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C d e f g is a zero diversity ordering of note that

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Unformatted text preview: ries generalize the notion of causal theories de ned for general networks of multivalued relations [8]. According to the de nition in [8], theories speci ed in the form of cnfs would correspond to causal if there is an ordering of the symbols such that each bucket contains a single clause, and consequently has zero diversity. Note that even when a general theory is not zero-diversity it is better to put zero-diversity literals last in the ordering (so that they will be processed rst). Then, the size of the directional-extension is exponentially bounded in the number of literals having only strictly-positive diversities. In general, however, the parameter of interest is the diversity of the directional extension Ed (') rather than the diversity of '. De nition 5: (induced diversity ) The induced diversity of an ordering d, div 3(d), is the diversity of Ed (') along d, and the induced diversity of a theory, div 3, is the minimal induced diversity over all its orderings. Since div 3(d) bounds the added clauses generated from each bucket, we can trivially bound the size of Ed (') using div 3: for every d, jEd (')j  j'j + n 1 div 3(d). The problem is 13 that even for a given ordering d, div 3(d) is not polynomially computable, and therefore, the bound is not useful. It can, however, be used for some special cases for which the bound is precomputable. We will use two principles to bound div 3. 1. Identify cases where div = div 3 can be polynomially recognizable. 2. Identify cases where all the added resolvents are super uous, giving the notion of zerodiversity a broader interpretation. For most theories and most orderings div 3(d) > div (d). A special counter example we observed are the zero-diversity theories for which div 3(d) = div (d) = 0. We next identify a subclass of diversity-1 theories whose div 3 remains 1. Theorem 9: A theory ' = '(Q1; :::; Qn), has div 3  1 and is therefore tractable, if each symbol Qi satis es one of the following conditions: a. it appears only negatively; b. it appears only positively; c. it appears in exactly two clauses. Proof: The proof is by induction on the number of symbols. Clearly, the diversity of the top literal is at most 1. If it is of zero diversity, no clause is added during processing; if it is of diversity 1, then at most one clause is added. Assume it is added to bucket Qj . If Qj is a single-sign symbol, it will remain so, namely, the diversity of its bucket will be zero. Else, since there are at most two clauses containing Qj , and one of these was at the bucket of Qn , the current bucket of Qj (after processing Qn ) cannot contain more than two clauses. The diversity of Qj is therefore 1. We can now assume that after processing Qn ; :::; Qi the induced diversity is at most 1, and we can show that processing Qi01 will leave the diversity at most 1. The argument is identical to the base case of the induction. 2 The set of theories in example 3 has div 3 = 1. Note though, that we can easily create exa...
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