Abstract
Fuzzy sets and rough sets are well-known approaches to incomplete or imprecise data. In the paper we compare two formalizations of these sets within one of the largest repositories of computer-checked mathematical knowledge – the Mizar Mathematical Library. Although the motivation was quite similar in both developments, these approaches – proposed by us – vary significantly. Paradoxically, it appeared that fuzzy sets are much closer to the set theory implemented within the Mizar library, while in order to make more feasible view for rough sets we had to choose relational structures as a basic framework. The formal development, although counting approximately 15 thousand lines of source code, is by no means closed – it allows both for further generalizations, building on top of the existing knowledge, and even merging of these approaches. The paper is illustrated with selected examples of definitions, theorems, and proofs taken from rough and fuzzy set theory formulated in the Mizar language.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
Bryniarski, E.: Formal conception of rough sets. Fundamenta Informaticae 27(2/3), 109–136 (1996)
Dubois, D., Prade, H.: Operations on fuzzy numbers. International Journal of System Sciences 9(6), 613–626 (1978)
Dubois, D., Prade, H.: Rough fuzzy sets and fuzzy rough sets. International Journal of General Systems 17(2–3), 191–209 (1990)
Grabowski, A.: The formal construction of fuzzy numbers. Formalized Mathematics 22(4), 313–319 (2014)
Grabowski, A.: On the computer certification of fuzzy numbers. In: Ganzha, M., Maciaszek, L., Paprzycki, M. (eds.) Proceedings of Federated Conference on Computer Science and Information Systems, FedCSIS 2013, pp. 51–54 (2013)
Grabowski, A.: Automated discovery of properties of rough sets. Fundamenta Informaticae 128(1-2), 65–79 (2013)
Grabowski, A.: On the computer-assisted reasoning about rough sets. In: Dunin-Kęplicz, B., Jankowski, A., Szczuka, M. (eds.) Monitoring, Security and Rescue Techniques in Multiagent Systems. Advances in Soft Computing, vol. 28, pp. 215–226 (2005)
Grabowski, A., Jastrzębska, M.: Rough set theory from a math-assistant perspective. In: Kryszkiewicz, M., Peters, J.F., Rybiński, H., Skowron, A. (eds.) RSEISP 2007. LNCS (LNAI), vol. 4585, pp. 152–161. Springer, Heidelberg (2007)
Grabowski, A., Korniłowicz, A., Naumowicz, A.: Mizar in a nutshell. Journal of Formalized Reasoning 3(2), 153–245 (2010)
Kacprzak, M., Kosiński, W.: On lattice structure and implications on ordered fuzzy numbers. In: Proc. of EUSFLAT 2011, pp. 267–274 (2011)
Kawahara, Y., Furusawa, H.: An algebraic formalization of fuzzy relations. Fuzzy Sets and Systems 101, 125–135 (1999)
Klir, G.J.: Fuzzy arithmetic with requisite constraints. Fuzzy Sets and Systems 91, 165–175 (1997)
Koleśnik, R., Prokopowicz, P., Kosiński, W.: Fuzzy calculator – useful tool for programming with fuzzy algebra. In: Rutkowski, L., Siekmann, J.H., Tadeusiewicz, R., Zadeh, L.A. (eds.) ICAISC 2004. LNCS (LNAI), vol. 3070, pp. 320–325. Springer, Heidelberg (2004)
Korniłowicz, A.: On rewriting rules in Mizar. Journal of Automated Reasoning 50(2), 203–210 (2013)
Kosiński, W., Prokopowicz, P., Ślęzak, D.: Fuzzy numbers with algebraic operations: algorithmic approach. In: Kłopotek, M., Wierzchoń, S.T., Michalewicz, M. (eds.) Intelligent Information Systems, IIS 2002, Poland, pp. 311–320 (2002)
Kosiński, W., Prokopowicz, P., Ślęzak, D.: Ordered fuzzy numbers. Bulletin of the Polish Academy of Sciences, Sér. Sci. Math. 51(3), 327–338 (2003)
Mitsuishi, T., Endou, N., Shidama, Y.: The concept of fuzzy set and membership function and basic properties of fuzzy set operation. Formalized Mathematics 9(2), 351–356 (2001)
Moore, R., Lodwick, W.: Interval analysis and fuzzy set theory. Fuzzy Sets and Systems 135(1), 5–9 (2003)
Naumowicz, A., Korniłowicz, A.: A brief overview of Mizar. In: Berghofer, S., Nipkow, T., Urban, C., Wenzel, M. (eds.) TPHOLs 2009. LNCS, vol. 5674, pp. 67–72. Springer, Heidelberg (2009)
Pawlak, Z.: Rough Sets: Theoretical Aspects of Reasoning about Data. Kluwer, Dordrecht (1991)
Pąk, K.: Methods of lemma extraction in natural deduction proofs. Journal of Automated Reasoning 50(2), 217–228 (2013)
Urban, J., Sutcliffe, G.: Automated reasoning and presentation support for formalizing mathematics in Mizar. In: Autexier, S., Calmet, J., Delahaye, D., Ion, P.D.F., Rideau, L., Rioboo, R., Sexton, A.P. (eds.) AISC 2010. LNCS, vol. 6167, pp. 132–146. Springer, Heidelberg (2010)
Wiedijk, F.: Formal proof – getting started. Notices of the AMS 55(11), 1408–1414 (2008)
Zadeh, L.: Fuzzy sets. Information and Control 8(3), 338–353 (1965)
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2015 Springer International Publishing Switzerland
About this paper
Cite this paper
Grabowski, A., Mitsuishi, T. (2015). Initial Comparison of Formal Approaches to Fuzzy and Rough Sets. In: Rutkowski, L., Korytkowski, M., Scherer, R., Tadeusiewicz, R., Zadeh, L., Zurada, J. (eds) Artificial Intelligence and Soft Computing. ICAISC 2015. Lecture Notes in Computer Science(), vol 9119. Springer, Cham. https://doi.org/10.1007/978-3-319-19324-3_15
Download citation
DOI: https://doi.org/10.1007/978-3-319-19324-3_15
Publisher Name: Springer, Cham
Print ISBN: 978-3-319-19323-6
Online ISBN: 978-3-319-19324-3
eBook Packages: Computer ScienceComputer Science (R0)