On Certain Classes of Irregular Languages

Abstract

Objective of this paper is to prove certain regularity and irregularity conditions in languages determined by a set of integer vectors called distribution vectors of the number of letters in words over a finite alphabet. Each language over the finite alphabet uniquely determines its proprietary set of distribution vectors and vice versa, i.e., each set of vectors is associated with a language having this set of distribution vectors. A single necessary condition for the language regularity was considered associated with the concept of Z+-plane (sets of points with non-negative integer coordinates lying on a plane in the affine space). The condition is that a set of distribution vectors determined by any regular language could be represented as a finite union of the Z+-planes. Certain sufficient irregularity conditions associated with the distribution vector properties were proven. Based on this, classes of irregular languages could be identified. These classes are determined by a set of vectors (points) that could not be represented as a finite union of the Z+-planes; by a set of vectors containing vectors with arbitrarily high values of each coordinate and having certain restrictions on the difference between maximum and minimum values of the coordinates; by a set of vectors called the sparse sets. A method is proposed for building such sets using strictly convex and strictly increasing numerical sequences. These sufficient irregularity conditions are based on the Myhill --- Nerode theorem, which is known in the formal languages' theory. Examples of applying the proved theorems to the analysis of languages' regularity/irregularity are presented