Generalization of Clauses Containing Cross Connections

In the area of inductive learning, generalization is the main operation, and the usual definition of induction is based on logical implication. Plotkin's well-known technique for computing least general generalizations of clauses under θ-subsumption sometimes produces results which are too general with respect to implication. Muggleton has shown that this problem only occurs in one type of generalization of recursive clauses, called an indirect root. Idestam-Almquist presented a technique, called recursive anti-unification, to compute indirect roots of clauses. However there exist cases for which recursive anti-unification does not work, for example, the clauses which contain a structure called a cross connection. In this paper, we develop a technique for computing indirect roots of Horn clauses. We first introduce a relation equivalent to the implication, called θ-proof, which is syntactically defined, using resolution and θ-subsumption. This leads to an algorithm, the J-algorithm, for computing indirect roots of clauses. The roots of clauses containing cross connections can be computed by the J-algorithm. We also prove that the output from the algorithm is a generalization under implication of the input.