On A. V. Anisimov's problem for finding a polynomial algorithm checking inclusion of context-free languages in group languages

The work investigates the problem of whether a context-free language is a subset of a group language. A.~V. Anisimov has shown that the problem of determining the unambiguity of finite automata is a special case of this problem. Then the question of finding polynomial algorithm verifying the inclusion of context-free languages in group languages naturally arises. The article focuses on this open problem. For the purpose, the paper describes an unconventional method of description of context-free languages, namely a representation with the help of a finite digraph whose arcs are labelled with a specially defined monoid $\mathcal{U}$. Also, we define a semiring ${\mathcal{S}}_{\mathcal{U}}$ whose elements are the set $2^{\mathcal{U}}$ of all subsets of $\mathcal{U}$ and with operations - product and union of the elements of $2^{\mathcal{U}}$. The described algorithm executes no more than $O(n^3)$ operations in ${\mathcal{S}}_{\mathcal{U}}$.