On a Class of Multiple-Valued Logic Functions with Truncated Sum, Differential Product and Not Operations

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Truncated sum (TSUM for short) is useful for MV-PLA's realization. This paper introduces a new class of multiple-valued logic functions that are expressed by truncated sum, differential product (DPRODUCT for short), NOT and variables, where TSUM(x, y)=min(x+y, p-1) and DPRODUCT(x, y)=max(x+y-(p-1), 0) is newly defined as the product that is derived by applying De Morgan's laws to TSUM. We call the functions T-functions. First, this paper clarifies that a set of T-functions is not a lattice. It clarifies that Lukasiewicz implication can be expressed by TSUM and NOT. It guarantees that a set of p-valued T-functions is not complete but complete with constants. Next, the speculations of the number of T-functions for less than ten radixes are derived. For eleven or more radix p, a speculation of the number of p-valued T-functions is shown. Moreover, it compares the T-functions with B-functions. The B-functions have been defined as the functions expressed by MAX, MIN, NOT and variables. As a result, it shows that a set of T-functions includes a set of B-functions. Finally, an inclusion relation among these functional sets and normality condition is shown.
社団法人電子情報通信学会の論文
1994-05-25