On Rough Set Logics Based on Similarity Relations

• Autorzy: Polkowski L. , Semeniuk-Polkowska M.
• Języki publikacji: EN

Abstrakty

EN

In this paper, dedicated to Professor Solomon Marcus on the occasion of His 80th birthday, we discuss the idea of intensional many-valued logic reflecting the logical content of rough set approach to analysis and treatment of uncertainty. In constructing the variety of logics presented in the paper, we make use of a certain kind of tolerance (similarity) relations called rough mereological tolerances. A study of tolerance relations that arise in rough set environments was initiated in 1994, with the paper [23], in which basic ideas pertaining to tolerance relations in the rough set framework were pointed to. The analysis of the role tolerance relations may play in machine learning based on rough set-theoretic ideas was carried out by Professor Solomon Marcus in His seminal paper, written during His stay in Warsaw in December of the year 1994. At the same time the first author had first ideas related to the applicability of ideas of mereology in the rough set analysis of uncertainty. In a later analysis it has turned out that mereological approach has led to a development of a new paradigm in reasoning under uncertainty, called rough mereology, proposed by Lech Polkowski and Andrzej Skowron. Within this paradigm, one is able to construct a variety of tolerance relations. Those tolerance relations, induced by rough mereological constructs called rough inclusions, serve as a basis for constructing a variety of logics, called rough mereological logics, that are related to the inherent structure of any rough set universe. In this paper, we introduce gradually all essential and necessary notions from the area of rough set theory, mereology and rough mereology, and then we discuss tolerance relations induced by rough inclusions along with some methods for inducing rough inclusions with desired properties. The paper culminates with a discussion of intensional logics based on rough mereological tolerance relations. In this way, we explore one of so many paths in scientific research, that have been either pointed to or threaded by Professor Solomon Marcus.

Słowa kluczowe

EN

rough mereology; logics for rough sets; similarity relation

• Wydawca: IOS Press
• Czasopismo: Fundamenta Informaticae
• Rocznik: 2005
• Tom: Vol. 64, nr 1-4
• Strony: 379--390
• Opis fizyczny: Bibliogr. 32 poz.

Twórcy

autor

Polkowski L.

• Polish–Japanese Institute of Information Technology Koszykowa 86, 02-008 Warszawa, Poland
• Department of Mathematics and Computer Science University Warmia and Mazury, Żołnierska 14a, 10-560 Olsztyn, Poland

autor

Semeniuk-Polkowska M.

• Chair of Formal Linguistics, Warsaw University, Browarna 8/10, 00-901 Warszawa, Poland

Bibliografia

• [1] van Benthem, J.: A Manual of Intensional Logic. CSLI Stanford University, 1988.
• [2] Čech, E.: Topologicke prostory. In: Eduard __ech. Topological Spaces (Frolik, Z., Ed.), Akademia, Praha, 1966.
• [3] Dubois, D., Grzymala-Busse, J., Inuiguchi, M., Polkowski, L. (eds.):Rough Sets and Fuzzy Sets, LNCS vol. 3135, Springer Verlag, Berlin, 2004.
• [4] Dubois, D., Prade, H.: Necessity measures and the resolution principle. IEEE Trans. Systems, Man, Cybernetics 17, 1997, pp. 111–127.
• [5] Frege, G.:Grundgesetze der Arithmetik II.Verlag Hermann Pohle, Jena, 1903.
• [6] De Laguna, T.:Point, line, surface as sets of solids. J. Philosophy 19, 1922, pp. 449–461.
• [7] Leśniewski, S.: On the foundations of mathematics. Topoi 2, 1982, pp. 7–52.
• [8] Leśniewski, S.:Grundz __ge eines neuen Systems der Grundlagen der Mathematik.Fundamenta Mathematicae 24, 1929, pp. 242–251.
• [9] Ling, C.-H.:Representation of associative functions. Publ.Math. Debrecen 12, 1965, pp. 189–212.
• [10] Łukasiewicz, J.: Jan Łukasiewicz. Selected Works (Borkowski, L., Ed.), North Holland and Polish Scientific Publishers, Amsterdam and Warsaw, 1970.
• [11] Łukasiewicz, J.:Die Logischen Grundlagen der Wahrscheinlichtkeitsrechnung, Kraków, 1913.
• [12] Marcus, S.:Tolerance rough sets, __ech topologies, learning processes. Bull. Polish Acad. Sci. Tech. 42, 1994, 471–487.
• [13] Menger, K.: Statistical metrics.Proc. Natl. Acad.Sci. USA 28, 1942, 535–537.
• [14] Montague, R.: Philosophical Writings of R. Montague. Thomason, R., Ed. Yale Univ. Press, New Haven Conn., 1972.
• [15] Orłowska, E.: Modal logics in the theory of information systems, Z.Math. Logik Grund. Math. 35, 1989, pp. 559–572.
• [16] Pawlak, Z.: Rough Sets: Theoretical Aspects of Reasoning about Data.Kluwer, 1991.
• [17] Pawlak, Z.:Rough sets. Intern. J. Comp. Inf. Sci. 11, 1982, pp. 341–356.
• [18] Poincaré_, H.: Science et Hypothèse.Paris, 1905.
• [19] Polkowski, L.: A note on 3–valued rough set logic with applications to decision rules. Fundamenta Informaticae 61:1, 2004, pp. 37–45.
• [20] Polkowski, L.: Rough mereology: A rough set paradigm for unifying rough set theory and fuzzy set theory. Fundamenta Informaticae 54: 1, 2003, pp. 67–88.
• [21] Polkowski, L.:.Rough Sets. Mathematical Foundations. Physica–Verlag, Heidelberg, 2002.
• [22] Polkowski, L., Skowron, A.: Rough mereology: a new paradigm for approximate reasoning. International Journal of Approximate Reasoning 15: 4, 1997, pp. 333–365.
• [23] Polkowski, L., Skowron, A., Żytkow, J.: Rough foundations for rough sets, in: Proceedings RSSC’94. The 3rd Workshop on Rough Sets and Soft Computing. San Jose, CA, 1994, pp. 142-149.
• [24] Rosser, J.B., Turquette, A. R.: Many–Valued Logics. North Holland, Amsterdam, 1958.
• [25] Tarski, A.: Les fondements de la géomé_trie des corps, In: Supplement to Ann. Soc. Polon. Math., Cracow, 1929, pp. 29–33.
• [26] Vakarelov, D.: Information systems, similarity relations and modal logics, in: Orłowska, E., Ed. Incomplete Information: Rough Set Analysis. Physica–Verlag, Heidelberg, 1998, pp. 492–550.
• [27] Zadeh, L.A.:Toward a theory of fuzzy information granulation and its certainty in human reasoning and fuzzy logic.Fuzzy Sets Syst.90, 2001, pp. 111–127.
• [28] Zadeh, L.A.:Fuzzy sets as a basis for a theory of possibility.Fuzzy Sets Syst.1(1), 1978, 3–28.
• [29] Zadeh, L.A.: Similarity relations and fuzzy orderings. Information Sciences 3, 1971, pp. 177-200.
• [30] L. A. Zadeh, L.A.:Fuzzy sets.Information and Control 8, 1965, pp. 338–353.
• [31] Zadeh, L.A., Kacprzyk, J.(eds.): (Computing with Words in Information/Intelligent Systems vol. 1. Physica, 1999.
• [32] Zeeman, E. C.: The topology of the brain and the visual perception. In: Fort, K. M. (ed.), Topology of 3–manifolds and Selected Topics.Prentice Hall, Englewood Cliffs, 1965, pp. 240–256