Residuated Algebraic Structures in the Vicinity of Pre-rough Algebra and Decidability

• Autorzy: Lin Zhe , Chakraborty Mihir Kumar , Ma Minghui

Identyfikatory

DOI

10.3233/FI-2021-2023

• Języki publikacji: EN

Abstrakty

EN

Varieties of topological quasi-Boolean algebras in the vicinity of pre-rough algebras [28, 29] are expanded to residuated algebraic structures by introducing a new implication operation and its residual in these structures. Sequent calculi for some classes of residuated algebraic structures are established. These sequent calculi have the strong finite model property which yields the decidability of the word problem for corresponding classes of algebraic structures.

Słowa kluczowe

EN

pre-rough algebra; residuation; word problem; finite model property; decidability

• Wydawca: IOS Press
• Czasopismo: Fundamenta Informaticae
• Rocznik: 2021
• Tom: Vol. 179, nr 3
• Strony: 239--274
• Opis fizyczny: Bibliogr. 37 poz., rys., tab.

Twórcy

autor

Lin Zhe

• Department of Philosophy, Xiamen University, Xiamen, China

autor

Chakraborty Mihir Kumar

• School of Cognitive Science, Jadavpur University, Kolkata, India

autor

Ma Minghui

• Department of Philosophy, Sun Yat-sen University, Guangzhou, China

Bibliografia

• [1] Banerjee M, Chakraborty M. A category for rough sets. Foundations of Computing and Decision Sciences, 1993. 18(3-4):167-180.
• [2] Banerjee M, Chakraborty M. Rough algebra. Bulletin of Polish Academy of Sciences: Mathematics, 1993. 41(4):293-297.
• [3] Banerjee M, Chakraborty M. Rough sets through algebraic logic. Fundamenta Informaticae, 1996. 28(3-4):211-221. doi:10.3233/FI-1996-283401.
• [4] Banerjee M. Rough sets and 3-valued Łukasiewicz logic. Fundamenta Informaticae, 1997. 31:213-220. doi:10.3233/FI-1997-313401.
• [5] Banerjee M, Chakraborty M. Foundations of vagueness: a category-theoretic approach. Electronic Notes in Theoretical Computer Science, 1993. 82(4):10-19. doi:10.1016/S1571-0661(04)80701-1.
• [6] Banerjee M, Yao Y. A categorial basis for granular computing. In: Rough Sets, Fuzzy Sets, Data Mining and Granular Computing, LNCS 4482. Springer Berlin Heidelberg, 2007 pp. 427-434. ISBN: 9783540725299. doi:10.1007/978-3-540-72530-5_51.
• [7] Buszkowski W. Some decision problems in the theory of syntactic categories. Zeitschrift für mathematische Logik und Grundlagen der Mathematik, 1982. 28(33-38):539-548. doi:10.1002/malq.19820283308.
• [8] Buszkowski W. Interpolation and FEP for logics of residuated algebras. Logic Journal of IGPL, 2011. 19(3):437-454. doi:10.1093/jigpal/jzp094.
• [9] Chvalovský K. Undecidability of consequence relation in full non-associative Lambek calculus. The Journal of Symbolic Logic, 2015. 80(2):567-586. doi:10.1017/jsl.2014.39.
• [10] Cosmadakis S. The word and generator problems for lattices. Information and Computation, 1988. 77:192-217. doi:10.1016/0890-5401(88)90048-x.
• [11] Diker M. Categories of rough sets and textures. Theoretical Computer Science, 2013. 488:46-65. doi:10.1016/j.tcs.2012.12.020.
• [12] Evans T. The word problem for abstract algebras. Journal of the London Mathematical Society, 1951. 26:64-71. doi:10.1112/jlms/s1-26.1.64.
• [13] Eklund P, Galán M. Monads can be rough. In: Rough Sets and Current Trends in Computing, LNCS 4259, Springer Berlin Heidelberg, 2006 pp. 77-84. ISBN: 9783540476931. doi:doi.org/10.1007/11908029_9.
• [14] Galatos N. The undecidability of the word problem for distributive residuated lattices. In: Ordered Algebraic Structures, Developments in Mathematics, vol 7. Springer Boston MA, 2002 pp. 231-243. ISBN: 9781441952257. doi:10.1007/978-1-4757-3627-4_12.
• [15] Galatos N, Jipsen P, Kowalski T, Ono H. Residuated Lattices: An Algebraic Glimpse at Substructural Logics. Springer, 2007. ISBN: 9780444521415.
• [16] Gehrke M, Harding J. Bounded lattice expansions. Journal of Algebra, 2001. 238:345-371. doi:10.1006/jabr.2000.8622.
• [17] Hughes GE, Cresswell MJ. A New Introduction to Modal Logic, 1996. Routledge, London. ISBN:9780415125994. 2nd edition.
• [18] Ma M, Chakraborty M. Covering-based rough sets and modal logics. part I. International Journal of Approximate Reasoning, 2016. 77:55-65. doi:10.1016/j.ijar.2016.06.002.
• [19] Ma M, Chakraborty M. Covering-based rough sets and modal logics. part II, International Journal of Approximate Reasoning, 2018. 95:113-123. doi:10.1016/j.ijar.2018.02.002.
• [20] Markov M. On the impossibility of certain algorithms in the theory of associative systems. Dokl. Akad. Nauk SSSR (N.S.), 1947. 55:583-586.
• [21] McKinsey JCC. The decision problem for some classes of sentences. The Journal Symbolic Logic, 1963. 8:61-76. doi:10.2307/2268172.
• [22] Novikov PS. On the algorithmic unsolvability of the word problem in group theory. In: American Mathematical Society Translations, Series 2. American Mathematical Society, 1958 pp. 1-122. ISBN:9780821817094. doi:10.1090/trans2/009/01.
• [23] Pawlak Z. Rough sets. International Journal of Computing and Information Sciences, 1982. 11(5):341-356.
• [24] Pagliani P. Rough set theory and logic-algebraic structures. In: Incomplete Information: Rough Set Analysis, Physica Heidelberg, 1998 pp. 109-190. ISBN: 9783790824575. doi:10.1007/978-3-7908-1888-8_6.
• [25] Pomykala JA. Approximation operations in approximation space. Bulletin of Polish Academy of Sciences: Mathematics, 1987. 35:653-662.
• [26] Post EL. Recursive unsolvability of a problem of Thue. The Journal of Symbolic Logic, 1947. 13:l-11. doi:10.2307/2267170.
• [27] Rasiowa H. An Algebraic Approach to Non-Classical Logics. North-Holland Publishing, 1974. ISBN: 9780720422641.
• [28] Saha A, Sen J, Chakraborty M. Algebraic structures in the vicinity of pre-rough algebra and their logics. Information Science, 2014. 282:296-320. doi:10.1016/j.ins.2014.06.004.
• [29] Saha A, Sen J, Chakraborty M. Algebraic structures in the vicinity of pre-rough algebra and their logics II. Information Science, 2016. 333:44-60. doi:10.1016/j.ins.2015.11.018.
• [30] Samanta P, Chakraborty M. Covering based approaches to rough sets and implication lattices. In: Rough Sets, Fuzzy Sets, Data Mining and Granular Computing, RSFDGrC 2009, LNCS 5908. Springer, 2009 pp. 127-134. ISBN: 9783642106453. doi:10.1007/978-3-642-10646-0_15.
• [31] Takeuchi K. The word problem for free distributive lattices. Journal of the Mathematical Society of Japan, 1969. 21(3):330-333. doi:10.2969/jmsj/02130330.
• [32] Touraille A. The word problem for Heyting algebras. Algebra Universalis, 1987. 24 (1-2):120-127. doi:10.1007/bf01188389.
• [33] Yao YY. Relational interpretations of neighborhood operators and rough set approximation operators, Information Sciences 111 (1-4) (1998) 239-259.
• [34] Yao Y. Constructive and algebraic methods of theory of rough sets. Information Sciences, 1998. 109:21-47. doi:10.1016/s0020-0255(98)00012-7.
• [35] Yao Y, Yao B. Covering based rough set approximations. Information Sciences, 2012. 200:91-107. doi:10.1016/j.ins.2012.02.065.
• [36] Zhu W. Generalized rough sets based on relations. Information Sciences, 2007. 177(22):4997-5011. doi:10.1016/j.ins.2007.05.037.
• [37] Zhang XH, Dai JH. Rough impliecation operation and residuated based fuzzy logic system. The journal of China Universities of Posts and Telecommunications, 2013. 20:109-112. doi:10.1016/s1005-8885(13) 60237-x.

Uwagi

Opracowanie rekordu ze środków MNiSW, umowa Nr 461252 w ramach programu "Społeczna odpowiedzialność nauki" - moduł: Popularyzacja nauki i promocja sportu (2021).