From Leśniewski, Łukasiewicz, Tarski to Pawlak : Enriching Rough Set Based Data Analysis. A Retrospective Survey

• Autorzy: Polkowski L. T.

Identyfikatory

DOI

10.3233/FI-2017-1570

• Języki publikacji: EN

Abstrakty

EN

This work is dedicated to Profesor Andrzej Ehrenfeucht, the eleve of the Warsaw School of Logic and Mathematics on the occasion of His 85th Birthday. We propose to exploit certain of the milestone ideas created by this School and to apply them to data analysis in the framework of the rough set theory proposed by Professor Zdzisław Pawlak. To wit, we apply the idea of fractional truth states due to Jan Łukasiewicz, mereology created by Stanisław Leśniewski and the betweenness relation used by Alfred Tarski as one of primitive predicates in His axiomatization of Euclidean geometry. These ideas applied in problems of approximate reasoning permit us to formalize calculus of granules of knowledge and use it in preprocessing of data before applying a classification algorithm. Introduction of a mereological version of betweenness relation to data allows for partitioning of data into the kernel and the residuum, both sub-data sets providing a faithful representation of the whole data set and reducing the size of data without any essential loss of accuracy of classification. In the process of algorithmic construction of the partition of data into the kernel and the residuum, we exploit the Dual Indiscernibility Matrix which further allows us to introduce notions of a pair classifier and, more generally, k-classifier yet to be studied.

Słowa kluczowe

EN

truth values; mereology; betweenness relation; rough sets; shuffled simplices; dual indiscernibility matrix; kernel; residuum; pair classifier

• Wydawca: IOS Press
• Czasopismo: Fundamenta Informaticae
• Rocznik: 2017
• Tom: Vol. 154, nr 1/4
• Strony: 343--358
• Opis fizyczny: Bibliogr. 18 poz., tab.

Twórcy

autor

Polkowski L. T.

• Polish-Japanese Academy IT, Koszykowa str. 86, 02-008 Warszawa, Poland

Bibliografia

• [1] Artiemjew P, Nowak B, Polkowski, L. The Boosting and Bootstrap Ensembles for the Pair Classifier based on the Dual Indiscernibility Matrix. In: Wang G. et al. (eds.) Thriving Rough Sets. Studies in Computational Intelligence 708, SpringerIntl. Publishing, Cham, 2017 pp. 425–439. doi:10.1007/978-3-319-54966-8 20.
• [2] van Benthem J. The Logic of Time. Reidel, Dordrecht, 1983
• [3] Blumer A, Ehrenfeucht A, Haussler D, Warmuth MK. Learnability and the Vapnik-Chervonenkis dimension, J. ACM 1989;36(4):929–965. doi:10.1145/76359.76371.
• [4] Casati R, Varzi AC. Parts and Places: The Structure of Spatial Representations. MIT Press, Cambridge MA 1999. ISBN:9780262032667.
• [5] Hájek P. Metamathematics of Fuzzy Logic. Springer Netherlands, Dordrecht, 1998. doi:10.1007/978-94-011-5300-3.
• [6] Loemker L (ed.), Leibniz GW. Philosophical Papers and Letters, 2nd ed. D.Reidel, Dordrecht, 1969
• [7] Leśniewski S. Podstawy Ogólnej Teoryi Mnogości (Foundations of General Set Theory). Published by the Polish Scientific Circle in Moscow as no. 2 in Matematics and Nature Sciences Section. Popławski Printing Ed., Moscow, 1916; see also Leśniewski, S.: On the foundations of mathematics. Topoi 2, 1982 pp. 7–52.
• [8] Łukasiewicz J. Die logischen Grundlagen der Wahrscheinlichkeitsrechnung. Kraków: Spółka Wydawnicza Polska, 1913. Translation: Logical Foundations of Probability Theory, in Selected Works, ed. L. Borkowski. Amsterdam: North-Holland, 1970 pp. 16–63.
• [9] Pawlak Z. Rough sets. International Journal of Computer and Information Sciences 1982;11(5):341-356. doi:10.1007/BF01001956 Cite this article as: Pawlak, Z. International Journal of Computer and Information Sciences (1982) 11: 341. doi:10.1007/BF01001956.
• [10] Pawlak Z. Rough Sets: Theoretical Aspects of Reasoning about Data. Kluwer, Dordrecht, 1991. ISBN:0792314727, 9780792314721.
• [11] Polkowski L. Approximate Reasoning by Parts. An Introduction to Rough Mereology. ISRL 20, Springer Intl. Publishing, Cham, 2011. doi:10.1007/978-3-642-22279-5.
• [12] Polkowski L. Mereology in engineering and computer science. In Calosi, C., Graziani, P. (eds.): Mereology and the Sciences. Springer Synthese Library 371. Springer Intl. Publishing, Cham, 2014 pp. 217–292. doi:10.1007/978-3-319-05356-1 10.
• [13] Polkowski L. Betweenness, Łukasiewicz Rough Inclusions, Euclidean Representations in Information Systems, Hyper–granules, Conflict Resolution. In : Proceedings CS& P 2015, Rzeszow University, Rzeszow, Poland, 2015. URL http://ceur-ws.org/Vol-1492/.
• [14] Polkowski L, Artiemjew P. Granular Computing in Decision Approximation. An Application of Rough Mereology. ISRL 77, Springer Intl. Publishing, Cham, 2015. doi:10.1007/978-3-319-12880-1.
• [15] Polkowski L, Nowak B. Betweenness, Łukasiewicz Rough Inclusions, Euclidean Representations in Information Systems, Hyper–granules, Conflict Resolution. Fundamenta Informaticae 2016;147(2-3):337-352. doi:10.3233/FI-2016-1411.
• [16] Semeniuk-Polkowska M, Polkowski L. On the problem of boundaries from mereology and rough meeology points of view. Fundamenta Informaticae 2014;133(2-3):241–255. doi:10.3233/FI-2014-1074.
• [17] Tarski A, Givant S. The Tarski system of geometry. The Bulletin of Symbolic Logic 1999;5(2):175–214. doi:10.2307/421089.
• [18] UC Irvine Machine Learning Repository; URL http://archive.ics.uci.edu/ml/

Uwagi

Opracowanie ze środków MNiSW w ramach umowy 812/P-DUN/2016 na działalność upowszechniającą naukę (zadania 2017).