Daniel Ruiz-Aguilera

The enumeration of certain classes of logical connectives and aggregation functions defined on a finite chain has garnered significant attention in recent years, particularly due to its usefulness in various applied domains such as image processing and decision-making. However, in some instances, the sheer magnitude of this enumeration makes its exact value less critical. Instead, the focus often shifts to understanding its asymptotic growth order. This perspective is valuable for anticipating and planning computational costs, and more importantly, for assessing the restrictiveness of properties imposed on a class of logical operators. Consequently, this paper delves into the asymptotic behavior of several expressions already proposed in the literature, mainly in the enumeration of discrete negations, discrete implications and discrete aggregation functions and some subclasses. Additionally, in this paper a measure is proposed to quantify the degree of restrictiveness associated with an additional property within a class of logical operators.

Discrete implications have been studied for almost two decades as those operators needed to perform inference processes when dealing with qualitative information from a finite chain. However, it is known that by means of some adequate transformations, fuzzy logic operators defined in [0,1] can generate the corresponding discrete operators. Thus, an immediate question arises: do we need to study discrete implications or is it enough to study implications defined in [0,1] and then to discretize them? The answer must rely on the preservation of the additional properties of fuzzy implication functions through these discretization methods. In this paper, for two specific methods based on the ceiling and floor functions it is proved that most of the additional properties are not preserved in general, showing that the preservation of the additional properties depends directly on the properties of the underlying operators considered in the discretization. Thus, sufficient, and for some properties necessary, conditions to guarantee the preservation are presented.

Many computer vision algorithms which are not robust to noise incorporate a noise removal stage in their workflow to avoid distortions in the final result. In the last decade, many filters for salt-and-pepper noise removal have been proposed. In this paper, a novel filter based on the weighted arithmetic mean aggregation function and the fuzzy mathematical morphology is proposed. The performance of the proposed filter is highly competitive when compared with other state-of-the-art filters regardless of the amount of salt-and-pepper noise present in the image, achieving notable results for any noise density from 5% to 98%. A statistical analysis based on some objective restoration measures supports that this filter surpasses several state-of-the-art filters for most of the noise levels considered in the comparison experiments.

In fuzzy rules based systems, fuzzy implication functions are usually considered to model fuzzy conditionals and to perform forward and backward inferences. These processes are guaranteed by the fulfilment of the Modus Ponens and Modus Tollens properties by the fuzzy implication function with respect to the considered conjunction and fuzzy negation. In this paper, we investigate which residual implications derived from uninorms satisfy both Modus Ponens and Modus Tollens properties with respect to the same t-norm and a fuzzy negation simultaneously. The most usual classes of uninorms are considered and many solutions are obtained which allow to model the fuzzy conditionals in a fuzzy rules based systems (and perform backward and forward inferences) with a unique residual implication derived from a uninorm.