Fuzzy reasoning and fuzzy relational equations (Q1094394)

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)].