Finding minimal generalizations for unions of pattern languages and its application to inductive inference from positive data

Abstract

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 patterns that defines a union containing S. The syntactic notion of minimality enables us to efficiently compute a candidate for a semantically minimal concept. We present a general methodology for designing an efficient algorithm to find a k-mmg. Under some conditions an mmg can be used as an appropriate hypothesis for inductive inference from positive data. As results several classes of unions of pattern languages are shown to be polynomial time inferable from positive data.