Finding Minimal Generalizations for Unions of Pattern Languages and Its Application to Inductive Inference from Positive Data.

A pattern is a string of constant symbols and variables. The language defined by a pattern p is the set of constant strings obtained from p by substituting nonempty constant strings for variables in p. In this paper we are concerning with polynomial time inference from positive data of the class of unions of a bounded number of pattern languages. We introduce a syntactic notion of minimal multiple generalizations (mmg for short) to study the inferability of classes of unions. If a pattern p is obtained from another pattern q by substituting nonempty patterns for variables in q, q is said to be more general than p. A set of patterns defines a union of their languages. A set Q of patterns is said to be more general than a set P of patterns if for any pattern p in P there exists a more general pattern q in Q than p. Clearly more general set of patterns defines larger unions. A k-minimal multiple generalization (k-mmg) of a set S of strings is a minimally general set of at most k pattern..