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On path-controlled insertion–deletion systems

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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.

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Notes

  1. The shuffle operation, denoted by , is defined recursively by

    $$\begin{aligned} (au\sqcup \mathchoice{}{}{}{}\sqcup \,\, bv) = a(u \sqcup \mathchoice{}{}{}{}\sqcup \,\, bv) \cup b(au \sqcup \mathchoice{}{}{}{}\sqcup \,\, v) \end{aligned}$$

    and \((u \sqcup \mathchoice{}{}{}{}\sqcup \,\, \lambda ) = (\lambda \sqcup \mathchoice{}{}{}{}\sqcup \,\, u) = \{u\},\) where \(u,v \in \varSigma ^*\) and \(a,b \in \varSigma \).

  2. A type-0 grammar G is usually specified by a quadruple (N, T, P, S) consisting of a nonterminal alphabet N, a terminal alphabet T, a finite set of (production) rules P and a start symbol \(S\in N\). Rules are written in the form \(\alpha \rightarrow \beta \), \(\alpha ,\beta \in (N\cup T)^*\). This defines a rewrite relation \(\Rightarrow _G\subseteq (N\cup T)^*\times (N\cup T)^*\), with \(u\Rightarrow _Gv\) if v is obtained from u by replacing the subword \(\alpha \) by \(\beta \), for some \(\alpha \rightarrow \beta \in P\). The reflexive transitive closure \(\Rightarrow _G^*\) can be used to define the semantics of G—the language of G—collecting all \(w\in T^*\) with \(S\Rightarrow _G^*w\).

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Acknowledgements

Some part of the work done by the second author was during his visit to University of Trier, Germany, in December 2016. The possibility to use some overhead money from the DFG Grant FE 560/6-1 to finance this visit is gratefully acknowledged.

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Authors and Affiliations

  1. Fachbereich 4 – Abteilung Informatikwissenschaften, CIRT, Universität Trier, 54286, Trier, Germany

    Henning Fernau

  2. School of Computer Science and Engineering, VIT University, Vellore, 632 014, India

    Lakshmanan Kuppusamy

  3. School of Information Technology and Engineering, VIT University, Vellore, 632 014, India

    Indhumathi Raman

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  1. Henning Fernau
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  2. Lakshmanan Kuppusamy
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  3. Indhumathi Raman
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Correspondence to Indhumathi Raman.

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Fernau, H., Kuppusamy, L. & Raman, I. On path-controlled insertion–deletion systems. Acta Informatica 56, 35–59 (2019). https://doi.org/10.1007/s00236-018-0312-2

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