Some Closed Classes of Three-Valued Logic Generated by Periodic Symmetric Functions

Closed classes of three-valued logic generated by periodic symmetric funtions that equal 1 in tuples from {1, 2} and equal 0 on the rest tuples are considered. Criteria for bases existence and finite bases existence for these classes is obtained. The problem of the bases existence for some families of closed classes of the three-valued logic functions is considered in the paper. E. Post [1] (see also, for instance, [2]) described all closed classes of Boolean functions and showed that each such class has a finite basis. This result is not extendable to the case of k-valued logics for k ≥ 3. Ju. I. Janov and A.A.Muchnik [3] (see also, for instance, [2]) showed that for all k ≥ 3 the set Pk (here Pk is the set of all functions of the k-valued logic) contains closed classes having a countable basis, and those having no basis. The generating systems for classes from these examples consist of symmetric functions that take values from the set {0, 1} and equal to zero on tuples containing at least one zero component. Similar classes have been described in [4–6]. Criteria of basis existence and finite basis existence for these classes have been obtained. In [7] some closed classes, generated by symmetric periodic function with bounded period have been investigated. Criterium of finite basis existence has been obtain and it was shown that if such class has no finite basis, it has no any basis. This paper deals with closed classes of three-valued logic, generated by symmetric periodic functions with increasing period. Criteria of basis existence and finite basis existence has been obtained. Let R be the set of all functions that take values from the set {0, 1} and equal zero on tuples containing at least one zero. R = {f(x1, . . . , xn) | ((∀α̃)((α̃ ∈ {0, 1, 2} ) → (f(α̃) ∈ {0, 1})))& ((∀α̃)((α̃ ∈ {0, 1, 2}\{1, 2}) → (f(α̃) = 0)))} In this paper we deal with some subclasses of the class R. Any function that does not change with variable relabeling is called symmetric. Denote by S the set of all symmetric functions from R. The set of all tuples that can be obtained from each other by component permutation is called a layer. With L(e, d) we denote a layer from {1, 2} containing e 1s ∗National Research University Higher School of Economics All necessary definitions can be found in [2, 4–6].