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

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.