On the Kanazawa--Salvati Conjecture
The language $\mathrm{MIX}$ consists of all words over a three-letter alphabet that have an equal number of occurrences of each letter. It is also the word problem of $\mathbb{Z}^2$ with respect to a suitable choice of generators. The Kanazawa--Salvati conjecture states that $\mathrm{MIX}$ is not a well-nested multiple context-free language. Every well-nested multiple context-free language is an indexed language. We reduce the conjecture to an explicit combinatorial problem about tuples of words, which is easier to state than the original formulation in terms of arbitrary well-nested multiple context-free grammars. More generally, for every surjective monoid homomorphism $ψ\colon Σ^* \to \mathbb{Z}^d$, we define a family of well-nested multiple context-free grammars $G_ψ[r]$ for $r \geq 1$, each of which generates a sublanguage of $ψ^{-1}(\mathbf{0})$. We prove that every well-nested multiple context-free sublanguage of $ψ^{-1}(\mathbf{0})$ is contained in $L(G_ψ[r])$ for some $r \geq 1$. Using this family, we prove that the four-letter analogue $\mathrm{MIX}_4$, which is a word problem of $\mathbb{Z}^3$, is not a well-nested multiple context-free language. The proof reduces this claim to a result of Bishop--Elder--Evetts--Gallot--Levine stating that the two-letter analogue $\mathrm{MIX}_2$ is not generated by any non-branching multiple context-free grammar. The Kanazawa--Salvati conjecture itself remains open.