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On a Class of MultipleValued Logic Functions with Truncated Sum, Differential Product and Not Operations
Yutaka HATA Kazuharu YAMATO
Publication
IEICE TRANSACTIONS on Information and Systems
Vol.E77D
No.5
pp.567573 Publication Date: 1994/05/25 Online ISSN:
DOI: Print ISSN: 09168532 Type of Manuscript: PAPER Category: Computer Hardware and Design Keyword: computer hardware and design, multiplevalued logic, truncated sum, completeness, number of the logic functions,
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Summary:
Truncated sum (TSUM for short) is useful for MVPLA's realization. This paper introduces a new class of multiplevalued 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 Tfunctios. First, this paper clarifies that a set of Tfunctions is not a lattice. It clarifies that Lukasiewicz implication can be expressed by TSUM and NOT. It guarantees that a set of pvalued Tfunctios is not complete but complete with constants. Next, the speculations of the number of Tfunctions for less than ten radixes are derived. For eleven or more radix p, a speculation of the number of pvalued Tfunctions is shown. Moreover, it compares the Tfunctions with Bfunctions. The Bfunctions have been defined as the functions expressed by MAX, MIN, NOT and variables. As a result, it shows that a set of Tfunctions includes a set of Bfunctions. Finally, an inclusion relation among these functional sets and normality condition is shown.

