Abstract
In this paper, we will visit Rough Set Theory and the Alternative Set Theory (AST) and elaborate a few selected concepts of them using the means of higher-order fuzzy logic (this is usually called Fuzzy Type Theory). We will show that the basic notions of rough set theory have already been included in AST. Using fuzzy type theory, we generalize basic concepts of rough set theory and the topological concepts of AST to become the concepts of the fuzzy set theory. We will give mostly syntactic proofs of the main properties and relations among all the considered concepts, thus showing that they are universally valid.
Highlights
This is a theoretical paper, in which we will visit the well known Rough Set Theory and less known Alternative Set Theory (AST) and show that the basic notions of rough set theory have been included already in AST.Recall that after establishing the rough set theory [1], generalization to its fuzzy version soon appeared
This paper focuses on a few concepts of two, originally unrelated theories: the Rough Set Theory and the Alternative Set Theory
It turns out that the topology in AST has been developed using the notion of indiscernibility relation, which is the leading notion in rough set theory
Summary
This is a theoretical paper, in which we will visit the well known Rough Set Theory and less known Alternative Set Theory (AST) and show that the basic notions of rough set theory have been included already in AST (though their motivation was different). Recall that after establishing the rough set theory [1], generalization to its fuzzy version soon appeared (see [2] and the citations therein) This suggests the idea that classical and fuzzy rough set theories can be developed as one formal theory using the formalism of mathematical fuzzy logic. Using formalism of FTT, we will unify rough set and fuzzy rough set theories into one formal system Their concepts can be distinguished only semantically in a model. Using formalism of FTT, we will show the equivalence of the concepts of rough set theory with some of the topological concepts introduced earlier in AST. We argue that the use of the language of (mathematical) fuzzy logic suggests more in-depth insight into the character of the discussed concepts and helps us to better understand why the proved properties hold.
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