Fuzzy reasoning and fuzzy relational equations
type: Item
instance of: scholarly article
title: Fuzzy reasoning and fuzzy relational equations
zbMATH Open document ID: 0629.94031
DOI: 10.1016/S0165-0114(86)80032-X
author: Wang Ming Wu
published in: Fuzzy Sets and Systems
publication date: 1986-01-01T00:00:00Z
review text: The paper covers an interesting approach to two basic schemes of reasoning with fuzzy premises namely modus ponens and modus tollens, \[ \begin{alignedat}{2} &A && B \\ &\underline{A_ i \to B_ i,\quad i=1,2,\ldots,N} &\quad& \underline{A_ i \to B_ i,\quad i=1,2,\ldots,N} \\ &B && A \end{alignedat} \] where \(A\), \(A_ i\), \(B\), \(B_ i\) denote fuzzy sets defined in two universes of discourse, \({\mathcal A}\) and \({\mathcal B}\), respectively. \(B\) and \(A\) are fuzzy sets to be determined by these schemes of reasoning. In the sequel, the reasoning schemes are reformulated in terms of fuzzy relational equations with sup-t composition, \(B=AtR\), \(A=BtR^T\), where \(R: \mathcal A\times\mathcal B\to [0,1]\), \(T\) denotes its transpose while \(t\) stands for \(t\)-norm. Some criteria concerning a choice of a suitable composition and implication operator are also proposed and discussed. The studies conveyed in this paper are strongly related to some earlier findings in the field of fuzzy relational equations [cf. the reviewer, J. Math. Anal. Appl. 107, 520--536 (1985; Zbl 0581.04003)] as well as their direct impacts to reasoning schemes [cf. the reviewer, Fuzzy Sets Syst. 16, 163--175 (1985; Zbl 0601.68061)].
reviewed by: Witold Pedrycz
Mathematics Subject Classification ID: 94D05
Mathematics Subject Classification ID: 03B52
zbMATH DE Number: 4025346
zbMATH Keywords: fuzzy reasoning
zbMATH Keywords: modus ponens
zbMATH Keywords: modus tollens
zbMATH Keywords: fuzzy relational equations
zbMATH Keywords: sup-t composition
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 4025346