Adding Matrix Control: Insertion-Deletion Systems with Substitutions III

Henning Fernau,Martin Vu

doi:10.3390/a14050131

Abstract

Insertion-deletion systems have been introduced as a formalism to model operations that find their counterparts in ideas of bio-computing, more specifically, when using DNA or RNA strings and biological mechanisms that work on these strings. So-called matrix control has been introduced to insertion-deletion systems in order to enable writing short program fragments. We discuss substitutions as a further type of operation, added to matrix insertion-deletion systems. For such systems, we additionally discuss the effect of appearance checking. This way, we obtain new characterizations of the family of context-sensitive and the family of recursively enumerable languages. Not much context is needed for systems with appearance checking to reach computational completeness. This also suggests that bio-computers may run rather traditionally written programs, as our simulations also show how Turing machines, like any other computational device, can be simulated by certain matrix insertion-deletion-substitution systems.

Highlights

Systems with Substitutions III.Insertion-deletion systems, or ins-del systems for short, are well-established as computational devices and as a research topic within Formal Languages throughout the past nearly 30 years, starting off with the PhD thesis of Lila Kari [1]
Insertion rules add a substring to a string, given a specified left and right context, while deletion rules remove a substring from a string, again taking a specified left and right context into consideration
Because there is no interaction between two symbols of different paths and /or cycles, it is clear that there is a derivation that applies all of the matrices used in the derivation of am b in the same order, but in which all insertions and deletions corresponding to the path P occur right of the final b

Summary

Introduction

It can be argued that the potentially most error-prone part of a bio-computing implementation of ins-del-sub systems are context checks concerning the site where the operation (be it an insertion, a deletion or a substitution operation) may be applied. It should be noted that all of our computational completeness proofs are constructive, which means that algorithms exist that will turn any program written in some high-level programming language of your choice into an ins-del-sub system that executes this program based on insertion, deletion, or substitution operations. One clearly sees a trade-off in our results between the necessity to have larger contexts and the necessity to have appearance checks Because it is not clear which of these mechanisms is really harder to implement when it comes to build real bio-computers, it appears to be reasonable to study the general possibilities of these mechanisms, paving the way to future generations of new computing devices

Definitions

Matrix Grammars

Insertion-Deletion Systems

Combining Ideas

Adding Substitutions

Appearance Checking

A Normal Form Theorem

One-Sided Context Dependence

DEL1 SUB

Computational Incompleteness

Context-Free Substitutions Do Not Always Help

One-Sided Substitutions