Well quasi-orders and regular languages

Abstract

An extension of Myhill's theorem of automata theory, due to Ehrenfeucht et al. [4] shows that a subsetX of a semigroupsS is recognizable if and only ifX is closed with respect to a monotone well quasi-order onS. In this paper we prove that a similar extension of Nerode's theorem is not possible by showing that there exist non-regular languages on a binary alphabet which are closed with respect to a right-monotone well quasi-order. We give then some additional conditions under which a setX S closed with respect to a right-monotone well quasi-order becomes recognizable. We prove the following main proposition: A subsetX ofS is recognizable if and only ifX is closed with respect to two well quasi-orders<=1 and<=2 which are right-monotone and left-monotone, respectively. Some corollaries and applications are given. Moreover, we consider the family ℱ of all languages which are closed with respect to a right-monotone well quasi-order on a finitely generated free monoid. We prove that ℱ is closed under rational operations, intersection, inverse morphisms and direct non-erasing morphisms. This implies that ℱ is closed under faithful rational transductions. Finally we prove that the languages in ℱ satisfy a suitable ‘pumping’ lemma and that ℱ contains languages which are not recursively enumerable.