Minimizing of the only-insertion insdel systems

Abstract

A more recent branch of natural computing is DNA computing. At the theoretical level, DNA computing is powerful. This is due to the fact that DNA structure and processing suggest a series of new data structures and operations, and to the fact of the massive parallelism. The insertion-deletion system (insdel system) is a DNA computing model based on two genetic operations: insertion and deletion which, working together, are very powerful, leading to characterizations of recursively enumerable languages. When designing an insdel computer, it is natural to try to keep the underlying model as simple as possible. One idea is to use either only insertion operations or only deletion operations. By helping with a weak coding and a morphism, the family INS ⁷₄ DEL ⁰₀ is equal to the family ofrecursively enumerable languages. It is an open problem proposed by Martin-Videet al. on whether or not the parameters 4 and 7 appearing here can be replaced by smaller numbers. In this paper, our positive answer to this question is that INS ⁴₂ DEL ⁰₀ can also play the same role as insertion and deletion. We suppose that the 1NS ⁴₂ DEL ⁰₀ may be the least only-insertion insdel system in this situation. We will give some reasons supporting this conjecture in our paper.