A; if w(d) = w, then the size of the corresponding theory is O(n 1 3w ). The bucket of A0 contains1 clauses de ned on k literals only. For the set of k 0 1 k01 C symbols there are at most B @ A subsets of i symbols. Each subset can be associated Proof: i 8 with at most 2i clauses (i.e., each symbol can appear either positive or negative in a clause), and A can be also positive or negative. Therefore we can have at most 0 1 XB k01 C i 21 @ A 2 = 2 1 3k0 : k01 i=0 1 i (1) clauses. Clearly, if the parent set is bounded by w, the extension is bounded by O(n 1 3w ). 2 Directional resolution applied along an ordering d to a theory having graph G adds new clauses and, accordingly, changes the interaction graph. The concept of induced graph will be de ned to re ect those changes. De nition 2: Given a graph G and an ordering d, the graph generated by recursively connecting the parents of G, in a reverse order of d, is called the induced graph of G w.r.t. d, and is denoted by Id (G). The width of Id (G) is denoted by w3 (d) and is called the induced width of G w.r.t. d. The graph in Figure 2a, for example, has width 2 along the ordering A; B; C; D; E (Figure 2b). Its induced graph is given in Figure 2c. The induced width of G equals 2. Lemma 2: Given a theory ' and an ordering d, the interaction graph G(Ed (')) of the directional extension of ' along d is a subgraph of Id (G(')). Proof: The proof is done by induction on the variables along the ordering d. The induction hypothesis is that all the arcs incident to Qn ; :::; Qi in the G(Ed (')) appear also in Id (G(')). The claim is true for Qn , since its connectivity is the same in both graphs. namely, if (Qi01; Qj ), j < i 0 1, is an arc in G(Ed (')), then it also belongs to Id (G(')). There are two cases: either Qi01 and Qj appeared in the same clause of the initial theory, Assume that the claim is true for Qn ; :::; Qi and we will show that it holds also for Qi01, ', so they are connected in G(') and, therefore, also in Id (G(')), or a clause containing both symbols was added during directional resolution. In the second case, that clause was obtained while processing some bucket Qt ; t > i 0 1. Since Qi01 and Qj appeared in the bucket of Qt, each must be connected to Qt in G(Ed (')) and, by the induction hypothesis, 9 they will also be connected in Id (G(')). Therefore, Qi01 and Qj would become...
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This note was uploaded on 09/17/2013 for the course PMATH 330 taught by Professor W. alabama during the Spring '09 term at Waterloo.

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