Circular Languages Generated by Complete Splicing Systems and Pure Unitary Languages

TL;DR

This paper characterizes the class of circular languages generated by complete circular splicing systems, linking them to pure unitary languages, and explores their computational power and regularity properties.

Contribution

It provides a complete characterization of circular languages generated by complete systems using pure unitary languages and clarifies their computational capabilities.

Findings

A circular language is generated by a complete system iff its linearization is a pure unitary language.

Complete systems have the same power as finite simple systems, which generate non-regular context-free languages.

The class of generated languages can be decided for regularity and characterized precisely.

Abstract

Circular splicing systems are a formal model of a generative mechanism of circular words, inspired by a recombinant behaviour of circular DNA. Some unanswered questions are related to the computational power of such systems, and finding a characterization of the class of circular languages generated by circular splicing systems is still an open problem. In this paper we solve this problem for complete systems, which are special finite circular splicing systems. We show that a circular language L is generated by a complete system if and only if the set Lin(L) of all words corresponding to L is a pure unitary language generated by a set closed under the conjugacy relation. The class of pure unitary languages was introduced by A. Ehrenfeucht, D. Haussler, G. Rozenberg in 1983, as a subclass of the class of context-free languages, together with a characterization of regular pure unitary languages by means of a decidable property. As a direct consequence, we characterize (regular) circular languages generated by complete systems. We can also decide whether the language generated by a complete system is regular. Finally, we point out that complete systems have the same computational power as finite simple systems, an easy type of circular splicing system defined in the literature from the very beginning, when only one rule is allowed. From our results on complete systems, it follows that finite simple systems generate a class of context-free languages containing non-regular languages, showing the incorrectness of a longstanding result on simple systems.