On the Power of Generalized Forbidding Insertion-Deletion Systems

Abstract

We consider generalized forbidding insertion-deletion systems (GFID) where each insertion-deletion rule is associated with a set $\mathcal {F}$ of words and the rule can be applied to a string only if every word of $\mathcal {F}$ is not a subword of the string. The parameters in the size $(k;n,i',i'';m,j',j'')$ of a GFID system denote (from left to right) the maximum length of a word in $\mathcal {F}$, the maximal length of an insertion string, the maximal length of the left context for insertion, the maximal length of the right context for insertion; the last three parameters follow a similar pattern with respect to deletion. We show that GFID systems of sizes $(k;n,i',i'';m,j',j'')$, where $k=2$ and $n\,+\,i'\,+\,i''=m\,+\,j'\,+\,j''=2$, with $n,m> 0$ and $i',i'',j',j''\in \{0,1\}$, describe all recursively enumerable languages, by explaining algorithms that transform a given type-0 grammar in some normal form to a GFID system of the required size.