International Journal of Intelligent Systems:Special Issue on “Fuzzy Bags, Cardinality,and Quantification”Miguel Delgado,† Daniel Sanchez,∗ Marıa-Amparo Vila‡

Department of Computer Science and Artificial Intelligence of the Universityof Granada, E.T.S.I.I.T., C/ Periodista Daniel Saucedo Aranda s/n, 18071Granada, Spain

Fuzzy cardinality concerns the assessment of the number of elements in afuzzy set. There are different approaches to fuzzy cardinality, dating back to theearly works of Zadeh, De Luca and Termini, Dubois and Prade, and Wygralak,among others. But the most important ones can be classified into scalar and fuzzyapproaches. The first one yields a scalar number (either integer or real) as the result,whereas the second one represents the cardinality of a fuzzy set as a fuzzy subset ofthe nonnegative integers.

The cardinality of a fuzzy set is on the basis of two other important issuesin fuzzy sets: fuzzy bags and fuzzy quantification. Bags (also called multisets) arealgebraic set-like structures that allow for repeated elements. Usually, this informa-tion is summarized as a set of pairs (item, number of times it appears). Fuzzy bagswhere introduced by Yager as fuzzy extensions of the notion of bag in which eachelement can appear more than once, with different degrees. To obtain the summary,a measure of the fuzzy cardinality of the set of instances of each element is needed.

Fuzzy quantifiers represent imprecise quantities or percentages, such as “ap-proximately between 4 and 6,” “around 20%,” or “most.” They generalize the usual“exists” and “forall” quantifiers of first-order logic, allowing us to represent themeaning of expressions like “Most of the students are tall.” The evaluation of suchquantified sentences requires determining the compatibility degree between the car-dinality of the fuzzy set of young students and the linguistic quantifier “most.” Earlyapproaches to this problem are due to Zadeh, Dubois and Prade, Yager, and Sugeno,among others.

Fuzzy cardinality and the closely related issues of fuzzy bags and fuzzy quan-tification are on the basis of many basic operations associated to most of the moreimportant and promising areas of application of fuzzy sets. A few of them are

∗Author to whom all correspondence should be addressed: e-mail: daniel@decsai.ugr.es.†mdelgado@ugr.es.‡vila@decsai.urg.es.

INTERNATIONAL JOURNAL OF INTELLIGENT SYSTEMS, VOL. 24, 569–571 (2009)C© 2009 Wiley Periodicals, Inc. Published online in Wiley InterScience(www.interscience.wiley.com). • DOI 10.1002/int.20356

570 DELGADO, SANCHEZ, AND VILA

• Calculation of frequencies in fuzzy data, text, and web mining, and fuzzy statistics.• Modeling linguistic quantifiers in fuzzy description logics (a key tool to build fuzzy

ontologies).• Measures of precision, recall, and similarity in fuzzy information retrieval.• Representation of the result of a flexible query in a fuzzy relational database.• Calculation of the area (number of pixels) of an imprecise region in an image. Also,

calculation of histograms for fuzzy colors.• Representation of rules in a database for robot navigation.• Information fusion, data summarization, and aggregation guided by quantifiers

(i.e., “obtain the final degree as the degree provided by most of the experts”).

Finding the most suitable cardinality measure, bag structure or quantificationmodel for each application is a key point. However, in most cases, classical standardsolutions are employed, that are not always the best solution for the specific appli-cation being developed. Hence, research on these topics is crucial and has motivatedthe present issue. The seven papers presented here show some of the more recentdevelopments and applications of these basic and very important technologies.

The first, “Fuzzy Quantification: Probabilistic Approaches and Their Applica-tion in Information Retrieval and Mobile Robotics,” by F. Dıaz-Hermida, A. Bugarın,P. Carinena, M. Mucientes, D. E. Losada, and S. Barro, presents a review of thestate of the art in fuzzy quantification for modeling linguistic expressions, and itshows practical applications in two demanding domains as information retrieval andtemporal fuzzy control in mobile robotics. In the first case, quantification is em-ployed to enrich the query language, whereas in the second one, two fuzzy controlsystems for the implementation of obstacle avoidance and wall following behaviorsare described.

In “Scalar and Fuzzy Cardinalities of Crisp and Fuzzy Multisets” by J. Casas-novas and F. Rossello, an axiomatic definition of scalar and fuzzy cardinalities offinite crisp and fuzzy bags is provided. The motivation behind this work is to providea suitable definition of cardinalities for fuzzy bags as a basis for a fuzzy versionof membrane computing, a formal computation paradigm with applications, for ex-ample, in developing formal models of biochemical processes involving chemicalcompounds represented by reactives.

In “Generalized Fuzzy Quantifiers and the Modeling of Fuzzy Branching Quan-tification,” I. Glockner extends his previous Determiner Fuzzification Scheme (DFS)theory of fuzzy quantification, that incorporates several types of fuzzy generalizedquantifiers, to deal with expressions involving branching quantification. DFS theoryis, in our view, the most recent and powerful tool for fuzzy quantification available.Branching quantification appears in expressions like “Most men and most womenadmire each other.” This work enriches the expressive power of fuzzy quantificationand provides a tool for the unsolved problem of linguistic data summarization withreciprocal summarizers, describing factors that are correlated or associated witheach other.

“Crisp Bags, One Dimensional Fuzzy Bags, Multidimensional Fuzzy Bags:Comparing Successful and Unsuccessful User’s Web Behavior,” by G. Meghabghab,shows an application of fuzzy bags for the study of user’s web behavior. The fact thatan user can visit the same web several times in a search process makes it necessary to

International Journal of Intelligent Systems DOI 10.1002/int

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employ multiset structures to register the information needed to study the behaviorof the user. To do so, the paper proposes novel measures of distance between bagsand the new notion of two dimensional bags that seems very promising for manyother applications.

In “The Set of Fuzzy Relative Integers and Fuzzy Bags,” D. Rocacher andP. Bosc extend fuzzy positive integers to fuzzy relative integers by incorporatingfuzzy negative integers, to characterize the difference operation between bags interms of fuzzy arithmetic operations in the corresponding cardinalities. These resultsare potentially useful in many applications like fuzzy database querying, fuzzy datamining, summarization, defuzzification, and fuzzy information retrieval.

The paper “An Extended Characterization of Fuzzy Bags,” M. Delgado, M. J.Martın-Bautista, D. Sanchez, and M. A. Vila introduces a characterization of fuzzybags by means of fuzzy correspondences between sets of objects and properties.In this view, classical bags, called here s-bags, are summaries of that information.This characterization allows for an interpretation of ordinary operations in s-bags asfixed union, intersection and complement of correspondences, providing a powerfulstructure with higher expressive power. In addition, an alternative formulation ofoperations on s-bags is provided in absence of the information provided by thecorrespondence, by extending the notion of IC-bags proposed by C. Chakrabarty.The new ideas are of direct application in fuzzy database querying and the definitionof fuzzy integers.

The last article, “On Nonstrict Archimedean Triangular Norms, Hamming Dis-tances, and Cardinalities of Fuzzy Sets,” is by M. Wygralak, one of the most relevantpeople in the study of fuzzy cardinality over the last 20 years. In this short note,nonstrict Archimedean t-norms are expressed in the language of Hamming dis-tances, and this new view is applied to generalized FGCounts and sigma counts, twowell-known measures of the cardinality of a fuzzy set that belong to the cardinalitytheory of fuzzy sets with triangular norms.

International Journal of Intelligent Systems DOI 10.1002/int