Inclusion of regular and linear languages in group languages
(1312.0190)Abstract
Let $\Sigma = X\cup X{-1} = { x1 ,x2 ,..., xm ,x1{-1} ,x2{-1} ,..., xm{-1} }$ 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 $\mathfrak{L} (G)$.
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