On The Computing Powers Of L-Reductions Of Insertion Languages

We investigate the computing power of the following language operation %: Given two languages L-1 over Sigma and L-2 over Gamma with Gamma subset of Sigma, we consider the language operation L-1%L-2 = {u(0)u(1) ... u(n) vertical bar there exists u = u(0)v(1)u(1) ... v(n)u(n) is an element of L-1 and there exists v(i), is an element of L-2(1 <= for all 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 vertical bar L-1 is an element of L-2, L-2 is an element of L-2}. Among many works concerning Dyck-reductions, for the family of recursively enumerable languages RE, it was shown that LIN{EQ} = R epsilon (Jantzen & Petersen, 1994) with EQ = {x(n)(x) over bar (n )vertical bar n is an element of N} and that min-LIN%{D-2 } = RE (Hirose & Okawa, 1996, and Latteux & Turakainen, 1990), where GIN - 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 is an element of INS30 and a Dyck language D, where INS*0 (INS30) 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 = RE, where D-2 denotes the Dyck language over binary alphabet. For context-free languages, we show that INS*0%F = RE, where D-2 is the family of finite sets. This also derives that INS30%F = CF with MIR = {x (x) over bar (R) vertical bar x is an element of {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 is an element of INS20 and a finite set F = {ab (b) over bar(a) over bar} vertical bar a is an element of 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. (C) 2020 Elsevier B.V. All rights reserved.