Formal Introduction to Fuzzy Implications

SummaryIn the article we present in the Mizar system the catalogue of nine basic fuzzy implications, used especially in the theory of fuzzy sets. This work is a continuation of the development of fuzzy sets in Mizar; it could be used to give a variety of more general operations, and also it could be a good starting point towards the formalization of fuzzy logic (together with t-norms and t-conorms, formalized previously).Institute of Informatics, University of Białystok, PolandMichał Baczyński and Balasubramaniam Jayaram. Fuzzy Implications. Springer Publishing Company, Incorporated, 2008. doi:10.1007/978-3-540-69082-5.Adam Grabowski. Basic formal properties of triangular norms and conorms. Formalized Mathematics, 25(2):93–100, 2017. doi:10.1515/forma-2017-0009.Adam Grabowski. The formal construction of fuzzy numbers. Formalized Mathematics, 22(4):321–327, 2014. doi:10.2478/forma-2014-0032.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 Magdalena Jastrzębska. Rough set theory from a math-assistant perspective. In Rough Sets and Intelligent Systems Paradigms, International Conference, RSEISP 2007, Warsaw, Poland, June 28–30, 2007, Proceedings, pages 152–161, 2007. doi:10.1007/978-3-540-73451-2_17.Adam Grabowski and Takashi Mitsuishi. Extending Formal Fuzzy Sets with Triangular Norms and Conorms, volume 642: Advances in Intelligent Systems and Computing, pages 176–187. Springer International Publishing, Cham, 2018. doi:10.1007/978-3-319-66824-6_16.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, Artur Korniłowicz, and Adam Naumowicz. Four decades of Mizar. Journal of Automated Reasoning, 55(3):191–198, 2015. doi:10.1007/s10817-015-9345-1.Takashi Mitsuishi, Noboru Endou, and Yasunari Shidama. The concept of fuzzy set and membership function and basic properties of fuzzy set operation. Formalized Mathematics, 9(2):351–356, 2001.Zdzisław Pawlak. Rough sets. International Journal of Parallel Programming, 11:341–356, 1982. doi:10.1007/BF01001956.Lotfi Zadeh. Fuzzy sets. Information and Control, 8(3):338–353, 1965.25324124

On Fuzzy Negations Generated by Fuzzy Implications

We continue in the Mizar system [2] the formalization of fuzzy implications according to the book of Baczynski and Jayaram “Fuzzy Implications” [1]. In this article we define fuzzy negations and show their connections with previously defined fuzzy implications [4] and [5] and triangular norms and conorms [6]. This can be seen as a step towards building a formal framework of fuzzy connectives [10]. We introduce formally Sugeno negation, boundary negations and show how these operators are pointwise ordered. This work is a continuation of the development of fuzzy sets [12], [3] in Mizar [7] started in [11] and partially described in [8]. This submission can be treated also as a part of a formal comparison of fuzzy and rough approaches to incomplete or uncertain information within the Mizar Mathematical Library [9].Institute of Informatics, University of Białystok, PolandMichał Baczynski and Balasubramaniam Jayaram. Fuzzy Implications. Springer Publishing Company, Incorporated, 2008. doi:10.1007/978-3-540-69082-5.Grzegorz Bancerek, Czesław Bylinski, Adam Grabowski, Artur Korniłowicz, Roman Matuszewski, Adam Naumowicz, Karol Pak, and Josef Urban. Mizar: State-of-the-art and beyond. In Manfred Kerber, Jacques Carette, Cezary Kaliszyk, Florian Rabe, and Volker Sorge, editors, Intelligent Computer Mathematics, volume 9150 of Lecture Notes in Computer Science, pages 261–279. Springer International Publishing, 2015. ISBN 978-3-319-20614-1. doi:10.1007/978-3-319-20615-8_17.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. Fundamental properties of fuzzy implications. Formalized Mathematics, 26(4):271–276, 2018. doi:10.2478/forma-2018-0023.Adam Grabowski. Basic formal properties of triangular norms and conorms. Formalized Mathematics, 25(2):93–100, 2017. doi:10.1515/forma-2017-0009Adam 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 and Takashi Mitsuishi. Extending Formal Fuzzy Sets with Triangular Norms and Conorms, volume 642: Advances in Intelligent Systems and Computing, pages 176–187. Springer International Publishing, Cham, 2018. doi:10.1007/978-3-319-66824-6_16.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.Petr Hájek. Metamathematics of Fuzzy Logic. Dordrecht: Kluwer, 1998.Takashi Mitsuishi, Noboru Endou, and Yasunari Shidama. The concept of fuzzy set and membership function and basic properties of fuzzy set operation. Formalized Mathematics, 9(2):351–356, 2001.Lotfi Zadeh. Fuzzy sets. Information and Control, 8(3):338–353, 1965. doi:10.1016/S0019-9958(65)90241-X.12112

Fitting aggregation operators to data

Theoretical advances in modelling aggregation of information produced a wide range of aggregation operators, applicable to almost every practical problem. The most important classes of aggregation operators include triangular norms, uninorms, generalised means and OWA operators.With such a variety, an important practical problem has emerged: how to fit the parameters/ weights of these families of aggregation operators to observed data? How to estimate quantitatively whether a given class of operators is suitable as a model in a given practical setting? Aggregation operators are rather special classes of functions, and thus they require specialised regression techniques, which would enforce important theoretical properties, like commutativity or associativity. My presentation will address this issue in detail, and will discuss various regression methods applicable specifically to t-norms, uninorms and generalised means. I will also demonstrate software implementing these regression techniques, which would allow practitioners to paste their data and obtain optimal parameters of the chosen family of operators.<br /

Interval-based uncertain reasoning

This thesis examines three interval based uncertain reasoning approaches: reasoning under interval constraints, reasoning using necessity and possibility functions, and reasoning with rough set theory. In all these approaches, intervals are used to characterize the uncertainty involved in a reasoning process when the available information is insufficient for single-valued truth evaluation functions. Approaches using interval constraints can be applied to both interval fuzzy sets and interval probabilities. The notion of interval triangular norms, or interval t-norms for short, is introduced and studied in both numeric and non-numeric settings. Algorithms for computing interval t-norms are proposed. Basic issues on the use of t-norms for approximate reasoning with interval fuzzy sets are studied. Inference rules for reasoning under interval constraints are investigated. In the second approach, a pair of necessity and possibility functions is used to bound the fuzzy truth values of propositions. Inference in this case is to narrow the gap between the pair of the functions. Inference rules are derived from the properties of necessity and possibility functions. The theory of rough sets is used to approximate truth values of propositions and to explore modal structures in many-valued logic. It offers an uncertain reasoning method complementary to the other two

Aggregation functions for typical hesitant fuzzy elements and the action of automorphisms

This work studies the aggregation operators on the set of all possible membership degrees of typical hesitant fuzzy sets, which we refer to as H, as well as the action of H-automorphisms which are defined over the set of all finite non-empty subsets of the unitary interval. In order to do so, the partial order ≤H, based on α-normalization, is introduced, leading to a comparison based on selecting the greatest membership degrees of the related fuzzy sets. Additionally, the idea of interval representation is extended to the context of typical hesitant aggregation functions named as the H-representation. As main contribution, we consider the class of finite hesitant triangular norms, studying their properties and analyzing the H-conjugate functions over such operators. © 2013 Elsevier Inc. All rights reserved.Peer Reviewe

CARS – A Spatio-Temporal BDI Recommender System: Time, Space and Uncertainty

International audienceAgent-based recommender systems have been exploited in the last years to provide informative suggestions to users, showing the advantage of exploiting components like beliefs, goals and trust in the recommenda-tions' computation. However, many real-world scenarios, like the traffic one, require the additional feature of representing and reasoning about spatial and temporal knowledge, considering also their vague connotation. This paper tackles this challenge and introduces CARS, a spatio-temporal agent-based recommender system based on the Belief-Desire-Intention (BDI) architecture. Our approach extends the BDI model with spatial and temporal information to represent and reason about fuzzy beliefs and desires dynamics. An experimental evaluation about spatio-temporal reasoning in the traffic domain is carried out using the NetLogo platform, showing the improvements our recommender system introduces to support agents in achieving their goals

Machine Learning Methods for Fuzzy Pattern Tree Induction

This thesis elaborates on a novel approach to fuzzy machine learning, that is, the combination of machine learning methods with mathematical tools for modeling and information processing based on fuzzy logic. More specifically, the thesis is devoted to so-called fuzzy pattern trees, a model class that has recently been introduced for representing dependencies between input and output variables in supervised learning tasks, such as classification and regression. Due to its hierarchical, modular structure and the use of different types of (nonlinear) aggregation operators, a fuzzy pattern tree has the ability to represent such dependencies in a very exible and compact way, thereby offering a reasonable balance between accuracy and model transparency. The focus of the thesis is on novel algorithms for pattern tree induction, i.e., for learning fuzzy pattern trees from observed data. In total, three new algorithms are introduced and compared to an existing method for the data-driven construction of pattern trees. While the first two algorithms are mainly geared toward an improvement of predictive accuracy, the last one focuses on eficiency aspects and seeks to make the learning process faster. The description and discussion of each algorithm is complemented with theoretical analyses and empirical studies in order to show the effectiveness of the proposed solutions