Finiteness of One-Valued Function Classes in Many-Valued Logic

Abstract

This paper addresses the theoretical issues in k-valued logic, which are crucial for developing solutions in various fields of science and technology. One of the fundamental issues is a complete description of the closed classes of functions of three-valued logic. The explicit description of closed classes in multivalued logic is an open problem. In this study, we consider a special case of the finite generation of all closed classes of three-valued logic through the operation of superposition. Previously, we considered the issue of the finite generation of classes containing a subset of single-variable functions. We have also provided a description of superlattices (lattices of lattices) containing a precomplete class of unary functions. The finite generation of these superlattices is proved. On the basis of these results, in this paper, we have proven that any class containing any of the precomplete classes from the set of single-valued functions is also finitely generated. The main result of this paper consists of three theorems on the finite generation of classes containing precomplete classes of single-valued functions and classes including all monotone unary functions. Thus, the obtained theoretical result provides easily verifiable criteria for the finiteness of classes of multivalued logic functions. It allows you to use simple procedures instead of cumbersome explicit constructs. The finite generation of overlattices allows the development of digital computing circuits that are crucial for practical applications. The proofs are based on an explicit description of these classes by an induction in the number of variables and essentially use the properties of functionally closed (Burle) classes of functions.