Declarative programs with implicit implication

In the presence of taxonomic information, there often exists implicit implication among atoms in an interpretation domain. A general framework is proposed for the discussion of an appropriate semantics for declarative programs with respect to such implicit implication. It is first assumed that the implicit implication can be predetermined and represented by a preorder on the interpretation domain. Under the consequent constraint that every interpretation must conform to the implicit implication, an appropriate model-theoretic semantics as well as its corresponding fixpoint semantics for declarative programs is described. Based on Kostler et al.'s (1993) foundation of fixpoint with subsumption, it is shown that, it the implicit-implication relation is, in addition, assumed to be a partial order, then the meaning of a program can be determined more efficiently by application of an immediate-consequence operator which involves only reduced representations, basically consisting only of their maximal elements, of subsets of the interpretation domain.