A substitution lemma for multiple context-free languages
Provides a new tool for proving non-multiple-context-freeness, addressing a gap where standard pumping lemmas fail for these languages.
The paper introduces a Substitution Lemma as a necessary condition for languages to be multiple context-free, and uses it to prove that several languages, including the word problem of the group F2×F2, are not multiple context-free. It also shows that groups with multiple context-free word problem have decidable rational subset membership.
We present a necessary condition for an infinite language to be multiple context-free, which we call a Substitution Lemma. We apply it to show a sample selection of languages are not multiple context-free, including the word problem of the group $F_2\times F_2$. We also show that groups with multiple context-free word problem have decidable rational subset membership problem. Our result contrasts with previous work showing that the standard pumping lemma for context-free languages cannot be generalised to multiple context-free languages, and that weak variants of generalised Ogden's lemma do not apply to multiple context-free languages.