Lattice Structure of Some Closed Classes for Non-binary Logic and Its Applications

Abstract

The paper provides a brief overview of modern applications of multi-valued logic models, where the design of heterogeneous computing systems with small computing units based on three-valued logic gives the mathematically better and more effective solution compared to binary models. It is necessary for applications to implement circuits comprised from 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 classes of such functions ensures that circuits with the desired operations can be produced from on 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 has been considered. It was 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 $T_2$ preserving two is considered. The closure operator $\mathcal{R}_1$ for which functions that differ only by dummy variables are considered to be equivalent is considered in this paper. A lattice is constructed for closed subclasses in $T_2 = \{f | f (2, \ldots , 2) = 2 \}$ – class of functions preserving two

The publication has been prepared with the support of the Russian Foundation for Basic Research according to the research project No.20-01-00575 A.