Comparing consecutive letter counts in multiple context-free languages

Abstract: Context-free grammars are not able to model cross-serial dependencies in natural languages. To overcome this issue, Seki et al. introduced a generalization called $m$-multiple context-free grammars ($m$-MCFGs), which deal with $m$-tuples of strings. We show that $m$-MCFGs are capable of comparing the number of consecutive occurrences of at most $2m$ different letters. In particular, the language ${a_1^{n_1} a_2^{n_2} \dots a_{k}^{n_{2m+1}} \mid n_1 \geq n_2 \geq \dots \geq n_{2m+1} \geq 0}$ is $(m+1)$-multiple context-free, but not $m$-multiple context-free.