Natural Computing

, Volume 17, Issue 2, pp 249–269 | Cite as

Investigations on the power of matrix insertion-deletion systems with small sizes

  • Henning Fernau
  • Lakshmanan Kuppusamy
  • Indhumathi Raman


Matrix insertion-deletion systems combine the idea of matrix control (a control mechanism well established in regulated rewriting) with that of insertion and deletion (as opposed to replacements). Given a matrix insertion-deletion system, the size of such a system is given by a septuple of integers \((k;n,i',i'';m,j',j'')\). The first integer k denotes the maximum number of rules in (length of) any matrix. The next three parameters \(n,i',i''\) denote the maximal length of the insertion string, the maximal length of the left context, and the maximal length of the right context of insertion rules, respectively. The last three parameters \(m,j',j''\) are similarly understood for deletion rules. In this paper, we improve on and complement previous computational completeness results for such systems, showing that matrix insertion-deletion systems of size (1) (3; 1, 0, 1; 1, 0, 1), (3; 1, 0, 1; 1, 1, 0), (3; 1, 1, 1; 1, 0, 0) and (3; 1, 0, 0; 1, 1, 1) (2) (2; 1, 0, 1; 2, 0, 0), (2; 2, 0, 0; 1, 0, 1), (2; 1, 1, 1; 1, 1, 0) and (2; 1, 1, 0; 1, 1, 1), are computationally complete. Further, we also discuss linear and metalinear languages and we show how to simulate grammars characterizing them by matrix insertion-deletion systems of size (3; 1, 1, 0; 1, 0, 0), (3; 1, 0, 1; 1, 0, 0), (2; 2, 1, 0; 1, 0, 0) and (2; 2, 0, 1; 1, 0, 0). We also generate non-semilinear languages using matrices of length three with context-free insertion and deletion rules.


Matrix ins-del systems Matrix control Descriptional complexity Computational completeness (Meta)linear languages 



We are grateful to Serghei Verlan for his comments on our previous UCNC version, and also for sharing with us his Perl tool (simulating graph-controlled ins-del systems) that facilitates checking some of our results and examples. Some extension part of the work was undertaken during the second author’s visit to University of Trier, Germany, in June 2016. Support of this visit by overhead money (from DFG grant FE 560/6-1) is gratefully acknowledged.


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© Springer Science+Business Media B.V., part of Springer Nature 2017

Authors and Affiliations

  1. 1.Fachbereich 4 - Abteilung Informatikwissenschaften, CIRTUniversität TrierTrierGermany
  2. 2.School of Computer Science and EngineeringVIT UniversityVelloreIndia
  3. 3.School of Information Technology and EngineeringVIT UniversityVelloreIndia

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