Full Symmetric Function in Partial K-Valued Logic

The functional completeness problems of the partial K-valued logic functions have a wide range of applications including cryptography and the real combinatorial circuits design. It includes the decision and construction for Sheffer functions in P, and the solution of the problems depends on determining all precomplete classes in P, and reduces to determine the minimal cover of the union of all precomplete classes in P. An n-ary function f is a Sheffer function if and only if f does not belong to any other precomplete classes in P. Hence, it is important to determine the minimal cover of the union of all precomplete classes on partial K-valued logic functions in on studying Sheffer Function. In this paper, some full symmetric function sets (m=k)are proved to be the component of the minimal cover of the union of all precomplete classes in P.