This paper studies the continuum between implicates and minimal implicates; and the continuum between explanations and minimally consistent explanations. The study is based on the approximation of the set of these objects. A general definition for approximated minimal implicates, called selective implicates, is presented. Three specific instances of selective implicates: query-based, ATMS and length-based are studied. Using the set of query-based minimal implicates, explanations are generated and the properties of these explanations are studied. The goal of these studies is to extract computationally feasible properties in order to achieve tractable abduction. The setting is the compiled approach using minimal implicates in Clause Management Systems.
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