In this paper we investigate the learnability of relations in Inductive Logic Programming, by using equality theories as background knowledge. We assume that a hypothesis and an observation are respectively a definite program and a set of ground literals. The targets of our learning algorithm are relations. By using equality theories as background knowledge we introduce tree structure into definite programs. The structure enable us to narrow the search space of hypothesis. We give pairs of a hypothesis language and a knowledge language in order to discuss the learnability of relations from the view point of inductive inference and PAC learning.

The elementary formal system (EFS, for short) is a kind of logic program which directly manipulates character strings. This paper outlines in brief the authors' studies on algorithmic learning theory developed in the framework of EFS's. We define two important classes of EFS's and a new hierarchy of various language classes. Then we discuss EFS's as logic programs. We show that EFS's form a good framework for inductive inference of languages by presenting model inference system for EFS's in Shapiro's sense. Using the framework we also show that inductive inference from positive data and PAC-learning are both much more powerful than they have been believed. We illustrate an application of our theoretical results to Molecular Biology.