I. Duntsch. A logic for rough sets. Theoretical Computer Science, 179:427--436, 1997.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....as a pair X ; X . These simple concepts form the basis of a theory that has demonstrated a variety of relations with other theoretical results.The collection of all rough subsets of U is shown to form a regular double Stone algebra, which serves as a model for three valued Lukasiewicz logic [9]. In Shafer s evidence theory, the beliefs and plausibility can be expressed by lower approximation and upper approximation, respectively [32, 33] With those concepts defined and calculi formed, RSDA further provides a group of definitions that are directly related with data mining problems. It ....

....just notice that a fine partition often preserve all the information of a coarse partition if the latter, represented as a relation, includes most of the elements of the former. Actually this is also the idea behind the notion of dependence aforementioned. Formally, and following the notation in [9], we can define the association rules as follows. Denote Q as fX 1 ; X 2 ; X s g, and d as fY 1 ; Y 2 ; Y t g (d can be a single attribute or a composite attribute which is a combination of a number of attributes) For each X i in Q , we define a set M i = fY j jX j [ Y j 6= ....

I. Duntsch. A logic for rough sets. Theoretical Computer Science, 179:427--436, 1997.

....again a subtopos of labelled forests. The Thetarst part of the paper introduces the concepts we need from topos theory, showing their interpretation in the topos of forests. Then, we use the topos of labelled trees in order to describe a categorical semantics for the complete SCCS as described in [8]. Other categorical characterizations of such a semantics have been proposed previously, but we stress the fact that we use the algebra of the topos in order to prove all the properties needed. The second part discusses a topos theoretical account of bisimulation. The characterization of the ....

....that since in the previous sections we discussed just non labelled constructions, we shall gradually introduce their labelled counterparts. The basic assumption of SCCS is that there is an abelian group Act = #A, 2, 1, # of actions (the reasons for such a choice are thoroughly discussed in [8]) We shall work from now onwards in the topos of trees labelled in A, and we shall write 2 also for the lifting to A # of the group product; in particular, 2 n : A # (n) A # (n) # A # (n) is de Thetaned by ### 1 , # 2 , # n #, ## 1 , # 2 , # n ## ## ## 1 2# 1 , # 2 2# ....

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, Calculi for synchrony and asynchrony, Theoretical Computer Science 25 (1983), 267#310.

.... these algebras can serve as semantic models for three valued 5 Table 2: Fisher s Iris data [18] Sepal Sepal Petal Petal Object length width length width Class s # 50 33 14 2 1 s # 46 34 14 3 1 s # 65 28 46 15 2 s # 62 22 45 15 2 s # 67 30 50 17 3 s # 64 28 56 22 3 L ukasiewicz logic [9]. The connections between algebra and logic of rough set systems are explored in some detail in [45] A different logical approach to the rough set model is given by [41] and [31] There, the lower approximation is considered a necessity operator # , and the upper approximation a possibility ....

Dntsch, I. (1997a). A logic for rough sets. Theoretical Computer Science, 179, 427--436.

....the intersection of the binary relations of the dual space of A defined according to [14] In Sections 2 and 3 the relation between our duality and Goldblatt s duality is made explicit. The duality we expose was discovered, in a slightly di#erent format, independently of us by C. Hartonas, see [16]. Moreover, a Stone type duality for positive modal algebras can be extracted from Abramsky [1] From this algebraic point of view, the present paper can also be seen as a contribution to the study of distributive lattices with operators. Notice that a Sahlqvist theorem for PML with the semantics ....

....lattices with operators by means of relational Priestley spaces. Dualities of this kind are considered by Cignoli, Lafalce and Petrovich [9] Petrovich [22] and Goldblatt [14] In all these papers, in contrast with the present one, the operators considered have no relation between them. In [16] the duality we present is also described in a slightly di#erent format. Also, a Stone type duality for positive modal algebras can be extracted from [1] a work dedicated to applications to Theoretical Computer Science. The topological duality we present has strong connections with the one given ....

C. Hartonas, Duality for Modal -Logics, Theoretical Computer Science, 202 (1-2), 1998, pp. 193-222.

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