Date: 03 Jun 2005

Declarative program theory with implicit implication

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In order to provide a general framework for deductive object-oriented representation systems, Akama's declarative program theory is extended under the assumptions that there exists implicit implication among elements of an interpretation domain and that this implicit implication can be represented by a preorder on the 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 Köstler et. al.'s foundation of fixpoint with subsumption, it is shown that if the implicit-implication relation is further assumed to be a partial order, then the meaning of a program can be determined more efficiently by the application of an immediate-consequence operator which involves only the reduced representations, basically consisting only of their maximal elements, of subsets of the interpretation domain.