$\mathcal {L}$-reduction computation revisited

Abstract

Let K and L be two languages over $\Sigma$ and $\Gamma$ (with $\Gamma \subset \Sigma$), respectively. Then, the L-reduction of K, denoted by $K\%\,L$, is defined by $\{ u_0u_1\cdots u_n \in (\Sigma - \Gamma )^* \mid u_0v_1u_1 \cdots v_nu_n \in K, \ v_i \in L \ (1\le i \le n) \}$. This is extended to language classes as follows: ${\mathcal {K}}\% {\mathcal {L}}=\{K\%L \mid K \in {\mathcal {K}}, \, L \in {\mathcal {L}} \}$. In this paper, we investigate the computing powers of $\mathcal {K}\%\,\mathcal {L}$ in which $\mathcal {K}$ ranges among various classes of $\mathcal {INS}^i_{\!\!j}$ and min-$\mathcal {LIN}$, while $\mathcal {L}$ is taken as $\mathcal {DYCK}$ and $\mathcal {F}$, where $\mathcal {INS}^i_{\!\!j}$: the class of insertion languages of weight (j, i), min-$\mathcal {LIN}$: the class of minimal linear languages, $\mathcal {DYCK}$: the class of Dyck languages, and $\mathcal {F}$: the class of finite languages. The obtained results include:

• $\mathcal {INS}^1_1\,\%\,\mathcal {DYCK}=\mathcal {RE}$

• $\mathcal {INS}^0_i\,\%\,\mathcal {F}= \mathcal {INS}^1_j\,\%\,\mathcal {F}=\mathcal {CF}$ (for $i\ge 3$ and $j\ge 1$)

• $\mathcal {INS}^0_2\,\%\,\mathcal {DYCK}=\mathcal {INS}^0_2$

• min-$\mathcal {LIN}\,\%\,\mathcal {F}_1=\mathcal {LIN}$

where $\mathcal {RE}$, $\mathcal {CF}$, $\mathcal {LIN}$, $\mathcal {F}_1$ are classes of recursively enumerable, of context-free, of linear languages, and of singleton languages over unary alphabet, respectively. Further, we provide a very simple alternative proof for the known result min-$\mathcal {LIN}\,\%\,\mathcal {DYCK}_2=\mathcal {RE}$. We also show that with a certain condition, for the class of context-sensitive languages $\mathcal {CS}$, there exists no $\mathcal {K}$ such that $\mathcal {K}\%\,\mathcal {DYCK}=\mathcal {CS}$, which is in marked contrast to the characterization results mentioned above for other classes in Chomsky hierarchy. It should be remarked from the viewpoint of molecular computing theory that the notion of L-reduction is naturally motivated by a molecular biological functioning well-known as RNA splicing occurring in most eukaryotic genes.