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The use of discrete probabilistic distributions is relevant to many practical tasks, especially in present-day situations where the data on distribution are insufficient and expert knowledge and evaluations are the only instruments for the restoration of probability distributions. However, in such cases, uncertainty arises, and it becomes necessary to build suitable approaches to overcome it. In this direction, this paper discusses a new approach of fuzzy binomial distributions (BDs) and their extensions. Four cases are considered: (1) When the elementary events are fuzzy. Based on this information, the probabilistic distribution of the corresponding fuzzy-random binomial variable is calculated. The conditions of restrictions on this distribution are obtained, and it is shown that these conditions depend on the ratio of success and failure of membership levels. The formulas for the generating function (GF) of the constructed distribution and the first and second order moments are also obtained. The Poisson distribution is calculated as the limit case of a fuzzy-random binomial experiment. (2) When the number of successes is of a fuzzy nature and is represented as a fuzzy subset of the set of possible success numbers. The formula for calculating the probability of convolution of binomial dependent fuzzy events is obtained, and the corresponding GF is built. As a result, the scheme for calculating the mathematical expectation of the number of fuzzy successes is defined. (3) When the spectrum of the extended distribution is fuzzy. The discussion is based on the concepts of a fuzzy-random event and its probability, as well as the notion of fuzzy random events independence. The fuzzy binomial upper distribution is specifically considered. In this case the fuzziness is represented by the membership levels of the binomial and non-binomial events of the complete failure complex. The GF of the constructed distribution and the first-order moment of the distribution are also calculated. Sufficient conditions for the existence of a limit distribution and a Poisson distribution are also obtained. (4) As is known, based on the analysis of lexical material, the linguistic spectrum of the statistical process of word-formation becomes two-component when switching to vocabulary. For this, two variants of the hybrid fuzzy-probabilistic process are constructed, which can be used in the analysis of the linguistic spectrum of the statistical process of word-formation. A fuzzy extension of standard Fuchs distribution is also presented, where the fuzziness is reflected in the growing numbers of failures. For better representation of the results, the examples of fuzzy BD are illustrated in each section. [ABSTRACT FROM AUTHOR]

This comprehensive, bird's view research note combines the state of the art, a brief presentation of the history and some original solutions, and position like views of some prospective future developments of one of the most relevant and interesting areas related to the use of fuzzy logic in database management systems, notably in its querying component, and – to some extent – in a broader issue of data and information management. We briefly summarize the roots of those new applications of fuzzy logic, more relevant proposals and development in the context of fuzzification of the basic relational database model, and then some of its further generalizations. We particularly focus on fuzzy querying as a human consistent and friendly way of retrieving information due to real human intentions and preferences expressed in natural language represented via fuzzy logic and possibility theory. We mention some extensions, notably fuzzy queries with linguistic quantifiers, and point their close relation to linguistic summaries. As for newer, prospective developments, we mainly focus on bipolar queries that can accomodate the users' intentions and preferences involving some sort of a required and desired, mandatory and optional, etc. conditions. We show various ways of handling such queries. We conclude with some brief position statements of our view on relevant and promising directions, and challenges. [ABSTRACT FROM AUTHOR]

Abstract: The concept of a bipolar query, meant as a database query that involves both negative and positive conditions, is discussed from the point of view of flexible database querying and modeling of more sophisticated intentions and preferences of the user. The aggregation of matching degrees against the negative and positive conditions to derive an overall matching degree is considered taking into account as the point of departure the Lacroix and Lavency''s (1987) approach and the semantics of these conditions. Three variants of formalization of the “and possibly” operator which governs the adopted aggregation mode of the negative and positive parts of a bipolar query are proposed. These variants take the form of logical formulas whose logical connectives may be modeled using fuzzy logic in many different ways. New recommendations as to the choice of such variants and sets of logical connectives, which preserve the assumed semantics of the “and possibly” operator, expressed as a number of postulated plausible, or desired, properties of aggregation, are formulated. [Copyright &y& Elsevier]

This paper is an improved and extended version of our previous work2 on typicality in terms of Atanassov's intuitionistic fuzzy sets (to be called A-IFSs, for short)3. We follow the line of reasoning known from psychological and cognitive sciences, in particular from linguistic experiments, and verify how those results work in the case of classification - a typical problem in computer science, decision sciences, etc. Our considerations concentrate on a typical example discussed in cognitive sciences - we investigate to which extent a linguistic representation in a psychological space (we start from nominal data - names are assigned to objects as labels) succeeds in predicting categories via A-IFSs. First, we consider a model of categories with a geometrical centroid model in which the similarity is defined in terms of a distance to centroids. Next, we verify if the extreme ideals, which are important in cognitive processes when categories are learnt in the presence of the alternative (contrast) category, give comparative results. Finally, we discuss if the 'reachable extreme ideals' and 'dominating frequency centres' give comparative results. We show that A-IFSs make it possible to reflect a positive and negative information via the concept of membership and non-membership. Although the paper presents ongoing research, the results obtained are promising and point out the usefulness and strength of A-IFSs as a tool to account for more aspects of vague data and information. [ABSTRACT FROM AUTHOR]

The fuzzy set applications in political science cover decision making, games, collective choice, representation and comparative politics. The earliest applications were largely theoretical and suggestive rather than empirically anchored and testable models. We discuss a representative sample of applications and suggest some problems in making the fuzzy sets more generally applicable in political science. [ABSTRACT FROM AUTHOR]

The theory of fuzzy sets has been applied to social choice primarily in the context where one is given a set of individual fuzzy preference relations and the aim is to find a non-fuzzy choice set of winners or best alternatives. In this article, we discuss the problem of composing multi-member deliberative bodies starting again from a set of individual fuzzy preference relations. We outline methods of aggregating these relations into a measure of how well each candidate represents each voter in terms of the latter's preferences. Our main goal is to show how the considerations discussed in the context of individual non-fuzzy complete and transitive preference relations can be extended into the domain of fuzzy preference relations. [ABSTRACT FROM AUTHOR]

In this paper a concise overview of the work that has been done by various researchers in the area of type-2 fuzzy logic is analyzed and discussed. Type-2 fuzzy systems have been widely applied in the fields of intelligent control, pattern recognition and classification, among others. The overview mainly focuses on past, present and future trends of type-2 fuzzy logic applications. Of utmost importance is the last part, outlining possible areas of applied research in type-2 FL in the future. The major contribution of the paper is briefing of the most relevant work in the area of type-2 fuzzy logic, including its theoretical and practical implications. As well as envisioning possible future works and trends in this area of research. We believe that this paper will provide a good platform for people interested in this area for their future research work. [ABSTRACT FROM AUTHOR]