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Although there have been many fuzzy or probabilistic relational database models proposed for representing and handling imprecise and uncertain information of objects in real-world applications, models combining the relevance and strength of both fuzzy set theory and probability theory appear sporadic. In this paper, we propose a new fuzzy and probabilistic relational database model where the imprecision of an attribute value is represented by a fuzzy set and the uncertainty of a relational tuple is represented by a probability interval. The mass assignment theory is employed to deal with the challenge of integration and computation of both fuzzy sets and probabilities in the same model. The conjunction and disjunction strategies to combine imprecise and uncertain information are introduced. Then the fundamental concepts of the classical relational database model are extended and generalized in this new model. The syntax and semantics of the selection operation are formally defined. Finally, the other important algebraic operations on imprecise attributes and uncertain tuples are developed. [ABSTRACT FROM AUTHOR]

Constraint Databases represent complex data by means of formulas described by constraints (equations, inequations or Boolean combinations of both). Commercial database management systems allow the storage and efficient retrieval of classic data, but for complex data a made-to-measure solution combined with expert systems for each type of problem are necessary. Therefore, in the same way as commercial solutions of relational databases permit storing and querying classic data, we propose an extension of the Selection Operator for complex data stored, and an extension of SQL language for the case where both classic and constraint data need to be managed. This extension shields the user from unnecessary details on how the information is stored and how the queries are evaluated, thereby enlarging the capacity of expressiveness for any commercial database management system. In order to minimize the selection time, a set of strategies have been proposed, which combine the advantages of relational algebra and constraint data representation. [ABSTRACT FROM AUTHOR]

Abstract: In social choice theory the Banks set is a well-established choice set for tournaments that consists of the undominated elements of the maximal subtournaments. For non-complete dominance relations J. Duggan proposed three possibilities to modify it. We develop relation-algebraic specifications to compute the Banks set, Duggan’s modifications, and variants of them. All these specifications are algorithmic and can directly be translated into the programming language of the computer algebra system RelView. We show that the system is well suited for computing and visualizing the Banks set, its modifications, and the objects to be associated with them. [Copyright &y& Elsevier]

Abstract: We formulate two open problems related to and, in a sense, suggested by the Reiterman–Tholen characterization of effective descent morphisms of topological spaces. [Copyright &y& Elsevier]

Abstract: More than 15 years ago, a set of qualitative spatial relations between oriented straight line segments (dipoles) was suggested by Schlieder. However, it turned out to be difficult to establish a sound constraint calculus based on these relations. In this paper, we present the results of a new investigation into dipole constraint calculi which uses algebraic methods to derive sound results on the composition of relations of dipole calculi. This new method, which we call condensed semantics, is based on an abstract symbolic model of a specific fragment of our domain. It is based on the fact that qualitative dipole relations are invariant under orientation preserving affine transformations. The dipole calculi allow for a straightforward representation of prototypical reasoning tasks for spatial agents. As an example, we show how to generate survey knowledge from local observations in a street network. The example illustrates the fast constraint-based reasoning capabilities of dipole calculi. We integrate our results into two reasoning tools which are publicly available. [Copyright &y& Elsevier]

Abstract: Forming closures of subsets of a set X is a technique that plays an important role in many scientific disciplines and there are many cryptomorphic mathematical structures that describe closures and their construction. One of them was introduced by Aumann in the year 1970 under the name contact relation. Using relation algebra, we generalize Aumann’s notion of a contact relation between X and its powerset and that of a closure operation on from powersets to general membership relations and their induced partial orders. We also investigate the relationship between contacts and closures in this general setting and present some applications. In particular, we investigate the connections between contacts, closures and topologies and use contacts to establish a one-to-one correspondence between the column intersections space and the row intersections space of arbitrary relations. [Copyright &y& Elsevier]

Abstract: Soft set theory, proposed by Molodtsov, has been regarded as an effective mathematical tool to deal with uncertainties. In this paper, first we prove that certain De Morgan’s law hold in soft set theory with respect to different operations on soft sets. Then, we discuss the basic properties of operations on soft sets such as intersection, extended intersection, restricted union and restricted difference. Moreover, we illustrate their interconnections between each other. Also we define the notion of restricted symmetric difference of soft sets and investigate its properties. The main purpose of this paper is to extend the theoretical aspect of operations on soft sets. [Copyright &y& Elsevier]

Abstract: The Gisin–Hughston–Jozsa–Wootters theorem plays an important role in analyzing various theories about quantum information, quantum communication, and quantum cryptography. It means that any purifications on the extended system which yield indistinguishable state ensembles on their subsystem should have a specific local unitary relation. In this Letter, we show that the local relation is also established even when the indistinguishability of state ensembles is extended to that of operator ensembles. [ABSTRACT FROM AUTHOR]

Abstract: We generalize stochastic Kripke models and Markov transition systems to stochastic right coalgebras. These are coalgebras for a functor with as an endofunctor on the category of analytic spaces, and is the subprobability functor. The modal operators are generalized through predicate liftings which are set-valued natural transformations involving the functor. Two states are equivalent iff they cannot be separated by a formula. This equivalence relation is used to construct a cospan for logical equivalent coalgebras under a separation condition for the set of predicate liftings. Consequently, behavioral and logical equivalence are really the same. From the cospan we construct a span. The central argument is a selection argument giving us the dynamics of a mediating coalgebra from the domains of the cospan. This construction is used to establish that behavioral equivalent coalgebras are bisimilar, yielding the equivalence of all three characterizations of a coalgebra’s behavior as in the case of Kripke models or Markov transition systems. [Copyright &y& Elsevier]