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

Yutaka HATA  Kazuharu YAMATO  

IEICE TRANSACTIONS on Information and Systems   Vol.E77-D   No.5   pp.567-573
Publication Date: 1994/05/25
Online ISSN: 
Print ISSN: 0916-8532
Type of Manuscript: PAPER
Category: Computer Hardware and Design
computer hardware and design,  multiple-valued logic,  truncated sum,  completeness,  number of the logic functions,  

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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 (xy, p1) and DPRODUCT (x, y)max (xy(p1), 0) is newly defined as the product that is derived by applying De Morgan's laws to TSUM. We call the functions T-functios. 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-functios 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.