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

TL;DR

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 U.

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

Figures (2)

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Theorems & Definitions (13)

• Definition 1.1
• Theorem 2.1
• Definition 3.1
• Definition 3.2
• Theorem 3.3
• Example 3.4
• Theorem 3.5
• Theorem 4.1
• Proposition 4.2
• Theorem 4.3