On infinite generativeness of quinary fractions in a class of probability transformers

Background. The object of the research is the expressibility of rational probabilities by transforming random variables with distributions from some initial set by Boolean functions. One of the important questions in the study of expressibility is the finite generation of the sets of distributions, i.e. the possibility of expressing all distributions from the required class using some finite set of initial distributions. Within the framework of this work, we investigate the finite generation of probabilities expressed by fivefold fractions when transforming random variables by the voting function. Materials and methods. To study the expressive capabilities of probability converters, methods are used that combine the theory of Boolean functions and mathematical analysis, as well as elementary number theory. Results. In this article, it is shown that transformations of random variables with distributions from a finite set using the voting function do not allow one to express all the probabilities written in fivefold fractions. Conclusions. The work proves the infinite generation of the class of rational probabilities under transformations by the voting function, which is a rather powerful probability transformer. Although the methods used do not carry over directly to other important transformative systems (for example, “conjunction, disjunction”), they provide a non-trivial example of the proven infinite generation of the probability class.