Inclusion of regular and linear languages in group languages

TL;DR

This paper investigates the inclusion problem of regular and linear languages within group languages, providing polynomial algorithms for testing such inclusion and exploring the relationship with context-free languages.

Contribution

It introduces polynomial algorithms to determine whether regular or linear languages are included in group languages, advancing understanding of language inclusion problems in group theory.

Findings

Polynomial algorithms for regular language inclusion in group languages

Polynomial algorithms for linear language inclusion in group languages

Finite set constructions characterize language inclusion in group languages

Abstract

Let $\Sigma = X\cup X^{-1} = \{ x_1 ,x_2 ,..., x_m ,x_1^{-1} ,x_2^{-1} ,..., x_m^{-1} \}$ and let $G$ be a group with set of generators $\Sigma$. Let $\mathfrak{L} (G) =\left\{ \left. \omega \in \Sigma^* \; \right\vert \;\omega \equiv e \; (\textrm{mod} \; G) \right\} \subseteq \Sigma^*$ be the group language representing $G$, where $\Sigma^*$ is a free monoid over $\Sigma$ and $e$ is the identity in $G$. The problem of determining whether a context-free language is subset of a group language is discussed. Polynomial algorithms are presented for testing whether a regular language, or a linear language is included in a group language. A few finite sets are built, such that each of them is included in the group language $\mathfrak{L} (G)$ if and only if the respective context-free language is included in $\mathfrak{L} (G)$.