On describing super-linear languages by matrix insertion–deletion systems

Abstract

Matrix insertion–deletion (matrix ins–del) systems combine the idea of matrix control (as established in regulated rewriting) with that of insertion and deletion operations (as opposed to replacements). There are matrix ins–del systems of small sizes that are known to describe linear languages but not even context-free languages. Our aim is to study the generative power of such matrix ins–del systems. In this regard, we consider six language classes, called super-linear languages between LIN and CFL, namely $${\mathrm {CLIN}}$$ , $${\mathrm {SLIN}}$$ , $${\mathrm {CSLIN}}$$ , $${\mathrm {SCLIN}}$$ , $${\mathrm {CSCLIN}}$$ and $${\mathrm {SCSLIN}}$$ obtained by applying concatenation or/and Kleene closure operation on linear languages. These classes deserve special attention due to the fact that $${\mathrm {LIN}}$$ is closed neither under concatenation nor under Kleene closure. In this paper, we discuss the context-free grammars that generate these language classes and also simulate them by matrix ins–del systems.