Rule-based models play a crucial role in scenarios that require transparency and accountable decision-making. However, they primarily consist of discrete parameters and structures, which presents challenges for scalability and optimization. In this work, we introduce a new rule-based classifier trained using gradient descent, in which the user can control the maximum number and length of the rules. For numerical partitions, the user can also control the partitions used with fuzzy sets, which also helps keep the number of partitions small. We perform a series of exhaustive experiments on $40$ datasets to show how this classifier performs in terms of accuracy and rule base size. Then, we compare our results with a genetic search that fits an equivalent classifier and with other explainable and non-explainable state-of-the-art classifiers. Our results show how our method can obtain compact rule bases that use significantly fewer patterns than other rule-based methods and perform better than other explainable classifiers.

Most fuzzy systems including fuzzy decision support and fuzzy control systems provide out-puts in the form of fuzzy sets that represent the inferred conclusions. Linguistic interpretation of such outputs often involves the use of linguistic approximation that assigns a linguistic label to a fuzzy set based on the predefined primary terms, linguistic modifiers and linguistic connectives. More generally, linguistic approximation can be formalized in the terms of the re-translation rules that correspond to the translation rules in ex-plicitation (e.g. simple, modifier, composite, quantification and qualification rules) in com-puting with words [Zadeh 1996]. However most existing methods of linguistic approximation use the simple, modifier and composite re-translation rules only. Although these methods can provide a sufficient approximation of simple fuzzy sets the approximation of more complex ones that are typical in many practical applications of fuzzy systems may be less satisfactory. Therefore the question arises why not use in linguistic ap-proximation also other re-translation rules corre-sponding to the translation rules in explicitation to advantage. In particular linguistic quantifica-tion may be desirable in situations where the conclusions interpreted as quantified linguistic propositions can be more informative and natu-ral. This paper presents some aspects of linguis-tic approximation in the context of the re-translation rules and proposes an approach to linguistic approximation with the use of quantifi-cation rules, i.e. quantified linguistic approxima-tion. Two methods of the quantified linguistic approximation are considered with the use of lin-guistic quantifiers based on the concepts of the non-fuzzy and fuzzy cardinalities of fuzzy sets. A number of examples are provided to illustrate the proposed approach.

In the present paper we use principles of fuzzy logic to develop a general model representing several processes in a system's operation characterized by a degree of vagueness and/or uncertainty. For this, the main stages of the corresponding process are represented as fuzzy subsets of a set of linguistic labels characterizing the system's performance at each stage. We also introduce three alternative measures of a fuzzy system's effectiveness connected to our general model. These measures include the system's total possibilistic uncertainty, the Shannon's entropy properly modified for use in a fuzzy environment and the "centroid" method in which the coordinates of the center of mass of the graph of the membership function involved provide an alternative measure of the system's performance. The advantages and disadvantages of the above measures are discussed and a combined use of them is suggested for achieving a worthy of credit mathematical analysis of the corresponding situation. An application is also developed for the Mathematical Modelling process illustrating the use of our results in practice.

This paper introduces the notion of qualitative belief assignment to model beliefs of human experts expressed in natural language (with linguistic labels). We show how qualitative beliefs can be efficiently combined using an extension of Dezert-Smarandache Theory (DSmT) of plausible and paradoxical quantitative reasoning to qualitative reasoning. We propose a new arithmetic on linguistic labels which allows a direct extension of classical DSm fusion rule or DSm Hybrid rules. An approximate qualitative PCR5 rule is also proposed jointly with a Qualitative Average Operator. We also show how crisp or interval mappings can be used to deal indirectly with linguistic labels. A very simple example is provided to illustrate our qualitative fusion rules.

Martin and Osswald \cite{Martin07} have recently proposed many generalizations of combination rules on quantitative beliefs in order to manage the conflict and to consider the specificity of the responses of the experts. Since the experts express themselves usually in natural language with linguistic labels, Smarandache and Dezert \cite{Li07} have introduced a mathematical framework for dealing directly also with qualitative beliefs. In this paper we recall some element of our previous works and propose the new combination rules, developed for the fusion of both qualitative or quantitative beliefs.

Abstract-- In this paper, we propose in Dezert-Smarandache Theory (DSmT) framework, a new probabilistic transformation, called DSmP, in order to build a subjective probability measure from any basic belief assignment defined on any model of the frame of discernment. Several examples are given to show how the DSmP transformation works and we compare it to main existing transformations proposed in the literature so far. We show the advantages of DSmP over classical transformations in term of Probabilistic Information Content (PIC). The direct extension of this transformation for dealing with qualitative belief assignments is also presented. In the theories of belief functions, Dempster-Shafer Theory (DST) [4], Transferable Belief Model (TBM) [11] or DSmT [6], [7], the mapping from the belief to the probability domain is a controversial issue.

Qualitative methods for reasoning under uncertainty have gained more and more attention by Information Fusion community, especially by the researchers and system designers working in the development of modern multi-source systems for defense, robotics and so on. This is because traditional methods based only on quantitative representation and analysis are not able to completely satisfy adequately the need of the development of science and technology integrating at higher fusion levels human beliefs and reports in complex systems. Therefore qualitative knowledge representation becomes more and more important and necessary in next generations of (semi) intelligent automatic and autonomous systems. For example, Wagner et al. [16] consider that although recent robots have powerful sensors and actuators, their abilities to show intelligent behavior is often limited because of lacking of appropriate spatial representation. Ranganathan et al. [11] describe a navigation system for a mobile robot which must execute motions in a building, the environment is represented by a topological model based on a Generalized Voronoi Graph (GVG) and by a set of visual landmarks. A qualitative self-localization method for indoor environment using a belt of ultrasonic sensors and a camera is proposed. Moratz et al. [6] point out that qualitative spatial reasoning(QSR) abstracts metrical details of the physical world, of which two main directions are topological reasoning about regions and reasoning about orientations of point configurations. So, because concrete problems need a combination of qualitative knowledge of orientation and qualitative knowledge of distance, they present a calculus based on ternary relations where they introduce a qualitative distance measurement based on two of the three points.

In this paper, we propose a simple arithmetic of linguistic labels which allows a direct extension of quantitative Belief Conditioning Rules (BCR) proposed in the DSmT [3, 4] framework to their qualitative counterpart. Qualitative beliefs assignments are well adapted for manipulated information expressed in natural language and usually reported by human expert or AIbased expert systems. A new method for computing directly with words (CW) for combining and conditioning qualitative information is presented. CW, more precisely computing with linguistic labels, is usually more vague, less precise than computing with numbers, but it is expected to offer a better robustness and flexibility for combining uncertain and conflicting human reports than computing with numbers because in most of cases human experts are less efficient to provide (and to justify) precise quantitative beliefs than qualitative beliefs. Before extending the quantitative DSmT-based conditioning rules to their qualitative counterparts, it will be necessary to define few but new important operators on linguistic labels and what is a qualitative belief assignment. Then we will show though simple examples how the combination of qualitative beliefs can be obtained in the DSmT framework.