The single loop representations of regular languages Original Research Article

A regular language of the form uv4w is called a single loop from the viewpoint of automata theory. It is known that every regular language can be expressed as (∪iϵΛuiviwi)∪F. where Λ is an index set, ui, wiϵX∗, viϵX+, i ϵ Λ, and F is a finite set of words. This expression is called an s-representation of that language. An s-representation is called disjoint if the union of it is disjoint. A language which has an s-representation with finite index is called an fs-representable language. This kind of languages is shown to be the semi-discrete languages. In this paper we give a classification of regular languages by the concept of single loops. We show that every fs-representable language can be expressed as a disjoint s-representation with finite index. We also show that the intersection of an fs-representable language with any context-free language is regular. The relationships between the languages of the form u4 v−, the non-fs-representable languages and codes are investigated for their own interests. We show that for u, vϵX4, u4vi being a code implies that it is not an fs-representable language.