Generalization of Clauses under Implication

TL;DR

This paper introduces T-implication, a decidable form of logical implication for clauses, enabling the generalization of recursive clauses in inductive learning, which was previously infeasible with standard implication.

Contribution

It proposes T-implication as a decidable alternative to implication, and provides methods to reduce generalizations under implication to theta-subsumption of clause expansions.

Findings

Existence of least general generalizations under T-implication for finite clause sets

Reduction of implication generalizations to theta-subsumption of clause expansions

Introduction of T-complete expansions for non-tautological clauses

Abstract

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.