Lattice Structure of Some Closed Classes for Three-Valued Logic and Its Applications

Abstract

This paper provides a brief overview of modern applications of nonbinary logic models, where the design of heterogeneous computing systems with small computing units based on three-valued logic produces a mathematically better and more effective solution compared to binary models. For application, it is necessary to implement circuits composed of chipsets, the operation of which is based on three-valued logic. To be able to implement such schemes, a fundamentally important theoretical problem must be solved: the problem of completeness of classes of functions of three-valued logic. From a practical point of view, the completeness of the class of such functions ensures that circuits with the desired operations can be produced from an arbitrary (finite) set of chipsets. In this paper, the closure operator on the set of functions of three-valued logic that strengthens the usual substitution operator is considered. It is shown that it is possible to recover the sublattice of closed classes in the general case of closure of functions with respect to the classical superposition operator. The problem of the lattice of closed classes for the class of functions T2 preserving two is considered. The closure operators R1 for the functions that differ only by dummy variables are considered equivalent. This operator is withiin the scope of interest of this paper. A lattice is constructed for closed subclasses in T2={f|f(2,…,2)=2}, a class of functions preserving two.

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

“…These classes do not include a choice function, a majority function, all single-valued functions, or a set of all single-valued functions precompleted for all single-valued functions. This paper continues the study of the theoretical problems of k-valued (in particular, ternary) logic [9,28]. As mentioned above, in two-valued logic, there are several sufficient conditions for finitely generated classes [29].…”

“…One of these conditions is the presence in the class (multivalued logic functions) of one-valued functions. In this paper, based on the previously proven results [9,25,28], we show that any class containing any of the precomplete classes of the set of unary functions is finitely generated. These conditions must be such that they can be easily verified.…”

“…In addition, k-valued logic is highly relevant in the modeling of complex dynamical systems, where causal relations play a crucial role [4]. It is also used in logical dynamical systems [5], the modern control and stabilization of unmanned systems [6], cyber-physical systems [7], data transmission [8,9], optimal network routing [10,11], and quantum cryptography [12]. A comprehensive analysis of the use of k-valued logic in various applications has been provided in the research paper [9].…”

“…It is also used in logical dynamical systems [5], the modern control and stabilization of unmanned systems [6], cyber-physical systems [7], data transmission [8,9], optimal network routing [10,11], and quantum cryptography [12]. A comprehensive analysis of the use of k-valued logic in various applications has been provided in the research paper [9]. Among the wide range of applications, data aggregation schemes stand out because the development of three-valued logic algorithms has proven to be drastically more efficient and cost-effective than two-valued logic [13,14].…”

“…Thus, although this paper is rather theoretical, all the results obtained here are significant in their respective fields of study and have been widely researched and studied in the field of computer science. In [9,28], a description of superlattices containing some precomplete class of unary functions is also given, and the problem of checking the finite generation of classes containing some subclass of a function of one variable is studied. It has been proven that overlattices can be finitely generated.…”