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In mathematical logic, proof compression by splitting is an algorithm that operates as a post-process on resolution proofs. It was proposed by Scott Cotton in his paper "Two Techniques for Minimizing Resolution Proof".〔Cotton, Scott. "Two Techniques for Minimizing Resolution Proofs". 13th International Conference on Theory and Applications of Satisfiability Testing, 2010.〕 The Splitting algorithm is based on the following observation: Given a proof of unsatisfiability and a variable , it is easy to re-arrange (split) the proof in a proof of and a proof of and the recombination of these two proofs (by an additional resolution step) may result in a proof smaller than the original. Note that applying Splitting in a proof using a variable does not invalidates a latter application of the algorithm using a differente variable . Actually, the method proposed by Cotton〔 generates a sequence of proofs , where each proof is the result of applying Splitting to . During the construction of the sequence, if a proof happens to be too large, is set to be the smallest proof in . For achieving a better compression/time ratio, a heuristic for variable selection is desirable. For this purpose, Cotton〔 defines the "additivity" of a resolution step (with antecedents and and resolvent ): : Then, for each variable , a score is calculated summing the additivity of all the resolution steps in with pivot together with the number of these resolution steps. Denoting each score calculated this way by , each variable is selected with a probability proportional to its score: : To split a proof of unsatisfiability in a proof of and a proof of , Cotton 〔 proposes the following: Let denote a literal and denote the resolvent of clauses and where and . Then, define the map on nodes in the resolution dag of : : Also, let be the empty clause in . Then, and are obtained by computing and , respectively. == Notes == 抄文引用元・出典: フリー百科事典『 ウィキペディア(Wikipedia)』 ■ウィキペディアで「Resolution proof compression by splitting」の詳細全文を読む スポンサード リンク
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