ϵ-bases of triadic logic operations

Basis is a functionally complete set of logic functions that contains no complete proper subset. Efficiently irreducible generating set, termed ε-basis, is an irreducible set of generators which guarantees an optimal implementation of every function, with respect to the number of literals in its formal expression. The notion of ε-basis is significant in the composition of functions, since the classical definition of basis does not consider the efficiency of implementation. In case of two-valued functions, an ε-basis consists of all unary function, constants, and only two binary functions which are freely chosen from certain classes. In this note, expanding the domain of basic operation from dyadic to triadic, we study the efficiency of sets of 3-input gates as basic operations. This expansion, evidently, induces a decrease of the complexity of functions (hence, the complexity of functional circuits to be designed). Gaining an evident merit in the complexity, we have to pay a price by considerably increasing the number of such generators for the multiple valued circuits. However, in the case of Boolean operations this number is still very small, and will certainly be useful to consider the approach in the practical circuit design. This note provides a criterion for the basic set of triadic operations to be efficiently irreducible. In the case of Boolean functions the five types of triadic operations which constitute an ε-basis are found and well classified. Finally, we show a typical example of generating set, which forms an ε-basis