Unavoidable sets and regularity of languages generated by (1,3)-circular splicing systems

Abstract

Circular splicing systems are a formal model of a generative mechanism of circular words, inspired by a recombinant behaviour of circular DNA. They are defined by a finite alphabet A, an initial set I of circular words and a set R of rules. Berstel, Boasson and Fagnot (2012) showed that if I is context-sensitive and R is finite, then the generated language is context-sensitive. Moreover, when I is contextfree and the rules are of a simple type (alphabetic splicing systems) the generated language is context-free. In this paper, we focus on the still unknown relations between regular languages and circular splicing systems with a finite initial set and a finite set of rules represented by a pair of letters ((1,3)-CSSH systems). We prove necessary conditions for (1,3)-CSSH systems generating regular languages. We introduce a special class of (1,3)-CSSH systems, hybrid systems, and we prove that if a hybrid system generates a regular language, then the full linearization of its initial set is unavoidable, a notion introduced by Ehrenfeucht, Haussler and Rozenberg (1983). Hybrid systems include two previously considered classes of (1,3)-CSSH systems: complete systems and transitive marked systems. Unavoidability of the full linearization of the initial set has been previously proved to characterize complete systems generating regular languages whereas transitive marked systems generating regular languages are characterized by a property of the set of rules. We conjecture that this property of the set of rules, along with unavoidability of the full linearization of the initial set, still characterizes hybrid systems generating regular languages.