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1
Content available remote The Structure of Multigranular Rough Sets
EN
We study multigranulation spaces of two equivalences. The lattice-theoretical properties of so-called “optimistic” and “pessimistic” multigranular approximation systems are given. We also consider the ordered sets of rough sets determined by these approximation pairs.
2
Content available remote Graphical Partitions and Graphical Relations
EN
We generalize the well-known correspondence between partitions and equivalence relations on a set to the case of graphs and hypergraphs. This is motivated by the role that partitions and equivalence relations play in Rough Set Theory and the results provide some of the foundations needed to develop a theory of rough graphs. We use one notion of a partition of a hypergraph, which we call a graphical partition, and we show how these structures correspond to relations on a hypergraph having additional properties. In the case of a hypergraph with only nodes and no edges these properties are exactly the usual reflexivity, symmetry and transitivity properties required for equivalence relations on a set. We present definitions for upper and lower approximations of a subgraph with respect to a graphical partition. These generalize the well-known approximations in Rough Set Theory. We establish fundamental properties of our generalized approximations and provide examples of these constructions on some graphs.
EN
This paper presents two approaches to determining estimates of an equivalence relation on the basis of pairwise comparisons with random errors. Obtaining such an estimate requires the solution of a discrete programming problem which minimizes the sum of the differences between the form of the relation and the comparisons. The problem is NP hard and can be solved with the use of exact algorithms for sets of moderate size, i.e. about 50 elements. In the case of larger sets, i.e. at least 200 comparisons for each element, it is necessary to apply heuristic algorithms. The paper presents results (a statistical preprocessing), which enable us to determine the optimal or a near-optimal solution with acceptable computational cost. They include: the development of a statistical procedure producing comparisons with low probabilities of errors and a heuristic algorithm based on such comparisons. The proposed approach guarantees the applicability of such estimators for any size of set.
4
Content available remote Equivalence Relations Defined by Numbers of Occurrences of Factors
EN
We study the question of what can be said about a word based on the numbers of occurrences of certain factors in it. We do this by defining a family of equivalence relations that generalize the so called k-abelian equivalence. The characterizations and answers we obtain are linear algebraic. We also use these equivalence relations to help us in solving some problems related to repetitions and palindromes, and to point out that some previous results about Sturmian words and k-abelian equivalence hold in a more general form.
5
Content available remote On Modal μ-Calculus in S5 and Applications
EN
We consider the μ-calculus over graphs where the accessibility relation is an equivalence (S5-graphs). We show that the vectorial μ-calculus model checking problem over arbitrary graphs reduces to the vectorial, existential μ-calculus model checking problem over S5 graphs. Moreover, we give a proof that satisfiability of μ-calculus in S5 is NP-complete, and by using S5 graphs we give a new proof that the satisfiability problem of the existential μ-calculus is also NP-complete. Finally we prove that on multimodal S5, in contrast with the monomodal case, the fixpoint hierarchy of the μ-calculus is infinite and the finite model property fails.
6
Content available remote An Improved Axiomatic Definition of Information Granulation
EN
To capture the uncertainty of information or knowledge in information systems, various information granulations, also known as knowledge granulations, have been proposed. Recently, several axiomatic definitions of information granulation have been introduced. In this paper, we try to improve these axiomatic definitions and give a universal construction of information granulation by relating information granulations with a class of functions of multiple variables. We show that the improved axiomatic definition has some concrete information granulations in the literature as instances.
EN
This paper presents a tool for handling equivalence relations in non-deterministic information systems. Some applications of equivalence relations are also shown. In a deterministic information system, it is possible to define an equivalence relation for any set of attributes. However, in a non-deterministic information system, some kinds of equivalence relations which we call possible equivalence relations are definable. This paper proposes two effective procedures producing all possible equivalence relations for any non-deterministic information system and any set of attributes. The details of algorithms, the implementation of algorithms and applications to the rule extraction, etc. are presented.
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