Characterization of t-norms on normal convex functions
Jie Sun
TL;DR
The paper addresses constructing $t$-norms on the lattice of convex normal functions $L$ for type-2 fuzzy rule-based systems. By analyzing a convolution ${\ast_{\vartriangle}}$ induced from base operations on $[0,1]$, it derives necessary and sufficient conditions for ${\ast_{\vartriangle}}$ to be a $t$-norm on $(L,⊑)$, linking these to the continuity properties of the underlying $\ast$ and $\vartriangle$. It provides a complete characterization, showing closure, monotonicity, and associativity hold under these conditions, and proposes a computationally efficient variant of convolution. Dual results for $t$-conorms via negation are also established, expanding design options for Type-2 RFSs and improving tractability. The work thus broadens the toolbox for combining convex normal functions in type-2 fuzzy inference while maintaining rigorous algebraic structure.
Abstract
Type-2 fuzzy set (T2 FS) were introduced by Zadeh in 1965, and the membership degrees of T2 FSs are type-1 fuzzy sets (T1 FSs). Owing to the fuzziness of membership degrees, T2 FSs can better model the uncertainty of real life, and thus, type-2 rule-based fuzzy systems (T2 RFSs) become hot research topics in recent decades. In T2 RFS, the compositional rule of inference is based on triangular norms (t-norms) defined on complete lattice $(L,\sqsubseteq)$ ( L is the set of all convex normal functions from [0,1] to [0,1], and , $\sqsubseteq$ is the so-called convolution order). Hence, the choice of t-norm on $(L,\sqsubseteq)$ may influence the performance of T2 RFS. Therefore, it is significant to broad the set of t-norms among which domain experts can choose most suitable one. To construct t-norms on $(L,\sqsubseteq)$, the mainstream method is convolution which is induced by two operators on the unit interval [0,1]. A key problem appears naturally, when convolution is a t-norm on $(L,\sqsubseteq)$. This paper has solved this problem completely. Moreover, note that the computational complexity of operators prevent the application of T2 RFSs. This paper also provides one kind of convolutions which are t-norms on $(L,\sqsubseteq)$ and extremely easy to calculate.