# Typed pattern languages and their learnability

## Abstract

In this paper, we extend patterns, introduced by Angluin [Ang80b], to typed patterns by introducing types into variables. A type is a recursive language and a variable of the type is substituted only with an element in the recursive language. This extension enhances the expressive power of patterns with preserving their good properties. First, we give a general learnability result for typed pattern languages. We show that if a class of types has finite elasticity then the typed pattern language is identifiable in the limit from positive data. Next, we give a useful tool to show the conservative learnability of typed pattern languages. That is, if an indexed family \({\cal L}\)of recursive languages has recursive finite thickness and the equivalence problem for \({\cal L}\) is decidable, then \({\cal L}\) is conservatively learnable from positive data. Using this tool, we consider the following classes of types: (1) the class of all strings over subsets of the alphabet, (2) the class of all untyped pattern languages, and (3) a class of *k*-bounded regular languages. We show that each of these typed pattern languages is conservatively learnable from positive data.

## Keywords

Infinite Sequence Equivalence Problem Regular Language Positive Data Pattern Language## Preview

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