The word problem of ℤ n is a multiple context-free language

Abstract

The word problem of a group {G=\langle\Sigma\rangle} can be defined as the set of formal words in {\Sigma^{*}} that represent the identity in G. When viewed as formal languages, this gives a strong connection between classes of groups and classes of formal languages. For example, Anīsīmov showed that a group is finite if and only if its word problem is a regular language, and Muller and Schupp showed that a group is virtually-free if and only if its word problem is a context-free language. Recently, Salvati showed that the word problem of {\mathbb{Z}^{2}} is a multiple context-free language, giving the first example of a natural word problem that is multiple context-free, but not context-free. We generalize Salvati’s result to show that the word problem of {\mathbb{Z}^{n}} is a multiple context-free language for any n.

Parallel Hyperedge Replacement String Languages

“…The genus 0 surface (the sphere) gives us the trivial group, and 1 (the torus), Z 2 (see for example [21]). We know both of these groups are regular, 2-MCF [16], respectively, so certainly PHRS (Theorem 3.17).…”

“…The multiple context-free (MCF) languages sit strictly in between the context-free and context-sensitive languages [29]. In 2015, a major breakthrough of Salvati was published, showing that the word problem of Z 2 is an MCF language [28], and in 2018, Ho extended this result to all finitely generated virtually Abelian groups [16]. This is interesting since the MCF languages are exactly the string languages generated by hyperedge replacement grammars [10,31].…”

Groups whose word problems are not semilinear

“…Meng-Che Ho [17] has recently shown that the word problem of Z n is MCF for all n. Hence by Lemma 3 all finitely generated virtually abelian groups have MCF word problems. We have the following corollary to Theorem 5.…”

Comparing consecutive letter counts in multiple context-free languages

“…A multiple context-free grammar is a quadruple G = (N, Σ, P, S), where N is a finite ranked set of non-terminals, Σ is an alphabet, P is a finite set of production rules over (N, Σ) and S ∈ N (1) is the start symbol. We call G m-multiple context-free or a m-MCFG, if the rank of all non-terminals is at most m.…”