Efficiently irreducible bases in multiple-valued logic

Basis is a functionally complete set of multiple-valued logic functions that is irreducible, i.e. contains no complete proper subsets. Functional completeness of a set means that for any function in MVL there exists a formula over this set that implements it. However, this classical definition of basis does not consider the efficiency of implementation, particularly, it does not guarantee the existence of an efficient implementation regarding the complexity of formal expressions. In this note the notion of efficiently irreducible basis is introduced, and is termed ε-basis. A criterion for the basic set of operations to be efficiently irreducible is given. In the cases of Boolean and ternary logic functions complete enumeration and description of ε-bases are presented