On the problems of extracting the root from a given finite language

Boris Melnikov, Aleksandra Melnikova

Abstract

The article considers finite languages only. Based on the standard definition of the product (concatenation), the non-negative degree of the language is introduced. Root extraction is the inverse operation to it, and it can be defined in several different ways. Despite the simplicity of the formulation of the problem, the authors could not find any description of it in the literature (as well as on the Internet), including even its formulation. Despite this, we believe that all the material of the paper (including the proven theorem) about a possible change in the root-answer it is very easy, and it is puzzling that there are no formulations of such tasks in the monographs known to the authors. Most of the material in this paper is devoted to the simplest version of the formulation, i.e., the root of the 2-th degree for the 1-letter alphabet; but many of the topics of the paper are generalized to more complex cases. Apparently, for a possible future description of a polynomial solution algorithm, at least one of the described statements of the root extraction tasks first needs to really analyze in detail such a special case, that is: either describe the necessary polynomial algorithm, or, conversely, show that the problem belongs to the class of NPcomplete problems. Thus, in this paper we do not propose a polynomial algorithm for the considered problems; however, the models described here should help in constructing appropriate heuristic algorithms for solving them. A detailed description of the possible further application of such heuristic algorithms is beyond the scope of this article, but several arguments about such a possible application can be carried out already now. Firstly, the possibility of using such algorithms in cryptography and cryptanalysis is quite obvious. Secondly, even if there is a polynomial algorithm for the considered root extraction problem, it seems to be quite complex, and should be called a “intractable” or a “hard problem”. Thirdly, the considered problem is easily reduced to an NP-complete problem from the field of graph theory, i.e. the problem of finding a clique; at the same time, the opposite information has not been proven; we also mention the problem of set covering.

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