Vector modeling of atomic linguistic elements taking into account their dual algebraic essence

Abstract

In modern linguistics and mathematics the task of mathematical formalization of natural language for the purpose of its further computer processing is becoming more and more relevant. This topic is in demand, primarily, in artificial intelligence systems, which are becoming increasingly widespread both in science and in everyday life. Therefore, the efforts of scientists are aimed at finding a mathematical apparatus that would meet the needs of an interdisciplinary research direction associated with the symbiosis of mathematical and humanitarian knowledge. Human thought processes, which are reflected in speech phenomena, are traditionally formalized by the apparatus of mathematical logic. But at the current level of development of linguistic research, the tools of classical logic are insufficient. The mathematical apparatus of vector logical algebra, which is a development of the algebra of finite predicates, makes it possible to develop computer linguistics. This article shows the fundamental possibility of mathematically formalizing the simplest speech constructions using the basic operations of vector logical algebra. Using the example of Ukrainian and English, which are representatives of different language groups, it is proven that for all simple word combinations of natural language the axioms of the logical field are fulfilled. It does not matter whether these word-combinations consist of main or secondary members of the sentence, nor what type of grammatical connection is used in these word combinations. Such formalization allows us to represent natural language in the form of algebraic formulas and consider its constructions as elements of a vector space.