Sharpness of the (k+N+3) bound for permutation closure of multiple context-free languages

Determine whether the upper bound (k+N+3) stated in Theorem A for the permutation closure C^N(L) of a k-multiple context-free language L is optimal. Specifically, ascertain the minimal multiple context-free rank M(k,N) such that, for every k-multiple context-free language L, the language C^N(L) is M(k,N)-multiple context-free, and decide whether M(k,N)=k+N+3 or a strictly smaller bound suffices (e.g., in the case k=1 and N=3).

Background

The paper proves that for any positive integers k and N, the permutation closure C^N(L) of a k-multiple context-free language L is (k+N+3)-multiple context-free (Theorem A). This establishes a general upper bound on the multiple context-free rank needed to recognize such permutation closures.

In the conclusion, the authors explicitly note that it is unknown whether this upper bound is sharp. They highlight the example k=1 and N=3, where their result gives an upper bound of 7, while a lower bound of 2 is known because context-free languages are not closed under C^3. The open problem asks for determining the optimal rank bound—either confirming the necessity of k+N+3 in general or improving it.