## Abstract

We investigate the computing power of the following language operation %: Given two languages L_{1} over Σ and L_{2} over Γ with Γ⊂Σ, we consider the language operation L_{1}%L_{2}={u_{0}u_{1}⋯u_{n}|∃u=u_{0}v_{1}u_{1}⋯v_{n}u_{n}∈L_{1} and ∃v_{i}∈L_{2}(1≤∀i≤n)}. In this case we say that L(=L_{1}%L_{2}) is the L_{2}-reduction of L_{1}. This is extended to the language families as follows: L_{1}%L_{2}={L_{1}%L_{2}|L_{1}∈L_{1},L_{2}∈L_{2}}. Among many works concerning Dyck-reductions, for the family of recursively enumerable languages RE, it was shown that LIN%{EQ}=RE (Jantzen & Petersen, 1994) with EQ={x^{n}x‾^{n}|n∈N} and that min-LIN%{D_{2}}=RE (Hirose & Okawa, 1996, and Latteux & Turakainen, 1990), where LIN and min-LIN are the families of linear and minimal linear context-free languages, respectively. In this paper, we show that each recursively enumerable language L can be represented in the form L=K%D, for some K∈INS_{3}^{0} and a Dyck language D, where INS_{⁎}^{0} (INS_{3}^{0}) denotes the family of insertion languages (insertion languages where the maximum length of the string to be inserted is 3). We can refine it as INS_{⁎}^{0}%{D_{2}}=RE, where D_{2} denotes the Dyck language over binary alphabet. For context-free languages, we show that INS_{3}^{0}%F=CF, where F is the family of finite sets. This also derives that INS_{⁎}^{0}%{MIR}=CF with MIR={xx‾^{R}|x∈{0,1}^{⁎}}. Further, for regular languages, it is shown that each regular language R can be represented in the form R=K%F, for some K∈INS_{2}^{0} and a finite set F={abb‾a‾|a∈V}. We also present some results which characterize the computability and properties of L in the framework of L_{2}-reduction of L_{1}. It is intriguing to note that, from the DNA computing point of view, the notion of L-reduction is naturally motivated by a molecular biological functioning well-known as DNA(RNA) splicing occurring in most eukaryotic genes.

Original language | English |
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Journal | Theoretical Computer Science |

DOIs | |

Publication status | Accepted/In press - 2020 |

## Keywords

- Context-free languages
- Dyck-reduction
- Insertion systems and languages
- Recursively enumerable languages

## ASJC Scopus subject areas

- Theoretical Computer Science
- Computer Science(all)