On Modularity, Indistinguishability and Generalized Metrics: Duality and Aggregation Gabriel Jaume-Martin, Francisco Javier Talavera, Jorge Elorza, Oscar Valero Mathematics, 2026 In this paper, we prove that, on the one hand, a duality relationship between different types of modular T-transitive relations and the reciprocal modular generalized metrics exists and, on the other hand, that, based on this duality, a construction of functions that aggregate modular T-transitive relations can be made from functions aggregating generalized metrics. Furthermore, we provide a guide for the families of that type of functions that can be used for the aforementioned purpose. Finally, illustrative examples of how to create such functions via the duality are given.
On the Use of Modular Quasi-Metrics and Possibility Theory for Heterogeneous Multi-Robot Systems José Guerrero, Gabriel Jaume-Martin, María-del-Mar Bibiloni-Femenias, Javier Antich, Oscar Valero Frontiers in Artificial Intelligence and Applications, 2024 This paper is focused on the “multi-robot task allocation” problem where a team of heterogeneous robots must select the best next task to perform at any given moment. Several approaches have been recently proposed to address this problem. Among them, the Response Threshold Methods (RTMs) should be high-lighted, which are inspired by swarm intelligence. In RTMs, each robot has a stimulus associated with each task that represents how suitable that task is for it and a parameter, called threshold, that modulates the response tendency of the robot to the stimulus. The robot selects the best next task to execute based on a probability that depends on the stimulus, the threshold and the task the robot is currently executing. In recent years, some works have demonstrated that possibility theory (specifically, fuzzy Markov chains) is a better tool than probability theory for describing robot transitions. This paper continues this research line and presents a mathematical model based on the use of modular quasi-metrics, possibility theory and RTMs in order to describe the behavior of multi-robot systems with heterogeneous agents/robots possessing distinct characteristics. Concretely, each robot will have a different threshold value. Moreover, a very simple communication mechanism among robots, compatible with the requirements of swarm intelligence, has been considered. So, each robot makes its decision taking into account the decision made by its teammates in a neighbourhood. In this approach, due to the system’s heterogeneity, the transitions possibilities will be modeled by means of modular quasi-metrics (a type of distances which depend on a parameter, such that the dissimilarity measure is relative to that parameter). To the best of our knowledge, this mathematical framework represents the first attempt to model this kind of systems.
