A short note on fuzzy relational inference systems

This paper is a short note contribution to the topic of fuzzy relational inference systems and the preservation of their desirable properties. It addresses the two main fuzzy relational inferences – compositional rule of inference (CRI) and the Bandler–Kohout subproduct (BK-subproduct) – and their combination with two fundamental fuzzy relational models of fuzzy rule bases, namely, the Mamdani–Assilian and the implicative models. The goal of this short note article is twofold. Firstly, we show that the robustness related to the combination of BK-subproduct and implicative fuzzy rule base model was not proven correctly in [24]. However, we will show that the result itself is still valid and a valid proof will be provided. Secondly, we shortly discuss the preservation of desirable properties of fuzzy inference systems and conclude that neither the above mentioned robustness nor any other computational advantages should automatically lead to a preference of the combinations of CRI with Mamdani–Assilian models or of the BK-subproduct with the implicative models

Order from non-associative operations

Algebraic structures are often converted to ordered structures to gain information about the algebra using the properties of partially ordered sets. Such studies have been predominantly undertaken for semigroups, using various proposed relations. This has led to a spate of works dealing with associative fuzzy logic connectives (FLCs) and the orders that they generate. One such relation, proposed by Clifford, is employed both for its generality as well as utility. In a recent work, Gupta and Jayaram classified the semigroups that yield a partial order through the relation. In this work, we characterise groupoids that would give a partial order by introducing a property called the Generalised Quasi-Projectivity. Further, for the groupoids that lead to an ordered set, we explore the monotonicity of the underlying groupoid operation on the obtained poset. Finally, in light of the above results, we explore the major non-associative fuzzy logic connectives along these lines, thus complementing and augmenting, already existing works in the literature. Our work also shows when an FLC from a given class of operations remains one even w.r.to the order generated from it

Interpolativity of at-least and at-most models of monotone fuzzy rule bases with multiple antecedent variables ☆

Of the many desirable properties of fuzzy inference systems, not all are known to co-exist. For instance, a fuzzy inference system should be interpolative, i.e., if a fuzzy set equivalent to one of the antecedent fuzzy sets appears on the input of such a system, the inferred output should be equivalent to the respective consequent fuzzy set. Similarly, the defuzzified output obtained from a system based on a monotone fuzzy rule base should be monotone, i.e., if x,x′x,x′ are crisp inputs to the system that are ordered x≤x′x≤x′, then, the corresponding defuzzified outputs from the system y,y′y,y′ should also be ordered accordingly, i.e., y≤y′y≤y′. However, the particular setting that ensures monotonicity need not simultaneously ensure interpolativity. Recently, Štěpnička and De Baets have investigated and demonstrated the co-existence of the above two properties in the case of fuzzy relational inference systems and single-input-single-output (SISO) rule bases. An extension of these results to the multiple-input-single-output (MISO) case is not straightforward due to the lack of a natural ordering in higher dimensions. In this work, we study the MISO case and show that similar results can be obtained when the monotone rule base is modeled based on at-most and at-least modifiers