JACIII Vol.24 No.6 pp. 774-784
doi: 10.20965/jaciii.2020.p0774


Bilattice Logic for Rough Sets

Yotaro Nakayama*, Seiki Akama**, and Tetsuya Murai***

*Nihon Unisys, Ltd.
1-1-1 Toyosu, Koto-ku, Tokyo 135-8560, Japan

**C-Republic, Inc.
1-20-1 Higashi-Yurigaoka, Asao-ku, Kawasaki-shi, Kanagawa 215-0012, Japan

***Chitose Institute of Science and Technology
758-65 Bibi, Chitose, Hokkaido 066-8655, Japan

May 11, 2019
September 29, 2020
November 20, 2020
bilattice, decision logic, four-valued logic, tableau calculi, variable precision rough set

Rough set theory is studied to manage uncertain and inconsistent information. Because Pawlak’s decision logic for rough sets is based on the classical two-valued logic, it is inconvenient for handling inconsistent information. We propose a bilattice logic as the deduction basis for the decision logic of rough sets to address inconsistent and ambiguous information. To enhance the decision logic to bilattice semantics, we introduce Variable Precision Rough Set (VPRS). As a deductive basis for bilattice decision logic, we define a consequence relation for Belnap’s four-valued semantics and provide a bilattice semantic tableau TB4 for a deduction system. We demonstrate the soundness and completeness of TB4 and enhance it with weak negation.

Cite this article as:
Y. Nakayama, S. Akama, and T. Murai, “Bilattice Logic for Rough Sets,” J. Adv. Comput. Intell. Intell. Inform., Vol.24 No.6, pp. 774-784, 2020.
Data files:
  1. [1] Z. Pawlak, “Rough Sets: Theoretical Aspects of Reasoning about Data,” Kluwer Academic Publishers, 1991.
  2. [2] W. Ziarko, “Variable precision rough set model,” J. of Computer and System Science, Vol.46, No.1, pp. 39-59, 1993.
  3. [3] R. M. Smullyan, “First-Order Logic,” Dover Books, 1995.
  4. [4] R. Hähnle, “Automated deduction in multiple-valued logics,” Oxford University Press, 1995.
  5. [5] M. D’Agostino, “Investigations into the Complexity of some Propositional calculi,” Oxford University Computing Laboratory, Programming Research Group, 1990.
  6. [6] T.-F. Fan, W.-C. Hu, and C.-J. Liau, “Decision logics for knowledge representation in data mining,” 25th Annual Int. Computer Software and Applications Conf. (COMPSAC 2011), pp. 626-631, 2001.
  7. [7] Y. Lin and L. Qing, “A Logical Method of Formalization for Granular Computing,” IEEE Int. Conf. on Granular Computing (GRC 2007), p. 22, 2007.
  8. [8] D. Ciucci and D. Dubois, “Three-Valued Logics, Uncertainty Management and Rough Sets,” Trans. on Rough Sets XVII, Lecture Notes in Computer Science, Vol.8375, pp. 1-32, 2014.
  9. [9] A. Avron and B. Konikowska, “Rough Sets and 3-Valued Logics,” Studia Logica, Vol.90, pp. 69-92, 2008.
  10. [10] B. Konikowska, “Three-Valued Logic for Reasoning about Covering-Based Rough Sets,” Rough Sets and Intelligent Systems, Intelligent Systems Reference Library, Vol.42, pp. 439-461, 1990.
  11. [11] Y. Nakayama, S. Akama, and T. Murai, “Deduction System for Decision Logic based on Partial Semantics,” The 11th Int. Conf. on Advances in Semantic Processing (SEMAPRO 2017), pp. 8-11, 2017.
  12. [12] A. Vitória, A. S. Andrzej, and J. Małuszynski, “Four-Valued Extension of Rough Sets,” Int. Conf. on Rough Sets and Knowledge Technology (RSKT 2008), pp. 106-114, 2008.
  13. [13] S. Akama, T. Murai, and Y. Kudo, “Reasoning with Rough Sets: Logical Approaches to Granularity-Based Framework,” Springer International Publishing, 2018.
  14. [14] N. D. Belnap, Jr., “A Useful Four-Valued Logic,” J. M. Dunn and G. Epstein (Eds.), “Modern Uses of Multiple-Valued Logic,” pp. 5-37, Reidel Publishing, 1977.
  15. [15] M. L. Ginsberg, “Multivalued logics: A uniform approach to reasoning in artificial intelligence,” Computer Intelligence, Vol.4, No.3, pp. 265-316, 1988.
  16. [16] O. Arieli and A. Avron, “Reasoning with logical bilattices,” J. of Logic, Language and Information, Vol.5, pp. 25-63, 1996.
  17. [17] R. Muskens, “On Partial and Paraconsistent Logics,” Notre Dame J. of Formal Logic, Vol.40, No.3, pp. 352-374, 1999.
  18. [18] S. Wintein and R. Muskens, “A calculus for Belnap’s logic in which each proof consists of two trees,” Logique et Analyse, Vol.55, No.220, pp. 643-656, 2012.
  19. [19] Z. Pawlak, “Rough Sets and Decision Algorithms,” 2nd Int. Conf. on Rough Sets and Current Trends in Computing (RSCTC 2000), pp. 30-45, 2000.
  20. [20] A. R. Anderson and J. N. D. Belnap, Jr., “Tautological Entailments,” Philosophical Studies: An Int. J. for Philosophy in the Analytic Tradition, Vol.13, No.1/2, pp. 9-24, 1962.
  21. [21] W. Ziarko, “Probabilistic Decision Tables in the Variable Precision Rough Set Model,” Computational Intelligence, Vol.17, No.3, pp. 593-603, 2001.
  22. [22] J. Van Benthem, “Partiality and Nonmonotonicity in Classical Logic,” Logique et Analyse, Vol.29, No.114, pp. 225-247, 1986.
  23. [23] S. Akama and Y. Nakayama, “Consequence relations in DRT,” Proc. of the 15th Int. Conf. on Computational Linguistics (COLING 1994), Volume 2, pp. 1114-1117, 1994.
  24. [24] Y. Nakayama, S. Akama, and T. Murai, “Deduction System for Decision Logic Based on Many-valued Logics,” Int. J. on Advances in Intelligent Systems, Vol.11, No.1/2, pp. 115-126, 2018.
  25. [25] Y. Nakayama, S. Akama, and T. Murai, “Four-valued Tableau Calculi for Decision Logic of Rough Set,” Knowledge-Based and Intelligent Information and Engineering Systems: Proc. of the 22nd Int. Conf. (KES-2018), pp. 383-392, 2018.
  26. [26] A. Avron, “Tableaux with Four Signs as a Unified Framework,” Int. Conf. on Automated Reasoning with Analytic Tableaux and Related Methods (TABLEAUX 2003), pp. 4-16, 2003.
  27. [27] J. Hintikka, “Form and content in quantification theory,” Acta Philosophica Fennica, Vol.8, pp. 7-55, 1955.

*This site is desgined based on HTML5 and CSS3 for modern browsers, e.g. Chrome, Firefox, Safari, Edge, Opera.

Last updated on Jul. 12, 2024