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arXiv · 2509.02117

A substitution lemma for multiple context-free languages

Andrew Duncan·Murray Elder·Lisa Frenkel·Mengfan Lyu
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Abstract

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.

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Andrew Duncan, Murray Elder, Lisa Frenkel, Mengfan Lyu. 2025-09-02. A substitution lemma for multiple context-free languages. https://arxiv.org/abs/2509.02117

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