Derivation languages, descriptional complexity measures and decision problems of a class of flat splicing systems

TL;DR

This paper explores the derivation languages of flat splicing systems, compares them with classical language families, and investigates their decision problems and generative capacities, revealing new insights into their computational properties.

Contribution

It introduces the concepts of Szilard and control languages for flat splicing systems, analyzes their relationships with Chomsky hierarchy, and establishes their generative and decision properties.

Findings

Szilard languages of flat splicing systems are incomparable with regular, context-free, and recursively enumerable languages.

Decidability results for subset relations involving Szilard languages and regular languages.

Homomorphic images of Szilard languages can generate various language classes, including regular, context-free, and recursively enumerable languages.

Abstract

In this paper, we associate the idea of derivation languages with flat splicing systems and compare the families of derivation languages (Szilard and control languages) of these systems with the family of languages in Chomsky hierarchy. We show that the family of Szilard languages of labeled flat finite splicing systems of type $( m, n)$ (i.e., $SZLS_{n, FIN}^{m}$ ) and $REG$, $CF$ and $CS$ are incomparable. Also, it is decidable whether or not $SZ_{n, FIN}^{m}(\mathscr{LS}) \subseteq R$ and $R \subseteq SZ_{n, FIN}^{m}(\mathscr{LS})$ for any regular language $R$ and labeled flat finite splicing system $\mathscr{LS}$. Also, any non-empty regular, non-empty context-free and recursively enumerable language can be obtained as homomorphic image of Szilard language of the labeled flat finite splicing systems of type $(1, 2), (2, 2)$ and $(4, 2)$ respectively. We also introduce the idea of control languages for labeled flat finite splicing systems and show that any non-empty regular and context-free language can be obtained as a control language of labeled flat finite splicing systems of type $(1,2)$ and $(2, 2)$ respectively. At the end, we show that any recursively enumerable language can be obtained as a control language of labeled flat finite splicing systems of type $(4,2)$ when $\lambda$-labeled rules are allowed.