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Generalized Quantifiers in the Context of Rough Set Semantics

Dutta S., Skowron A.

Fundamenta Informaticae

2015

Vol. 142, nr 1-4

213--236

Looking back to Prof. Zadeh’s paradigm of Computing withWords (CWW) [28, 29, 30], one can notice that the initial attempt of such an endeavour was to set up a basic vocabulary of linguistic words, and fix their semantics based on fuzzy sets. Then a grammar was proposed to generate compound linguistic expressions based on the primitive ones, and simultaneously based on the semantic interpretations of those basic linguistic expressions a general scheme for the semantics of the rest of linguistic expressions were proposed. Sentences involving linguistic quantifiers and vague predicates constitute a fragment of natural language. In this paper, we choose this fragment of the natural language, and explore the semantics from the perspective of rough sets [13, 14, 16, 17, 18, 21]. We fix a set of basic crisp quantifiers, mainly of proportional kind. A set of vague quantifiers are proposed to lie in a close vicinity of those crisp quantifiers in the sense that a particular vague quantifier can be visualized as a blurred, may be called rough, image of a set of crisp quantifiers. Semantics of the rest of the vague quantifiers can be obtained based on the subjective perception of the interrelations among the (vague) quantifiers.

Conditions with aggregates evaluated using gradual numbers

Lietard L., Rocacher D.

Control and Cybernetics

2009

Vol. 38, no 2

395-417

This paper is devoted to the evaluation of conditions involving aggregates in the context of flexible querying of relational databases. An example of such a condition is "the maximum salary of young employees is high", where the aggregate max applies to a fuzzy set of salaries. At first, we consider the evaluation of quantified statements where the aggregate count (the cardinality) is implicitly used. We extend this result to conditions involving the aggregates average, maximum or minimum. The contribution of this paper is to propose a new theoretical background for their evaluations based on an arithmetic on gradual numbers (Nƒ,Zƒ,Qƒ).