Abstract
The article focuses on a problem that naturally comes to an interest by joining two classical topics, namely the solvability of systems of fuzzy relational equations, and the partiality as a tool for modeling undefined values. As the first topic basically formally investigates the correctness of fuzzy rule-based systems by the investigation of the preservation of modus ponens, the other one allows dealing in an elegant algebraic way with distinct undefined values, in the particular case of the Dragonfly algebra, the focus is on missing values, their connection is straightforward. Indeed, dealing with missing values is rather omnipresent, and distinct expert rule-based systems are not immune to it so, such an investigation is highly desirable. What is not so straightforward are the results ensuring the solvability of such systems, i.e., the existence of safe models of the rules. Some previous results have been published and they relied on restrictions on the algebraic level. This article brings a new insight and investigates, what happens if we refuse to accept such a restriction. The answer is interesting as restricting the choice of the algebra does not seem to be critical but it imposes some restrictions on the sides of the consequents. Luckily, these restrictions are not so critical and restrictive from the application point of view.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
References
Avron, A., Konikowska, B.: Proof systems for reasoning about computation errors. Stud. Logica. 91(2), 273–293 (2009)
Bartl, E., Belohlavek, R., Vychodil, V.: Bivalent and other solutions of fuzzy relational equations via linguistic hedges. Fuzzy Sets Syst. 187(1), 103–112 (2012)
Běhounek, L., Daňková, M.: Variable-domain fuzzy sets - Part I: Representation. Fuzzy Sets Syst. 38, 1–18 (2020)
Běhounek, L., Daňková, M.: Variable-domain fuzzy sets - Part II: Apparatus. Fuzzy Sets Syst. 38, 19–43 (2020)
Burda, M.: Linguistic fuzzy logic in R. In: Proceedings of the IEEE International Conference on Fuzzy Systems, Istanbul, Turkey (2015)
Burda, M., Štěpnička, M.: lfl: an R package for linguistic fuzzy logic. Fuzzy Sets Syst. 431, 1–38 (2022). Logic and Related Topics
Běhounek, L., Dvořák, A.: Fuzzy relational modalities admitting truth-valueless propositions. Fuzzy Sets Syst. 388, 38–55 (2020)
Cao, N.: Solvability of fuzzy relational equations employing undefined values. In: The 11th Conference of the European Society for Fuzzy Logic and Technology (EUSFLAT 2019), pp. 227–234. Atlantis Press (2019)
Cao, N., Štěpnička, M.: Fuzzy relational equations employing dragonfly operations. In: 2019 11th International Conference on Knowledge and Systems Engineering (KSE), pp. 1–6. IEEE (2019)
Cao, N., Štěpnička, M.: Sufficient solvability conditions for systems of partial fuzzy relational equations. In: Lesot, M.-J., et al. (eds.) IPMU 2020. CCIS, vol. 1237, pp. 93–106. Springer, Cham (2020). https://doi.org/10.1007/978-3-030-50146-4_8
Cao, N., Štěpnička, M.: On solvability of systems of partial fuzzy relational equations. Fuzzy Sets Syst. 450, 87–117 (2022)
Cao, N., Štěpnička, M.: Preservation of properties of residuated algebraic structure by structures for the partial fuzzy set theory. Int. J. Approximate Reasoning 154, 1–26 (2023)
Ciucci, D., Dubois, D.: A map of dependencies among three-valued logics. Inf. Sci. 250, 162–177 (2013)
d’Allonnes, A.R., Lesot, M.J.: If I don’t know, should I infer? Reasoning around ignorance in a many-valued framework. In: Joint 17th World Congress of International Fuzzy Systems Association and 9th International Conference on Soft Computing and Intelligent Systems, IFSA-SCIS 2017, Otsu, Japan, 27–30 June 2017, pp. 1–6. IEEE (2017)
De Baets, B.: Analytical solution methods for fuzzy relational equations. In: Dubois, D., Prade, H. (eds.) The Handbook of Fuzzy Set Series, vol. 1, pp. 291–340. Academic Kluwer Publication, Boston (2000)
Di Nola, A., Sessa, S., Pedrycz, W., Sanchez, E.: Fuzzy Relation Equations and Their Applications to Knowledge Engineering. Kluwer, Boston (1989)
Gottwald, S.: Fuzzy control and fuzzy relation equations. A unified view as interpolation problem. In: IEEE Annual Meeting of the Fuzzy Information, Processing NAFIPS 2004, vol. 1, pp. 270–275. IEEE (2004)
Karpenko, A., Tomova, N.: Bochvar’s three-valued logic and literal paralogics: their lattice and functional equivalence. Logic Log. Philos. 26(2), 207–235 (2016)
Klir, G.J., Yuan, B.: Approximate solutions of systems of fuzzy relation equations. In: Proceedings of 1994 IEEE 3rd International Fuzzy Systems Conference, pp. 1452–1457. IEEE (1994)
Novák, V.: Fuzzy type theory with partial functions. Iran. J. Fuzzy Syst. 16(2), 1–16 (2019)
Perfilieva, I., Lehmke, S.: Correct models of fuzzy if-then rules are continuous. Fuzzy Sets Syst. 157, 3188–3197 (2006)
Perfilieva, I.: Fuzzy function as an approximate solution to a system of fuzzy relation equations. Fuzzy Sets Syst. 147(3), 363–383 (2004)
Perfilieva, I., Gottwald, S.: Solvability and approximate solvability of fuzzy relation equations. Int. J. Gen. Syst. 32(4), 361–372 (2003)
Prior, A.N.: Three-valued logic and future contingents. Philos. Q. 317–326 (1953)
Sanchez, E.: Resolution of composite fuzzy relation equations. Inf. Control 30, 38–48 (1976)
Štěpnička, M., Bodenhofer, U., Daňková, M., Novák, V.: Continuity issues of the implicational interpretation of fuzzy rules. Fuzzy Sets Syst. 161, 1959–1972 (2010)
Štěpnička, M., Cao, N., Běhounek, L., Burda, M., Dolný, A.: Missing values and dragonfly operations in fuzzy relational compositions. Int. J. Approximate Reasoning 113, 149–170 (2019)
Wangming, W.: Fuzzy reasoning and fuzzy relational equations. Fuzzy Sets Syst. 20(1), 67–78 (1986)
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2023 The Author(s), under exclusive license to Springer Nature Switzerland AG
About this paper
Cite this paper
Cao, N., Štěpnička, M. (2023). Partial Fuzzy Relational Equations and the Dragonfly Operations – What Happens If...?. In: Massanet, S., Montes, S., Ruiz-Aguilera, D., González-Hidalgo, M. (eds) Fuzzy Logic and Technology, and Aggregation Operators. EUSFLAT AGOP 2023 2023. Lecture Notes in Computer Science, vol 14069. Springer, Cham. https://doi.org/10.1007/978-3-031-39965-7_7
Download citation
DOI: https://doi.org/10.1007/978-3-031-39965-7_7
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-031-39964-0
Online ISBN: 978-3-031-39965-7
eBook Packages: Computer ScienceComputer Science (R0)