Graph-Controlled Insertion-Deletion Systems Generating Language Classes Beyond Linearity

Abstract

A regulated extension of an insertion-deletion system known as graph-controlled insertion-deletion (GCID) system has several components and each component contains some insertion-deletion rules. A rule is applied to a string in a component and the resultant string is moved to the target component specified in the rule. When resources are so limited (especially, when deletion is context-free) then GCID systems are not known to describe the class of recursively enumerable languages. Hence, it becomes interesting to find the descriptional complexity of such GCID systems of small sizes with respect to language classes below \(\mathrm {RE}\). To this end, we consider closure classes of linear languages. We show that whenever GCID systems describe \(\mathrm {LIN}\) with t components, we can extend this to GCID systems with just one more component to describe, for instance, 2-\(\mathrm {LIN}\) and with further addition of one more component, we can extend to GCID systems that describe the rational closure of \(\mathrm {LIN}\).