Completeness and normal form of multi-valued logical functions. Issue 14 (September 2020)

• Record Type: Journal Article
• Authors: Cheng, Daizhan; Liu, Zequn; Qi, Hongsheng
• Is Part Of: Journal of the Franklin Institute. Volume 357:Issue 14(2020)
• Page Start: 9871
• Page End: 9884
• Publication Date: 2020-09
• Subjects: Science -- Periodicals; Technology -- Periodicals; Patents -- United States -- Periodicals; 505
• DOI: 10.1016/j.jfranklin.2020.06.026
• Languages: English
• ISSNs: 0016-0032

Abstract

Abstract: Theory of completeness is essential for multi-valued logical functions. Using semi-tensor product (STP) of matrices, the algebraic form of k -valued logical functions is presented. Using algebraic form, a method is proposed to construct an adequate set of connectives (ASC), consisting of unary operators with conjunction/disjunction for k -valued logical functions, which can be used to express any k -valued logical functions. Based on it, two normal forms of k -valued logical functions are presented, which are extensions of the disjunctive normal form and conjunctive normal form of Boolean functions respectively. The ASC is then simplified to a condensed set. Finally, the normal forms are further extended to mix-valued logical functions.