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2020 | OriginalPaper | Buchkapitel

Sequences of Refinements of Rough Sets: Logical and Algebraic Aspects

verfasst von : Stefania Boffa, Brunella Gerla

Erschienen in: Transactions on Rough Sets XXII

Verlag: Springer Berlin Heidelberg

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Abstract

In this thesis, a generalization of the classical Rough set theory [83] is developed considering the so-called sequences of orthopairs that we define in [20] as special sequences of rough sets.

Mainly, our aim is to introduce some operations between sequences of orthopairs, and to discover how to generate them starting from the operations concerning standard rough sets (defined in [32]). Also, we prove several representation theorems representing the class of finite centered Kleene algebras with the interpolation property [31], and some classes of finite residuated lattices (more precisely, we consider Nelson algebras [87], Nelson lattices [23], IUML-algebras [73] and Kleene lattice with implication [27]) as sequences of orthopairs.

Moreover, as an application, we show that a sequence of orthopairs can be used to represent an examiner’s opinion on a number of candidates applying for a job, and we show that opinions of two or more examiners can be combined using operations between sequences of orthopairs in order to get a final decision on each candidate.

Finally, we provide the original modal logic \(SO_n\) with semantics based on sequences of orthopairs, and we employ it to describe the knowledge of an agent that increases over time, as new information is provided. Modal logic \(SO_n\) is characterized by the sequences \((\square _1, \ldots , \square _n)\) and \((\bigcirc _1, \ldots , \bigcirc _n)\) of n modal operators corresponding to a sequence \((t_1, \ldots , t_n)\) of consecutive times. Furthermore, the operator \(\square _i\) of \((\square _1, \ldots , \square _n)\) represents the knowledge of an agent at time \(t_i\), and it coincides with the necessity modal operator of S5 logic [29]. On the other hand, the main innovative aspect of modal logic \(SO_n\) is the presence of the sequence \((\bigcirc _1, \ldots , \bigcirc _n)\), since \(\bigcirc _i\) establishes whether an agent is interested in knowing a given fact at time \(t_i\).

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Vorheriges Kapitel ||-ROSETTA

Nächstes Kapitel A Study of Algebras and Logics of Rough Sets Based on Classical and Generalized Approximation Spaces

The equivalence relations coming from the same region and coming from the same city are defined on proper subsets of U, for there are missing data for some users.

We exclude the operations \(\odot _4\) and \(\hookrightarrow _4\), since they can not be obtained starting from operations between the orthopairs.

By Preposition 15, \(\square _i \varphi \wedge \square _i \psi = \square _i (\varphi \wedge \psi )\).

By Preposition 15, \(\triangle _i \varphi \wedge \triangle _i \psi = \triangle _i(\varphi \vee \psi )\).

Observe that this expression is equivalent to \((\square _i \varphi \setminus \triangle _i \psi \ \wedge \ \square _i \psi \setminus \triangle _i \varphi )\).

Aguzzoli, S., Boffa, S., Ciucci, D., Gerla, B.: Refinements of orthopairs and IUML-algebras. In: Flores, V., et al. (eds.) IJCRS 2016. LNCS (LNAI), vol. 9920, pp. 87–96. Springer, Cham (2016). https://doi.org/10.1007/978-3-319-47160-0_8View Article
Aguzzoli, S., Boffa, S., Ciucci, D., Gerla, B.: Finite iuml-algebras, finite forests and orthopairs. Fundamenta Informaticae 163, 139–163 (2018)MathSciNetMATHView Article
Aguzzoli, S., Flaminio, T., Marchioni, E.: Finite forests. Their algebras and logics, pp. 139–163 (2018)
Ahn, B., Cho, S., Kim, C.: The integrated methodology of rough set theory and artificial neural network for business failure prediction. Expert Syst. Appl. 18(2), 65–74 (2000)View Article
Atanassov, K., Gargov, G.: Elements of intuitionistic fuzzy logic. Part i. Fuzzy Sets Syst. 95(1), 39–52 (1998)MATHView Article
Aumann, R.J.: Agreeing to disagree. Ann. Stat. 1236–1239 (1976)
Bacharach, M., Gerard-Varet, L.A., Mongin, P., Shin, H.S.: Epistemic Logic and the Theory of Games and Decisions, vol. 20. Springer, Heidelberg (2012). https://doi.org/10.1007/978-1-4613-1139-3View ArticleMATH
Banerjee, M., Chakraborty, M.K.: Rough consequence and rough algebra. In: Ziarko, W.P. (ed.) Rough Sets, Fuzzy Sets and Knowledge Discovery. Workshops in Computin, pp. 196–207. Springer, Heidelberg (1994). https://doi.org/10.1007/978-1-4471-3238-7_24View Article
Banerjee, M., Chakraborty, M.K.: Algebras from rough sets (2004)
Belohlavek, R.: What is a fuzzy concept lattice? II. In: Kuznetsov, S.O., Ślęzak, D., Hepting, D.H., Mirkin, B.G. (eds.) RSFDGrC 2011. LNCS (LNAI), vol. 6743, pp. 19–26. Springer, Heidelberg (2011). https://doi.org/10.1007/978-3-642-21881-1_4View Article
Bezhanishvili, N., Gehrke, M.: Free Heyting algebras: revisited. In: Kurz, A., Lenisa, M., Tarlecki, A. (eds.) CALCO 2009. LNCS, vol. 5728, pp. 251–266. Springer, Heidelberg (2009). https://doi.org/10.1007/978-3-642-03741-2_18View Article
Bialynicki-Birula, A., Rasiowa, H.: On the representation of quasi-Boolean algebras (1957)
Bianchi, M.: A temporal semantics for nilpotent minimum logic. Int. J. Approx. Reason. 55(1), 391–401 (2014)MathSciNetMATHView Article
Birkhoff, G.: Lattice Theory, vol. 25. American Mathematical Society (1940)
Birkhoff, G., et al.: Rings of sets. Duke Math. J. 3(3), 443–454 (1937)MathSciNetMATHView Article
Blackburn, P., De Rijke, M., Venema, Y.: Modal Logic: Graph. Darst, vol. 53. Cambridge University Press, Cambridge (2002)MATH
Boffa, S., De Maio, C., Di Nola, A., Fenza, G., Ferraioli, A.R., Loia, V.: Unifying fuzzy concept lattice construction methods. In: 2016 IEEE International Conference on Fuzzy Systems (FUZZ-IEEE), pp. 209–216. IEEE (2016)
Boffa, S., De Maio, C., Gerla, B., Parente, M.: Context-aware advertisment recommendation on twitter through rough sets. In: 2018 IEEE International Conference on Fuzzy Systems (FUZZ-IEEE), pp. 1–8. IEEE (2018)
Boffa, S., Gerla, B.: Sequences of orthopairs given by refinements of coverings. In: Petrosino, A., Loia, V., Pedrycz, W. (eds.) WILF 2016. LNCS (LNAI), vol. 10147, pp. 95–105. Springer, Cham (2017). https://doi.org/10.1007/978-3-319-52962-2_8View Article
Boffa, S., Gerla, B.: Kleene algebras as sequences of orthopairs. In: Kacprzyk, J., Szmidt, E., Zadrożny, S., Atanassov, K.T., Krawczak, M. (eds.) IWIFSGN/EUSFLAT -2017. AISC, vol. 641, pp. 235–248. Springer, Cham (2018). https://doi.org/10.1007/978-3-319-66830-7_22View Article
Boffa, S., Gerla, B.: How to merge opinions by using operations between sequences of orthopairs. In: 2019 IEEE International Conference on Fuzzy Systems (FUZZ-IEEE). IEEE (2019)
Brignole, D., Monteiro, A.: Caracterisation des algèbres de nelson par des egalités. I. Proce. Japan Acad. 43(4), 279–283 (1967)MATHView Article
Busaniche, M., Cignoli, R.: Constructive logic with strong negation as a substructural logic. J. Logic Comput. 20(4), 761–793 (2008)MathSciNetMATHView Article
Calegari, S., Ciucci, D.: Granular computing applied to ontologies. Int. J. Approx. Reason. 51(4), 391–409 (2010)MATHView Article
Campagner, A., Ciucci, D.: Three-way and semi-supervised decision tree learning based on orthopartitions. In: Medina, J., et al. (eds.) IPMU 2018. CCIS, vol. 854, pp. 748–759. Springer, Cham (2018). https://doi.org/10.1007/978-3-319-91476-3_61View Article
Carpineto, C., Romano, G.: Concept Data Analysis: Theory and Applications. Wiley, Hoboken (2004)MATHView Article
Castiglioni, J.L., Celani, S.A., San Martín, H.J.: Kleene algebras with implication. Algebra Universalis 77(4), 375–393 (2017). https://doi.org/10.1007/s00012-017-0433-4MathSciNetView ArticleMATH
Celani, S.: Bounded distributive lattices with fusion and implication. Southeast Asian Bull. Math. 28(6) (2004)
Chellas, B.F.: Modal Logic: An Introduction. Cambridge University Press, Cambridge (1980)MATHView Article
Cignoli, R.: Injective De Morgan and Kleene algebras. In: Proceedings of the American Mathematical Society, pp. 269–278 (1975)
Cignoli, R.: The class of Kleene algebras satisfying an interpolation property and nelson algebras. Algebra Universalis 23(3), 262–292 (1986). https://doi.org/10.1007/BF01230621MathSciNetView ArticleMATH
Ciucci, D.: Orthopairs: a simple and widely used way to model uncertainty. Fundamenta Informaticae 108(3–4), 287–304 (2011)MathSciNetMATHView Article
Ciucci, D.: Orthopairs and granular computing. Granul. Comput. 1(3), 159–170 (2016). https://doi.org/10.1007/s41066-015-0013-yView Article
Ciucci, D., Dubois, D.: Three-valued logics, uncertainty management and rough sets. In: Peters, J.F., Skowron, A. (eds.) Transactions on Rough Sets XVII. LNCS, vol. 8375, pp. 1–32. Springer, Heidelberg (2014). https://doi.org/10.1007/978-3-642-54756-0_1View ArticleMATH
Ciucci, D., Mihálydeák, T., Csajbók, Z.E.: On definability and approximations in partial approximation spaces. In: Miao, D., Pedrycz, W., Ślȩzak, D., Peters, G., Hu, Q., Wang, R. (eds.) RSKT 2014. LNCS (LNAI), vol. 8818, pp. 15–26. Springer, Cham (2014). https://doi.org/10.1007/978-3-319-11740-9_2View ArticleMATH
Ciucci, D., Yao, Y.: Advances in rough set theory (2011)
Crawford, J.M., Etherington, D.W.: A non-deterministic semantics for tractable inference. In: AAAI/IAAI, pp. 286–291 (1998)
Cresswell, M.J., Hughes, G.E.: A New Introduction to Modal Logic. Routledge, Abingdon (2012)MATHView Article
Csajbók, Z.: Approximation of sets based on partial covering. Theoret. Comput. Sci. 412(42), 5820–5833 (2011)MathSciNetMATHView Article
Dai, J.H.: Rough 3-valued algebras. Inf. Sci. 178(8), 1986–1996 (2008)MathSciNetMATHView Article
De Maio, C., Boffa, S.: Discovery of interesting users in twitter by using rough sets. In: Park, J.J., Loia, V., Choo, K.-K.R., Yi, G. (eds.) MUE/FutureTech -2018. LNEE, vol. 518, pp. 671–677. Springer, Singapore (2019). https://doi.org/10.1007/978-981-13-1328-8_86View Article
Ditmarsch, H., Halpern, J.Y., van der Hoek, W., Kooi, B.P.: Handbook of Epistemic Logic. College Publications, London (2015)MATH
Dubois, D., Prade, H.: Rough fuzzy sets and fuzzy rough sets. Int. J. Gener. Syst. 17(2–3), 191–209 (1990)MATHView Article
Dubois, D., Prade, H., Schockaert, S.: Generalized possibilistic logic: foundations and applications to qualitative reasoning about uncertainty. Artif. Intell. 252, 139–174 (2017)MathSciNetMATHView Article
Dwinger, P., Balbes, R.: Distributive lattices (1974)
Fagin, R., Halpern, J.Y., Moses, Y., Vardi, M.: Reasoning About Knowledge. MIT Press, Cambridge (2004)MATHView Article
Fidel, M.M.: An algebraic study of a propositional system of Nelson. Bull. Sect. Logic 7(2), 89 (1978)MathSciNetMATH
Friedl, M.A., Brodley, C.E.: Decision tree classification of land cover from remotely sensed data. Remote Sens. Environ. 61(3), 399–409 (1997)View Article
Fussner, W., Galatos, N.: Categories of models of r-mingle. arXiv preprint arXiv:1710.04256 (2017)
Ganter, B., Wille, R.: Formal Concept Analysis: Mathematical Foundations. Springer, Heidelberg (2012). https://doi.org/10.1007/978-3-642-59830-2View ArticleMATH
Garson, J.: Modal logic. In: Zalta, E.N. (ed.) The Stanford Encyclopedia of Philosophy. Metaphysics Research Lab, Stanford University, fall 2018 edn. (2018)
Golan, R.H., Ziarko, W.: A methodology for stock market analysis utilizing rough set theory. In: Proceedings of the IEEE/IAFE Computational Intelligence for Financial Engineering, pp. 32–40. IEEE (1995)
Grätzer, G., Wehrung, F.: Lattice Theory: Special Topics and Applications. Springer, Heidelberg (2016). https://doi.org/10.1007/978-3-319-44236-5View ArticleMATH
Grzymala-Busse, J.W.: LERS-A system for learning from examples based on rough sets. In: Słowiński, R. (ed.) Intelligent Decision Support. Series D: System Theory, Knowledge Engineering and Problem Solving, pp. 3–18. Springer, Heidelberg (1992). https://doi.org/10.1007/978-94-015-7975-9_1View Article
Haack, S.: Philosophy of Logics. Cambridge University Press, Cambridge (1978)MATHView Article
Halpern, J.Y., Moses, Y.: Knowledge and common knowledge in a distributed environment. J. ACM (JACM) 37, 549–587 (1990)MathSciNetMATHView Article
Hendricks, V.: New waves in epistemology (2008)
Hintikka, J.: Knowledge and Belief. An Introduction to the Logic of the Two Notions (1965)
Holliday, W.H.: Epistemic logic and epistemology. In: Hansson, S.O., Hendricks, V.F. (eds.) Introduction to Formal Philosophy. SUTP, pp. 351–369. Springer, Cham (2018). https://doi.org/10.1007/978-3-319-77434-3_17View Article
Järvinen, J.: Knowledge representation and rough sets. Turku Centre for Computer Science (1999)
Järvinen, J., Radeleczki, S.: Rough sets determined by tolerances. Int. J. Approx. Reason. 55(6), 1419–1438 (2014)MathSciNetMATHView Article
Järvinen, J., Radeleczki, S.: Representing regular pseudo complemented Kleene algebras by tolerance-based rough sets. J. Aust. Math. Soc. 105(1), 57–78 (2018)MathSciNetMATHView Article
Kalman, J.A.: Lattices with involution. Trans. Am. Math. Soc. 87(2), 485–491 (1958)MathSciNetMATHView Article
Krajicek, J., Krajíček, J., et al.: Bounded Arithmetic, Propositional Logic and Complexity Theory. Cambridge University Press, Cambridge (1995)MATHView Article
Krysiński, J.: Rough sets approach to the analysis of the structure-activity relationship of quaternary imidazolium compounds. Arzneimittelforschung 40(7), 795–799 (1990)
Kryszkiewicz, M.: Rules in incomplete information systems. Inf. Sci. 113(3–4), 271–292 (1999)MathSciNetMATHView Article
Ladner, R.E.: The computational complexity of provability in systems of modal propositional logic. SIAM J. Comput. 6(3), 467–480 (1977)MathSciNetMATHView Article
Lewis, C.I., Langford, C.H., Lamprecht, P.: Symbolic Logic. Dover Publications, New York (1959)
Liau, C.-J.: Modal reasoning and rough set theory. In: Giunchiglia, F. (ed.) AIMSA 1998. LNCS, vol. 1480, pp. 317–330. Springer, Heidelberg (1998). https://doi.org/10.1007/BFb0057455View Article
Lin, G., Liang, J., Qian, Y.: Multigranulation rough sets: from partition to covering. Inf. Sci. 241, 101–118 (2013)MathSciNetMATHView Article
Lin, T.Y., Liu, Q.: Rough approximate operators: axiomatic rough set theory. In: Ziarko, W.P. (ed.) Rough Sets, Fuzzy Sets and Knowledge Discovery. Workshops in Computing, pp. 256–260. Springer, Heidelberg (1994). https://doi.org/10.1007/978-1-4471-3238-7_31View Article
Liu, Y., Xu, J., Sun, L., Du, L.: Decisions tree learning method based on three-way decisions. In: Yao, Y., Hu, Q., Yu, H., Grzymala-Busse, J.W. (eds.) RSFDGrC 2015. LNCS (LNAI), vol. 9437, pp. 389–400. Springer, Cham (2015). https://doi.org/10.1007/978-3-319-25783-9_35View Article
Metcalfe, G., Montagna, F.: Substructural fuzzy logics. J. Symb. Logic 72(3), 834–864 (2007)MathSciNetMATHView Article
Moisil, G.C.: Recherches sur l’algèbre de la logique. Ann. Sci. Univ. Jassy 22, 1–117 (1935)
Monteiro, A.: Matrices de morgan caractéristiques pour le calcul propositionnel classique (1963)
Monteiro, A.: Construction des algébres de Nelson finies. Univ. (1964)
Mrozek, A.: Rough sets and some aspects of expert systems realigation. The 7th International Workshop on Expert Systems and Application (1987)
Nation, J.B.: Notes on lattice theory. In: Cambridge Studies in Advanced Mathematics, vol. 60 (1998)
Orlowska, E.: A logic of indiscernibility relations. In: Skowron, A. (ed.) SCT 1984. LNCS, vol. 208, pp. 177–186. Springer, Heidelberg (1985). https://doi.org/10.1007/3-540-16066-3_17View Article
Orlowska, E.: Logical aspects of learning concepts. Int. J. Approx. Reason. 2(4), 349–364 (1988)MATHView Article
Pagliani, P.: Rough set theory and logic-algebraic structures. In: Orlowska, E. (ed.) Incomplete information: Rough set analysis. Studies in Fuzziness and Soft Computing, vol. 13, pp. 109–190. Springer, Heidelberg (1998). https://doi.org/10.1007/978-3-7908-1888-8_6View Article
Pal, S.K., Skowron, A.: Rough-Fuzzy Hybridization: A New Trend in Decision Making. Springer, New York (1999)MATH
Pawlak, Z.: Rough sets. Int. J. Comput. Inf. Sci. 11(5), 341–356 (1982). https://doi.org/10.1007/BF01001956View ArticleMATH
Pawlak, Z.: Imprecise categories, approximations and rough sets. In: Pawlak, Z. (ed.) Rough sets. Theory and Decision Library. (Series D: System Theory, Knowledge Engineering and Problem Solving), pp. 9–32. Springer, Heidelberg (1991). https://doi.org/10.1007/978-94-011-3534-4_2View Article
Polkowski, L.: Rough Sets in Knowledge Discovery 2: Applications, Case Studies and Software Systems, vol. 19. Physica (2013)
Priestley, H.: Ordered sets and duality for distributive lattices. In: North-Holland Mathematics Studies, vol. 99, pp. 39–60. Elsevier (1984)
Rasiowa, H.: N-lattices and constructive logic with strong negation (1969)
Rasiowa, H.: An algebraic approach to non-classical logics, vol. 78. North-Holland, Amsterdam (1974)
Rasiowa, H.: Rough concepts and multiple valued logic. In: Proceedings of the 16th ISMVL, vol. 86, pp. 282–288 (1986)
Reeves, S., Clarke, M.: Logic for Computer Science. International Computer Science Series (1990)
Shi, K.: S-rough sets and its applications in diagnosis-recognition for disease. In: 2002 International Conference on Machine Learning and Cybernetics, Proceedings, vol. 1, pp. 50–54. IEEE (2002)
Skowron, A., Stepaniuk, J.: Tolerance approximation spaces. Fundamenta Informaticae 27(2,3), 245–253 (1996)
Slowinski, R., Vanderpooten, D.: A generalized definition of rough approximations based on similarity. IEEE Trans. Knowl. Data Eng. 12(2), 331–336 (2000)View Article
Slowinski, R., Zopounidis, C.: Application of the rough set approach to evaluation of bankruptcy risk. Intell. Syst. Account. Finan. Manag. 4(1), 27–41 (1995)View Article
Snyder, D.P.: Modal logic and its applications (1971)
Sobociński, B.: Axiomatization of a partial system of three-value calculus of propositions. Institute of Applied Logic (1952)
Spinks, M., Veroff, R.: Constructive logic with strong negation is a substructural logic. I. Studia Logica 88(3), 325–348 (2008). https://doi.org/10.1007/s11225-008-9113-xMathSciNetView ArticleMATH
Tripathy, B., Acharjya, D., Cynthya, V.: A framework for intelligent medical diagnosis using rough set with formal concept analysis. arXiv preprint arXiv:1301.6011 (2013)
Vakarelov, D.: Notes on n-lattices and constructive logic with strong negation. Stud. Logica 36(1–2), 109–125 (1977). https://doi.org/10.1007/BF02121118MathSciNetView ArticleMATH
Ward, M., Dilworth, R.P.: Residuated lattices. Trans. Am. Math. Soc. 45(3), 335–354 (1939)MathSciNetMATHView Article
Whitesitt, J.E.: Boolean Algebra and its Applications. Courier Corporation (1995)
Yahia, S.B., Jaoua, A.: Discovering knowledge from fuzzy concept lattice. In: Kandel, A., Last, M., Bunke, H. (eds.) Data Mining and Computational Intelligence. Studies in Fuzziness and Soft Computing, pp. 167–190. Springer, Heidelberg (2001). https://doi.org/10.1007/978-3-7908-1825-3_7View Article
Yao, Y.: Three-way decision: an interpretation of rules in rough set theory. In: Wen, P., Li, Y., Polkowski, L., Yao, Y., Tsumoto, S., Wang, G. (eds.) RSKT 2009. LNCS (LNAI), vol. 5589, pp. 642–649. Springer, Heidelberg (2009). https://doi.org/10.1007/978-3-642-02962-2_81View Article
Yao, Y.: Rough sets and three-way decisions. In: Ciucci, D., Wang, G., Mitra, S., Wu, W.-Z. (eds.) RSKT 2015. LNCS (LNAI), vol. 9436, pp. 62–73. Springer, Cham (2015). https://doi.org/10.1007/978-3-319-25754-9_6View Article
Yao, Y.: Two views of the theory of rough sets in finite universes. Int. J. Approx. Reason. 15(4), 291–317 (1996)MathSciNetMATHView Article
Yao, Y., Lin, T.Y.: Generalization of rough sets using modal logics. Intell. Autom. Soft Comput. 2(2), 103–119 (1996)View Article
Zhu, W.: Generalized rough sets based on relations. Inf. Sci. 177(22), 4997–5011 (2007)MathSciNetMATHView Article
Zhu, W., Wang, F.Y.: Reduction and axiomization of covering generalized rough sets. Inf. Sci. 152, 217–230 (2003)MathSciNetMATHView Article
Ziarko, W.: Variable precision rough set model. J. Comput. Syst. Sci. 46(1), 39–59 (1993)MathSciNetMATHView Article

Titel: Sequences of Refinements of Rough Sets: Logical and Algebraic Aspects
verfasst von: Stefania Boffa
Brunella Gerla
Verlag: Springer Berlin Heidelberg
Buch: Transactions on Rough Sets XXII
Print ISBN: 978-3-662-62797-6

Electronic ISBN: 978-3-662-62798-3

Copyright-Jahr: 2020
DOI: https://doi.org/10.1007/978-3-662-62798-3_3

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