Inductive inference of unbounded unions of pattern languages from positive data

Abstract

A pattern is a string consisting of constant symbols and variables. The language of a pattern is the set of constant strings obtained by substituting nonempty constant strings for variables in the pattern. For any fixed k, the class of unions of at most k pattern languages is already shown to be inferable from positive data. The class of all the unions of arbitrarily finitely many pattern languages is not inferable, because any constant string defines a singleton set consisting of itself, and the class of unions contains all the finite languages. A proper pattern is a pattern that contains at least one variable. The language of a proper pattern is infinite. In this paper, we consider the class of unions when patterns are restricted to be proper and show that the class is not inferable from positive data. A regular pattern is a pattern that contains at most one occurrence of every variable. When regular patterns are restricted not to contain more than I consecutive occurrences of constant symbols for some l, the class of unions is shown to be inferable from positive data.