Decision on Function of One Simple Separable Relation for the Minimal Covering of P*K

In the function structure theory of multi-logic, decision on Sheffer function is an important role. It contains structure and decision of full multi-logic and partial multi-logic. Its decision is closely related to decision of completeness of function which can be done by deciding the minimal covering of full multi-logic and partial-logic. By theory of completeness of partial multi-logic, we prove that function of one simple separable ralation is not minimal covering of P*K under the condition of m=2, σ=e .

The Determination on Minimal Covering of Regular Separable Function Sets in Partial Four-Valued Logic

In the structure theory of many-valued logic function, the decision and constitution of the Sheffer function is a very important problem, which is reduced to the decision of the minimal covering of precomplete sets in the many-valued logic function sets. According to the completeness theory in partial k-valued logic and the similar relationship theory among precomplete sets, in this paper, the methods of determination on the minimal covering of regular separable function sets are found out, and the minimal covering of regular separable function sets in partial four-valued logic are decided.

Some large classes of n-valued Sheffer functions

In 1967, Singer (11) gave 3 classes of n-valued two-place functors and proved that all these functors were Sheffer functions. Out of the n possible assignments needed to define a functor completely, Singer showed that it was sufficient to define 3n − 2, 3n − 2, and 2n assignments respectivelyfor the 3 classes. We shall enlarge Singer's classes to give functors of type Ia, type II and type III. For types Ia and III, it will be shown that it is sufficient to define 2n − 1 assignments and for type II we require 2n − 1 assignments to be defined and conditions on a further n/p1 assignments (where P1 is the least prime factor of n). These classes of functors include all of Singer's classes. We also introduce functors of type Ib, similar to those of type Ia, and show that for these itis sufficient to define 2n − 1 assignments to ensure the functor is a Sheffer function.

The Sheffer functions of 3-valued logic

In previous papers, Post, Webb, Götlind and the present author have described some Sheffer functions (in Swift's terminology, “independent binary generators”) in m-valued logic. Professor J. Dean Swift has recently isolated the symmetric Sheffer functions of 3-valued logic. In the present paper, we will prove some properties of Sheffer functions in m-valued logic and isolate all of the Sheffer functions of 3-valued logic.Before we proceed we will define some terms which we will find convenient. A set of functions in m-valued logic is functionally complete, if the set of the functions which can be defined explicitly from the functions of the set is exactly the set of all functions of m-valued logic. A function is functionally complete, if its unit set is functionally complete. A Sheffer function is a two-place functionally complete function. If i and j are truth values (1 i, j ≤ m), we will say i ~ j (D), if D is a decomposition of the truth values 1, …, m into 2 or more disjoint non-empty classes and i and j are elements of the same class. A binary function f(p, q) satisfies the substitution law for a decomposition D, if for any truth values h, i, j, k, whenever h ~ j (D) and i~k(D), then f(h, i) ~ f(j, k) (D). The function f(p,q) satisfies the co-substitution law for D, if for any truth values h, i, j, k, whenever f(h, i) ~ f(j, k) (D), then h ~ j (D) or i ~ k (D). We will say f(p, q) has the proper substitution property, if there is a decomposition of the truth values into less than m classes for which it satisfies the substitution law.