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 any finitely generated Abelian automatic group is explanatorily learnable. Furthermore, patterns are considered in which variables occur at most n times. The classes of the languages generated by such patterns as well as the bounded unions of such languages are, for finitely generated automatic groups, explanatorily learnable by an automatic learner. On the other hand, the unions of up to two arbitrary pattern languages over the integers cannot be learnt by any automatic learner. Furthermore, there is an algorithm which provides for every automatic group G, given by the automata describing the group, a learning algorithm MG such that either MG explanatorily learns all the pattern languages over that group or there is no such learner at all, even not a non-recursive one. For some automatic groups, non-learnability results of natural classes of pattern languages are provided.