Deciding Whether a Regular Language is Generated by a Splicing System

TL;DR

This paper proves that it is decidable whether a given regular language can be generated by a splicing system, providing an effective construction based on the language's syntactic monoid.

Contribution

It establishes a decision procedure for recognizing splicing-generated regular languages and constructs a splicing system from the syntactic monoid.

Findings

Decidability of whether a regular language is generated by a splicing system.

Construction of a splicing system from the syntactic monoid.

Effective method based on the size of the syntactic monoid.

Abstract

Splicing as a binary word/language operation is inspired by the DNA recombination under the action of restriction enzymes and ligases, and was first introduced by Tom Head in 1987. Shortly thereafter, it was proven that the languages generated by (finite) splicing systems form a proper subclass of the class of regular languages. However, the question of whether or not one can decide if a given regular language is generated by a splicing system remained open. In this paper we give a positive answer to this question. Namely, we prove that, if a language is generated by a splicing system, then it is also generated by a splicing system whose size is a function of the size of the syntactic monoid of the input language, and which can be effectively constructed.