Generalization of Clauses under Implication

In the area of inductive learning, generalization is a main operation, and the usual definition of induction is based on logical implication. Recently there has been a rising interest in clausal representation of knowledge in machine learning. Almost all inductive learning systems that perform generalization of clauses use the relation theta-subsumption instead of implication. The main reason is that there is a well-known and simple technique to compute least general generalizations under theta-subsumption, but not under implication. However generalization under theta-subsumption is inappropriate for learning recursive clauses, which is a crucial problem since recursion is the basic program structure of logic programs. We note that implication between clauses is undecidable, and we therefore introduce a stronger form of implication, called T-implication, which is decidable between clauses. We show that for every finite set of clauses there exists a least general generalization under T-implication. We describe a technique to reduce generalizations under implication of a clause to generalizations under theta-subsumption of what we call an expansion of the original clause. Moreover we show that for every non-tautological clause there exists a T-complete expansion, which means that every generalization under T-implication of the clause is reduced to a generalization under theta-subsumption of the expansion.Comment: See http://www.jair.org/ for any accompanying file

theta-subsumption for structural matching

Structural matching, originally introduced by Steven Vere, implements and formalizes the notion of a "most specific generalisation" of two productions, possibly in the presence of a background theory. Despite various studies in the mid-seventies and early eighties, several problems remained. These include the use of background knowledge, the nonuniqueness of most specific generalisations, and handling in-equalities. We show how Gordon Plotkin's notions of "least general generalisation" and "relative least general generalisation" defined on clauses can be adapted for use in structural matching such that the remaining problems disappear. Defining clauses as universally quantified disjunctions of literals and productions as existentially quantified conjunctions of literals, it is shown that the lattice on clauses imposed by theta-subsumption is order-isomorphic to the lattice on productions needed for structural matching.status: publishe

Minimal Generalizations under OI-Implication

The adoption of the object identity bias for weakening implication has lead to the definition of OI-implication, a generalization model for clausal spaces. In this paper, we investigate on the generalization hierarchy in the space ordered by OI-implication. The decidability of this relationship and the existence of minimal generalizations in the related search space is demonstrated. These results can be exploited for constructing refinement operators for incremental relational learning