Revising the logical foundations of inductive logic programming systems with ground reduced programs

In this paper we revise Muggleton’s theory of inverse entailment, which is the logical foundation of Progol, one of the most famous ILP systems. We first point out that the theory is incomplete in general. Secondly we prove that the theory is complete if the background knowledge given to the system is a ground reduced program, every training example is a ground unit clause, and the hypothesis space is the set of all definite clauses. The proof is obtained by showing that every ground reduced logic program is logically equivalent to the conjunction of all atoms in its least Herbrand model. As a corollary to this equivalence, we are finally able to improve the logical foundation of the GOLEM system.