Learning pattern languages over groups

This article studies the learnability of classes of pattern languages over automatic groups. It is shown that the class of bounded unions of pattern languages over finitely generated Abelian automatic groups is explanatorily learnable. For patterns in which variables occur at most n times, it is shown that the classes of languages generated by such patterns as well as their bounded unions are, for finitely generated automatic groups, explanatorily learnable by an automatic learner. In contrast, automatic learners cannot learn the unions of up to two arbitrary pattern languages over the integers. Furthermore, there is an algorithm which, given an automaton describing a group G, generates a learning algorithm MG such that either MG explanatorily learns all pattern languages over G or there is no learner for this set of languages at all, not even a non-recursive one. For some automatic groups, non-learnability results of natural classes of pattern languages are provided.