Least generalizations and greatest specializations of sets of clauses

The main operations in Inductive Logic Programming (ILP) are generalization and specialization, which only make sense in a generality order. In ILP, the three most impor-tant generality orders are subsumption, implication and implication relative to background knowledge. The two languages used most often are languages of clauses and languages of only Horn clauses. This gives a total of six dierent ordered languages. In this paper, we give a systematic treatment of the existence or non-existence of least generalizations and greatest specializations of nite sets of clauses in each of these six ordered sets. We survey results already obtained by others and also contribute some answers of our own. Our main new results are, rstly, the existence of a computable least generalization under implication of every nite set of clauses containing at least one non-tautologous function-free clause (among other, not necessarily function-free clauses). Secondly, we show that such a least generalization need not exist under relative implication, not even if both the set that is to be generalized and the background knowledge are function-free. Thirdly, we give a complete discussion of existence and non-existence of greatest specializations in each of the six ordered languages. 1