ILP systems which use some form of Inverse Entailment (IE) are based on clause refinement through a hypotheses space bounded by a most specific clause. In this paper we give a new analysis of refinement operators in this setting. In particular, Progol's refinement operator is revisited and discussed. It is known that Progol's refinement operator is incomplete with respect to the general subsumption order. We introduce a subsumption order relative to a most specific (bottom) clause. This subsumption order, unlike previously suggested orders, characterises Progol's refinement space. We study the properties of this subsumption order and show that ideal refinement operators exist for this order. It is shown that efficient operators can be implemented for least generalisation and greatest specialisation in the subsumption order relative to a bottom clause. We also study less restricted subsumption orders relative to a bottom clause and show how Progol's incompleteness can be addressed.
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