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# On Certain Classes of Irregular Languages

 Авторы: Belousov A.I., Ismagilov R.S., Filippova L.E. Опубликовано: 08.06.2020 Опубликовано в выпуске: #3(90)/2020 DOI: 10.18698/1812-3368-2020-3-30-43 Раздел: Математика | Рубрика: Вычислительная математика Ключевые слова: formal language, alphabet, regular language, distribution vector, irregularity condition, equivalence relation, equivalence relation index, sparse sets

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

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