# Which hypotheses can be found with inverse entailment?

## Abstract

In this paper we give a completeness theorem of an inductive inference rule *inverse entailment* proposed by Muggleton. Our main result is that a hypothesis clause *H* can be derived from an example *E* under a background theory *B* with inverse entailment iff *H* subsumes *E* relative to *B* in Plotkin's sense. The theory *B* can be any clausal theory, and the example *E* can be any clause which is neither a tautology nor implied by *B*. The derived hypothesis *H* is a clause which is not always definite. In order to prove the result we give a declarative semantics for arbitrary consistent clausal theories, and show that SB-resolution, which was originally introduced by Plotkin, is a complete procedural semantics. The completeness is shown as an extension of the completeness theorem of SLD-resolution. We also show that every hypothesis *H* derived with saturant generalization, proposed by Rouveirol, must subsume E w.r.t. *B* in Buntine's sense. Moreover we show that saturant generalization can be obtained from inverse entailment by giving some restriction to it.

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