On the regularity of circular splicing languages: a survey and new developments

Circular splicing has been introduced to model a specific recombinant behaviour of circular DNA, continuing the investigation initiated with linear splicing. In this paper we focus on the relationship between regular circular languages and languages generated by finite circular splicing systems. We survey the known results towards a characterization of the intersection between these two classes and provide new contributions on the open problem of finding this characterization. First, we exhibit a non-regular circular language generated by a circular simple system thus disproving a known result in this area. Then we give new results related to a restrictive class of circular splicing systems, the marked systems. Precisely, we review in a graph theoretical setting the recently obtained characterization of marked systems generating regular circular languages. In particular, we define a slight variant of marked systems, that is the g-marked systems, and we introduce the graph associated with a g-marked system. We show that a g-marked system generates a regular circular language if and only if its associated graph is a cograph. Furthermore, we prove that the class of g-marked systems generating regular circular languages is closed under a complement operation applied to systems. We also prove that marked systems with self-splicing generate only regular circular languages.