$O_n$ is an $n$-MCFL

TL;DR

This paper proves that the language $O_n$, which balances multiple pairs of symbols, is an n-dimensional multiple context-free language, using algebraic topology tools and a variant of the necklace splitting theorem.

Contribution

The paper provides two proofs confirming that $O_n$ is an n-MCFL, resolving a conjecture and advancing understanding of formal language classifications.

Findings

$O_n$ is an n-MCFL.

Two proofs established the conjecture.

Introduced a variant of the necklace splitting theorem.

Abstract

Commutative properties in formal languages pose problems at the frontier of computer science, computational linguistics and computational group theory. A prominent problem of this kind is the position of the language $O_n$, the language that contains the same number of letters $a_i$ and $\bar a_i$ with $1\leq i\leq n$, in the known classes of formal languages. It has recently been shown that $O_n$ is a Multiple Context-Free Language (MCFL). However the more precise conjecture of Nederhof that $O_n$ is an MCFL of dimension $n$ was left open. We present two proofs of this conjecture, both relying on tools from algebraic topology. On our way, we prove a variant of the necklace splitting theorem.