Abstract
This paper is an overview paper extended by new results, that deals with the relation of the basic notions of rough set theory and the topological notions introduced in the Alternative Set Theory (AST). Using the formalism of higher-order fuzzy logic (FTT), we prove syntactically that the main properties and relations among all the considered concepts hold both in classical as well as rough fuzzy set theory, and show that the basic concepts of rough sets are among some of the topological concepts of AST.
The paper has been supported from ERDF/ESF by the project “Centre for the development of Artificial Intelligence Methods for the Automotive Industry of the region” No. CZ.02.1.01/0.0/0.0/17-049/0008414.
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Notes
- 1.
By the abuse of language we will on syntactic level call the formula \(x_{o\alpha }\) also a fuzzy set.
- 2.
This means that, if the degree of the fuzzy equality \(z\approx t\) in a model is equal to 1 then \(z=t\).
- 3.
Note that in the fuzzy set theory, such a fuzzy set is called extensional (w.r.t. \(\approx \)).
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Novák, V. (2022). Rough Sets and Topology in AST: A Study Via Higher-Order Fuzzy Logic. In: Harmati, I.Á., Kóczy, L.T., Medina, J., Ramírez-Poussa, E. (eds) Computational Intelligence and Mathematics for Tackling Complex Problems 3. Studies in Computational Intelligence, vol 959. Springer, Cham. https://doi.org/10.1007/978-3-030-74970-5_15
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