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

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 $T_2$ preserving two is considered. The closure operators ${\mathcal{R}}_1$ 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 $T_2=\left\{f\right|f(2,\dots ,2)=2\}$, a class of functions preserving two.