Unformatted text preview: [6] is an orderingbased restricted resolution that can be described as follows. Given an arbitrary ordering of the propositional variables, we assign to each clause the index of the highest ordered literal in that clause. Then we resolve only clauses having the same index, and only on their highest literal. The result of this restriction is a systematic elimination of literals from the set of clauses that are candidates for future resolution. DPelimination also includes additional steps, one forcing unit resolution whenever possible and another dealing with the variables that appear only negatively (called allnegative) or only positively (called allpositive). Allpositive (allnegative) variables are assigned the value \true" (\false") and all clauses containing those variables are deleted from the theory. There are many other intermediate steps that can be introduced between the basic steps of eliminating the highest indexed variable (e.g., subsumption elimination). However, in this paper we focus on the ordered elimination step and refer to auxiliary steps only when necessary. We are interested not merely in achieving refutation, but also in the sum total of the clauses accumulated by this process, which constitutes an equivalent theory with useful computational features. Algorithm directional resolution (DR) (the core of DPelimination) is described in Figure 1. We call its output theory, Ed ('), the directional extension of '. The algorithm can be conveniently described using the notion of buckets partitioning the set of clauses in '. Given an ordering d = Q1; :::Qn, the bucket for Qi, bucketi, contains all the clauses containing Qi , that do not contain any symbol higher in the ordering. Given the theory ', algorithm directional resolution processes the buckets in a reverse order of d. When processing bucketi, it resolves over Qi all possible pairs of clauses in the bucket and inserts the resolvents into the appropriate lower buckets. Theorem 1: (model generation) Let ' be a cnf formula, d = Q1; :::; Qn an ordering, and Ed (') its directional extension. Then, if the extension is not empty, any model of ' can be generated in time O(jEd (')j) in a backtrackfree manner, consulting Ed ('), as follows: Step 1. Assign to Q1 a truth value that
4 directionalresolution Input: A cnf theory ', an ordering d = Q1 ; :::; Qn of its variables. A decision of whether ' is satisable. If it is, a theory Ed ('), equivalent to ', else an generate an ordered partition of the clauses, bucket1; :::; bucketn, where bucketi con Output: empty directional extension. 1.
Initialize: tains all the clauses whose highest literal is Qi . 2. For i = n to 1 do: 3. Resolve each pair f( _ Qi ); ( _ :Qi )g bucketi . If = _ is empty, return Ed (') = ;, the theory is not satisable; else, determine the index of and add it to the appropriate bucket. 4. Endfor. 5. Return Ed (') (= S bucket .
i i Figure 1: Algorithm directional resolution is consistent with clauses...
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 Spring '09
 W. Alabama
 Analysis of algorithms, Conjunctive normal form, Qn, directional resolution

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