Investigations on the power of matrix insertion-deletion systems with small sizes Henning Fernau 1 · Lakshmanan Kuppusamy 2 · Indhumathi Raman 3 © Springer Science+Business Media B.V., part of Springer Nature 2017 Abstract 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 ðk; n; i 0 ; i 00 ; m; j 0 ; j 00 Þ. The first integer k denotes the maximum number of rules in (length of) any matrix. The next three parameters n; i 0 ; i 00 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 0 ; j 00 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. Keywords Matrix ins-del systems · Matrix control · Descriptional complexity · Computational completeness · (Meta)linear languages 1 Introduction Inserting or deleting words in between parts of sentences often take place when processing natural languages; such insertions and deletions are usually based on context information. To some surprise, this is also happening in biology, especially, in DNA processing and in RNA editing (see Benne 1993; Biegler et al. 2007; Pa ˘un et al. 1998). Based on the insertion operation, Marcus (1969) introduced external contextual grammars as an attempt to mathemat- ically model natural language phenomena. A different variety of linguistically motivated contextual grammars are the semi-contextual grammars studied by Galiukschov (1981), which can be also viewed as insertion grammars. The deletion operation as a basis of a grammatical derivation process was introduced in Kari (1991), where the deletion was motivated as a variant of the right-quotient operation that does not necessarily happen at the right end of the string. Insertion and deletion together were first studied in Kari and Thierrin (1996). The corresponding grammatical mechanism is called insertion-deletion system (abbreviated as ins-del system). Informally, the insertion and deletion operations of an ins-del system are defined as follows: if a string g is inserted between two parts w 1 and w 2 of a string w 1 w 2 to get w 1 gw 2 , we call the operation insertion, whereas if a substring d is deleted from a string A preliminary version of this paper appeared in Proceedings of UCNC 2016, LNCS 9726, pp. 35–48, 2016. & Henning Fernau fernau@uni-trier.de Lakshmanan Kuppusamy klakshma@vit.ac.in Indhumathi Raman indhumathi.r@vit.ac.in 1 Fachbereich 4 - Abteilung Informatikwissenschaften, CIRT, Universita ¨t Trier, 54286 Trier, Germany 2 School of Computer Science and Engineering, VIT University, Vellore 632 014, India 3 School of Information Technology and Engineering, VIT University, Vellore 632 014, India 123 Natural Computing https://doi.org/10.1007/s11047-017-9656-8