Abstract
Technically put, a metaphor is a conceptual mapping between two domains, which allows one to better understand the target domain; as Lakoff and Núñes put it, the main function of a metaphor is to allow us to reason about relatively abstract domains using the inferential structure of relatively concrete domains. In the paper we would like to apply this idea of framing one domain through conceptual settings of another domain to rough set theory (RST). The main goal is to construe rough sets in terms of the following mathematical metaphor: RST is a modular set-arithmetic. That is, we would like to map/project modular arithmetic onto rough sets, and, as a consequence, to redefine the fundamental concepts/objects of RST. Specifically, we introduce new topological operators (which play a similar role as remainders in modular arithmetic), discuss their formal properties, and finally apply them to the problem of vagueness (which has been intertwined with RST since the 1980’s).
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Notes
- 1.
A very interesting discussion of these problems may be found in Krajewski [5].
- 2.
Although Chakraborty’s question makes perfect sense for abstract approximation spaces, the case of decision tables is a bit different: here the set X represents a decision attribute, which – although well known – still needs to be approximated by means of conditional attributes.
- 3.
\(X\uplus Y = (X\setminus Y) \cup (Y\setminus X)\).
- 4.
The modular representation is not – however – equivalent to a rough set, e.g., if \(\underline{X} = \emptyset \), then \((\underline{X},\overline{X})\) usually represents/approximates more than a single set. However, the modular representation is \((X,\mathbf d (X))\), which stands for X alone.
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We are greatly indebted to anonymous referees for their valuable comments and corrections.
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Wolski, M., Gomolińska, A. (2018). A Metaphor for Rough Set Theory: Modular Arithmetic. In: Nguyen, H., Ha, QT., Li, T., Przybyła-Kasperek, M. (eds) Rough Sets. IJCRS 2018. Lecture Notes in Computer Science(), vol 11103. Springer, Cham. https://doi.org/10.1007/978-3-319-99368-3_9
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