From Inverse Entailment to Inverse Subsumption

Modern explanatory ILP methods like Progol, Residue procedure, CF-induction, HAIL and Imparo use the principle of Inverse Entailment (IE). It is based on the fact that the negation of a hypothesis is derived from a prior background theory and the negation of examples. IE-based methods commonly compute a hypothesis in two steps: by first constructing an intermediate theory and next by generalizing its negation into the hypothesis with the inverse relation of entailment. In this paper, we focus on the sequence of intermediate theories that constructs a derivation from the background theory and the negation of examples to the negation of the hypothesis. We then show the negations of those derived theories in a sequence are represented with inverse subsumption. Using our result, inverse entailment can be reduced into inverse subsumption, while it preserves the completeness for finding hypotheses.