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

TL;DR

This paper explores the open problem of developing a polynomial-time algorithm to determine if a context-free language is included in a group language, introducing a novel language representation and an efficient algorithm.

Contribution

It proposes a new method for representing context-free languages using finite digraphs labeled with a monoid, and describes an algorithm with cubic complexity in the size of the input.

Findings

Algorithm executes no more than O(n^3) operations in the semiring.

Introduces a novel representation of context-free languages via labeled digraphs.

Addresses the open problem of polynomial algorithms for language inclusion.

Abstract

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 $\mathcal{S}_\mathcal{U}$.