On is an n-MCFL

Kilian Gebhardt (1) , Frédéric Meunier (2, 3) , Sylvain Salvati (4, 5, 6)

1 HTW Dresden - Hochschule für Technik und Wirtschaft [Dresden]
2 ENPC - École des Ponts ParisTech
3 CERMICS - Centre d'Enseignement et de Recherche en Mathématiques et Calcul Scientifique
4 Université de Lille
5 LINKS - Linking Dynamic Data
6 CRIStAL - Centre de Recherche en Informatique, Signal et Automatique de Lille - UMR 9189

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.

Domaines: Informatique et langage [cs.CL]; Topologie algébrique [math.AT]; Combinatoire [math.CO]; Théorie des groupes [math.GR]

Kilian Gebhardt, Frédéric Meunier, Sylvain Salvati. On is an n-MCFL. Journal of Computer and System Sciences, 2022, 127, pp.41-52. ⟨10.1016/j.jcss.2022.02.003⟩. ⟨hal-01771670v2⟩