On path-controlled insertion–deletion systems

Abstract

A graph-controlled insertion–deletion system is a regulated extension of an insertion–deletion system. It has several components and each component contains some insertion–deletion rules. These components are the vertices of a directed control graph. A transition is performed by any applicable rule in the current component on a string and the resultant string is then moved to the target component specified in the rule. This also describes the arcs of the control graph. Starting from an axiom in the initial component, strings thus move through the control graph. The language of the system is the set of all terminal strings collected in the final component. In this paper, we investigate a variant of the main question in this area: which combinations of size parameters (the maximum number of components, the maximal length of the insertion string, the maximal length of the left context for insertion, the maximal length of the right context for insertion; plus three similar restrictions with respect to deletion) are sufficient to maintain computational completeness of such restricted systems under the additional restriction that the (undirected) control graph is a path? Notice that these results also bear consequences for the domain of insertion–deletion P systems, improving on a number of previous results from the literature, concerning in particular the number of components (membranes) that are necessary for computational completeness results.