Distances between languages and reflexivity of relations

Abstract

In this paper, the notions of k-reflexivity and almost reflexivity of binary relations over the free monoid A* are introduced, which are natural extensions of the usual reflexivity. They can be defined relative to arbitrary distances over A*. The problems of deciding whether or not a relation is almost reflexive and whether or not it is k-reflexive for a given integer k, are studied. It is shown that both problems are unsolvable in the case of deterministic rational relations. Moreover, the latter problem remains undecidable even when an oracle asserting that the relation is almost reflexive is provided. On the other hand, for the subclass of recognizable relations, both problems are shown to be solvable: as a consequence, the distance between two rational languages can be effectively computed. Further decidability results concerning the intermediate class of synchronized rational relations are proved.