The ⊛-composition of fuzzy implications: Closures with respect to properties, powers and families

Recently, Vemuri and Jayaram proposed a novel method of generating fuzzy implications from a given pair of fuzzy implications. Viewing this as a binary operation ⊛ on the set II of fuzzy implications they obtained, for the first time, a monoid structure (I,⊛)(I,⊛) on the set II. Some algebraic aspects of (I,⊛)(I,⊛) had already been explored and hitherto unknown representation results for the Yager's families of fuzzy implications were obtained in [53] (N.R. Vemuri and B. Jayaram, Representations through a monoid on the set of fuzzy implications, fuzzy sets and systems, 247 (2014) 51–67). However, the properties of fuzzy implications generated or obtained using the ⊛-composition have not been explored. In this work, the preservation of the basic properties like neutrality, ordering and exchange principles , the functional equations that the obtained fuzzy implications satisfy, the powers w.r.t. ⊛ and their convergence, and the closures of some families of fuzzy implications w.r.t. the operation ⊛, specifically the families of (S,N)(S,N)-, R-, f- and g-implications, are studied. This study shows that the ⊛-composition carries over many of the desirable properties of the original fuzzy implications to the generated fuzzy implications and further, due to the associativity of the ⊛-composition one can obtain, often, infinitely many new fuzzy implications from a single fuzzy implication through self-composition w.r.t. the ⊛-composition

A map of dependencies among three-valued logics

International audienceThree-valued logics arise in several fields of computer science, both inspired by concrete problems (such as in the management of the null value in databases) and theoretical considerations. Several three-valued logics have been defined. They differ by their choice of basic connectives, hence also from a syntactic and proof-theoretic point of view. Different interpretations of the third truth value have also been suggested. They often carry an epistemic flavor. In this work, relationships between logical connectives on three-valued functions are explored. Existing theorems of functional completeness have laid bare some of these links, based on specific connectives. However we try to draw a map of such relationships between conjunctions, negations and implications that extend Boolean ones. It turns out that all reasonable connectives can be defined from a few of them and so all known three-valued logics appear as a fragment of only one logic. These results can be instrumental when choosing, for each application context, the appropriate fragment where the basic connectives make full sense, based on the appropriate meaning of the third truth-value

Lattice operations on fuzzy implications and the preservation of the exchange principle

In this work, we solve an open problem related to the preservation of the exchange principle (EP) of fuzzy implications under lattice operations ([3], Problem 3.1.). We show that generalizations of the commutativity of antecedents (CA) to a pair of fuzzy implications (I,J)(I,J), viz., the generalized exchange principle and the mutual exchangeability are sufficient conditions for the solution of the problem. Further, we determine conditions under which these become necessary too. Finally, we investigate the pairs of fuzzy implications from different families such that (EP) is preserved by the join and meet operations

Representations through a monoid on the set of fuzzy implications

Fuzzy implications are one of the most important fuzzy logic connectives. In this work, we conduct a systematic algebraic study on the set II of all fuzzy implications. To this end, we propose a binary operation, denoted by ⊛, which makes (I,⊛I,⊛) a non-idempotent monoid. While this operation does not give a group structure, we determine the largest subgroup SS of this monoid and using its representation define a group action of SS that partitions II into equivalence classes. Based on these equivalence classes, we obtain a hitherto unknown representations of the two main families of fuzzy implications, viz., the f- and g-implications

The *-composition -A Novel Generating Method of Fuzzy Implications: An Algebraic Study

Fuzzy implications are one of the two most important fuzzy logic connectives, the other being t-norms. They are a generalisation of the classical implication from two-valued logic to the multivalued setting. A binary operation I on [0; 1] is called a fuzzy implication if (i) I is decreasing in the first variable, (ii) I is increasing in the second variable, (iii) I(0; 0) = I(1; 1) = 1 and I(1; 0) = 0. The set of all fuzzy implications defined on [0; 1] is denoted by I. Fuzzy implications have many applications in fields like fuzzy control, approximate reasoning, decision making, multivalued logic, fuzzy image processing, etc. Their applicational value necessitates new ways of generating fuzzy implications that are fit for a specific task. The generating methods of fuzzy implications can be broadly categorised as in the following: (M1): From binary functions on [0; 1], typically other fuzzy logic connectives, viz., (S;N)-, R-, QL- implications, (M2): From unary functions on [0,1], typically monotonic functions, for instance, Yager’s f-, g- implications, or from fuzzy negations, (M3): From existing fuzzy implications

A Deep Study of Fuzzy Implications

This thesis contributes a deep study on the extensions of the IMPLY operator in classical binary logic to fuzzy logic, which are called fuzzy implications. After the introduction in Chapter 1 and basic notations about the fuzzy logic operators In Chapter 2 we first characterize In Chapter 3 S- and R- implications and then extensively investigate under which conditions QL-implications satisfy the thirteen fuzzy implication axioms. In Chapter 4 we develop the complete interrelationships between the eight supplementary axioms FI6-FI13 for fuzzy implications satisfying the five basic axioms FI1-FI15. We prove all the dependencies between the eight fuzzy implication axioms, and provide for each independent case a counter-example. The counter-examples provided in this chapter can be used in the applications that need different fuzzy implications satisfying different fuzzy implication axioms. In Chapter 5 we study proper S-, R- and QL-implications for an iterative boolean-like scheme of reasoning from classical binary logic in the frame of fuzzy logic. Namely, repeating antecedents $n$ times, the reasoning result will remain the same. To determine the proper S-, R- and QL-implications we get a full solution of the functional equation $I(x,y)=I(x,I(x,y))$, for all $x$, $y\in[0,1]$. In Chapter 6 we study for the most important t-norms, t-conorms and S-implications their robustness against different perturbations in a fuzzy rule-based system. We define and compare for these fuzzy logical operators the robustness measures against bounded unknown and uniform distributed perturbations respectively. In Chapter 7 we use a fuzzy implication $I$ to define a fuzzy $I$-adjunction in $\mathcal{F}(\mathbb{R}^{n})$. And then we study the conditions under which a fuzzy dilation which is defined from a conjunction $\mathcal{C}$ on the unit interval and a fuzzy erosion which is defined from a fuzzy implication $I^{'}$ to form a fuzzy $I$-adjunction. These conditions are essential in order that the fuzzification of the morphological operations of dilation, erosion, opening and closing obey similar properties as their algebraic counterparts. We find out that the adjointness between the conjunction $\mathcal{C}$ on the unit interval and the implication $I$ or the implication $I^{'}$ play important roles in such conditions

Implikacje rozmyte generowane z kopuł

Implikacje rozmyte sa jednymi z najwazniejszych spójników logiki rozmytej, które uogólniaja klasyczne implikacje dla klasycznej logiki na odcinek. Ponadto implikacje rozmyte odgrywaja wazna role w takich zastosowaniach jak wnioskowaniu przyblizonym, rozmytym rozpoznawaniu obrazu, problemach decyzyjnych, logice wielowartosciowej, itd. Celem nastepujacej dysertacji jest uporzadkowanie informacji o implikacjach rozmytych generowanych z dwuwartosciowych kopuł, badz z funkcji ogólniejszych (np. z semikopuł). Kopuły sa waznymi funkcjami w probabilistyce. Waznosc kopuł w rachunku prawdopodobienstwa wynika z twierdzenia Sklara. Rozdział I zawiera informacje wstepne dotyczace podstawowych spójników logicznych, kopuł, qausikopuł i semikopuł wraz z ich najwazniejszymi własnosciami oraz kilka przydatnych własnosci funkcji rzeczywistych. Rozdział II jest poswiecony rozwiazaniu równania Franka, który to dowód jest rzadko prezentowany w monografiach, ale t-normy Franka, które sa rozwiazaniem równania Franka, sa dosc czesto przytaczane w wielu pracach. Ponadto okazuje sie, ze wiele równan dla kopuł, wynikajacych z odpowiednich własnosci dla implikacji s-probabilistycznych, mozna rozwiazac wykorzystujac t-normy Franka. Dlatego tez prezentujemy pełny dowód rozwiazania równania Franka w wersji dla t-norm i dla kopuł. Rozdział III jest poswiecony omówieniu dwóch waznych klas implikacji. Pierwsza z nich sa implikacje indukowane z semikopuł. W rozdziale IV pokazano jak przy pomocy twierdzenia Sklara mozna otrzymac takie funkcje jak implikacje probabilistyczne, s-probabilistyczne, warunkowe, dualne oraz s-dualne. Ponadto przedstawiano podstawowe własnosci tych klas funkcji. W ostatni rozdziale V zaprezentowane sa nowe wyniki z pracy, uzyskane przez Autora we współpracy z M. Baczynskim, P. Grzegorzewskim, W. Niemyska oraz nieopublikowane wyniki uzyskane przez Autora. W skład tych wyników wchodza takie własnosci implikacji z rozdziału IV jak prawa kontrapozycji, prawo importacji, Tconditionality oraz przeciecia klas tych funkcji z innymi znanymi klasami implikacji rozmytych. W niniejszej pracy przyjeto konwencje, w której wszystkie rezultaty sa podane z odnosnikami do zródeł, z wyjatkiem nieopublikowanych rezultatów uzyskanych przez Autora, które sa podane bez odnosników

On Fuzzy Negations and Laws of Contraposition. Lattice of Fuzzy Negations

This the next article in the series formalizing the book of Baczyński and Jayaram “Fuzzy Implications”. We define the laws of contraposition connected with various fuzzy negations, and in order to make the cluster registration mechanism fully working, we construct some more non-classical examples of fuzzy implications. Finally, as the testbed of the reuse of lattice-theoretical approach, we introduce the lattice of fuzzy negations and show its basic properties.Faculty of Computer Science, University of Białystok, PolandMichał Baczyński and Balasubramaniam Jayaram. Fuzzy Implications. Springer Publishing Company, Incorporated, 2008. doi:10.1007/978-3-540-69082-5.Grzegorz Bancerek, Czesław Byliński, Adam Grabowski, Artur Korniłowicz, Roman Matuszewski, Adam Naumowicz, and Karol Pąk. The role of the Mizar Mathematical Library for interactive proof development in Mizar. Journal of Automated Reasoning, 61(1):9–32, 2018. doi:10.1007/s10817-017-9440-6.Józef Drewniak. Invariant fuzzy implications. Soft Computing, 10:506–513, 2006.Didier Dubois and Henri Prade. Fuzzy Sets and Systems: Theory and Applications. Academic Press, New York, 1980.Adam Grabowski. Formal introduction to fuzzy implications. Formalized Mathematics, 25(3):241–248, 2017. doi:10.1515/forma-2017-0023.Adam Grabowski. On fuzzy negations generated by fuzzy implications. Formalized Mathematics, 28(1):121–128, 2020. doi:10.2478/forma-2020-0011.Adam Grabowski. Fuzzy implications in the Mizar system. In 30th IEEE International Conference on Fuzzy Systems, FUZZ-IEEE 2021, Luxembourg, July 11–14, 2021, pages 1–6. IEEE, 2021. doi:10.1109/FUZZ45933.2021.9494593.Adam Grabowski. On the computer certification of fuzzy numbers. In M. Ganzha, L. Maciaszek, and M. Paprzycki, editors, 2013 Federated Conference on Computer Science and Information Systems (FedCSIS), Federated Conference on Computer Science and Information Systems, pages 51–54, 2013.Adam Grabowski. Lattice theory for rough sets – a case study with Mizar. Fundamenta Informaticae, 147(2–3):223–240, 2016. doi:10.3233/FI-2016-1406.Adam Grabowski and Takashi Mitsuishi. Initial comparison of formal approaches to fuzzy and rough sets. In Leszek Rutkowski, Marcin Korytkowski, Rafal Scherer, Ryszard Tadeusiewicz, Lotfi A. Zadeh, and Jacek M. Zurada, editors, Artificial Intelligence and Soft Computing – 14th International Conference, ICAISC 2015, Zakopane, Poland, June 14-18, 2015, Proceedings, Part I, volume 9119 of Lecture Notes in Computer Science, pages 160–171. Springer, 2015. doi:10.1007/978-3-319-19324-3_15.Adam Grabowski and Takashi Mitsuishi. Formalizing lattice-theoretical aspects of rough and fuzzy sets. In D. Ciucci, G. Wang, S. Mitra, and W.Z. Wu, editors, Rough Sets and Knowledge Technology – 10th International Conference held as part of the International Joint Conference on Rough Sets (IJCRS), Tianjin, PR China, November 20–23, 2015, Proceedings, volume 9436 of Lecture Notes in Artificial Intelligence, pages 347–356. Springer, 2015. doi:10.1007/978-3-319-25754-9_31.Adam Grabowski and Christoph Schwarzweller. On duplication in mathematical repositories. In Serge Autexier, Jacques Calmet, David Delahaye, Patrick D. F. Ion, Laurence Rideau, Renaud Rioboo, and Alan P. Sexton, editors, Intelligent Computer Mathematics, 10th International Conference, AISC 2010, 17th Symposium, Calculemus 2010, and 9th International Conference, MKM 2010, Paris, France, July 5–10, 2010. Proceedings, volume 6167 of Lecture Notes in Computer Science, pages 300–314. Springer, 2010. doi:10.1007/978-3-642-14128-7_26.Takashi Mitsuishi. Definition of centroid method as defuzzification. Formalized Mathematics, 30(2):125–134, 2022. doi:10.2478/forma-2022-0010.Takashi Mitsuishi. Isosceles triangular and isosceles trapezoidal membership functions using centroid method. Formalized Mathematics, 31:59–66, 2023. doi:10.2478/forma-2023-0006.Lotfi Zadeh. Fuzzy sets. Information and Control, 8(3):338–353, 1965. doi:10.1016/S0019-9958(65)90241-X.31115115