Interpretability, a silver lining to a fuzzy cloud

Bernadette Bouchon-Meunier

University of Paris 6 (LiP6) – Paris (France)

Abstract: To grasp the complexity of real-world situations is a key issue for all processes of decision-making and con-trol. Fuzzy set theory provides efficient solutions to deal with imprecise, incomplete and uncertain data per-taining to this complexity. In addition, the large scale of data available in all kinds of modern environments highlights the importance of the interpretability of re-sults in the management of data, be they videos, tem-poral data or web elements, for instance. We show vari-ous cases where choosing a fuzzy knowledge represen-tation provides interesting tools to summarize, inter-prete, annotate, analyse or score the available data or knowledge, even though we are facing huge and com-plex information.

Short-Bio: Bernadette Bouchon-Meunier is a director of research at the National Centre for Scientific Research, head of the department of Databases and Machine Learning in the Computer Science Laboratory of the University Paris 6 (LIP6). She is the Editor-in-Chief of the International Journal of Uncertainty, Fuzziness and Knowledge-based Systems, the (co)-editor of 24 books and the (co)-author of five. Co-executive director of the IPMU International Conference held every other year, she served as the FUZZ-IEEE 2010 and FUZZ-IEEE 2013 Program Chair, the IEEE Symposium Series on Computational Intelligence (SSCI 2011) General Chair and the FUZZ-IEEE 2012 Conference Chair. She is presently a member of the IEEE Computational Intelligence Society Adcom and the IEEE Women in Engineering Committee. She is an IEEE fellow and an International Fuzzy Systems Association fellow.

Image Processing and Classification using Extensions of Fuzzy Sets. First Successes

Humberto Bustince

Public University of Navarra – Pamplona (Spain)

Abstract: Since Zadeh first presented the concept of fuzzy set (or type I fuzzy set) in 1965, different types of fuzzy sets have been defined. Roughly speaking, the most important characteristics of all of them are:

• They are particular cases of the L-fuzzy sets defined by Goguen;
• Their motivation arises from theoretical problems and they have always been very effective for their solution.
• In the applied field, it is not always shown that the results obtained with them are better than those obtained with type I fuzzy sets. This consideration leads skeptics about these sets to argue as follows: when we use new types of sets, we have almost always to handle more information but the improvement in the results is not proportional to the amount of information that we use

In my opinion, the problem stated in the last item arises from the difficulty to build the best fuzzy set for a given application we are working in. This consideration has led me to consider the following objectives: (i) To study the sets that have been defined to solve the problem posed by Zadeh in 1971 about the difficulty for building the membership degrees of the elements to the set. We are going to call these sets extensions of the fuzzy sets. Among the extensions, we are going to work with interval-valued fuzzy sets or interval type 2 fuzzy sets. (ii) To show applications that can be found in the literature and where the use of interval-valued fuzzy sets provides better results than those obtained with fuzzy sets. We should remark that in the papers where this improvement is shown a comparison to the best fuzzy techniques for the considered problem is always carried out. In particular, I will present the new results obtained using interval-valued fuzzy sets in the following problems:

• Classification problems. The presented proposal allows outperforming two state-of-the-art fuzzy classifiers, namely, the FARC-HD method and the FURIA algorithm.
• Image processing. The adaptation of Huang and Wang algorithm to the interval-valued fuzzy setting has allowed proving that for some regions in ultrasound images segmentation is better than the one obtained with the same algorithm making use only of type I fuzzy sets.
• Decision Making. The theoretical developments about the construction of new admissible orders for intervals have allowed us to show that in decision making problems it is always necessary to use different orders and arrive at the choice of the best alternative by means of consensus techniques.

Short-Bio: Humberto Bustince received his Bs. C. degree on Physics from the Salamanca University, Spain, in 1983 and his Ph.D. degree in Mathematics from the Public University of Navarra, Pamplona, Spain, in 1994. He has been a teacher at the Public University of Navarra since 1991, and he is currently a Full Professor with the Department of Automatics and Computation. He served as subdirector of the Technical School for Industrial Engineering and Telecommunications from 01/01/2003 to 30/10/2008 and he was involved in the implantation of Computer Science courses at the Public University of Navarra. He is currently involved in teaching artificial intelligence for students of computer sciences.
Dr. Bustince has authored more than 80 journal papers ( Web of Knowledge), and more than 73 contributions to international conferences. He has also been co-author of four books on fuzzy theory and extensions of fuzzy sets.
Fellow of the IEEE Computational Intelligence Systems society. Member of the board of the European Society for Fuzzy Logic and Applications (EUSFLAT). Editor in chief of the Mathware&Soft Computing Magazine and of Notes on Intuitionistic Fuzzy Sets. Guest editor of the Fuzzy Sets and Systems journal. Member of the editorial board of the Journal of Intelligence&Fuzzy Systems, The International Journal of Computational Intelligence Systems and the Axioms journal.

On generalizing the Nullstellensatz and McNaughton’s Theorem for MV algebras

Antonio Di Nola

University of Salerno – Salerno (Italy)

Abstract: Classical algebraic geometry is one of the most successful areas of mathematics. However, there are good reasons to try to generalize algebraic geometry from its original domain (rings and fields) to more extended domains (universal algebra).
We believe that an algebro-geometric analysis of universal algebraic structures (varieties) may have other successful applications. From a logical point of view, it seems natural to look at varieties with a clear logical significance. For instance one can consider the variety of MV algebras, a kind of algebraic structures corresponding to many valued logic in the same sense in which Boolean algebras correspond to classical two valued logic. A fundamental result for MV algebras is McNaughton’s Theorem, which gives a perspicuous description of free MV algebras as algebras of polynomials.
In this talk we generalize in the sense of considering more general scenarios than usual MV polynomials (without coefficients) and usual evaluation of polynomials in [0,1]. Actually, the idea of a duality between (MV) algebraic objects and geometric objects can be realized in many ways. Every time we have an adjunction (V,I) between two categories, the composed operator VI becomes an endofunctor and its images constitute what can be called a “geometry”. In turn, one hopes that information about geometric objects can help understanding the algebraic side

Short-Bio: Antonio Di Nola is Full Professor of Mathematical Logic and Director of the Department of Mathematics of the University of Salerno.
Since the nineties he has been a leading proponent of the study of algebraic models of Lukasiewicz logic (MV-algebras), the most important among the many-valued logics. His contribution to the study of MV-algebras, witnessed by the seventeen citations of his works in the fundamental monograph “Algebraic foundations of many-valued reasoning”, includes: a functional representation theorem for all MV-algebras (aka Di Nola’s Representation Theorem); the discovery of categorical equivalences between categories of MV-algebras and categories of groups, rings, and semi-rings, profitably used in the literature of MV-algebras, the discovery of an equational axiomatisation of all varieties of MV-algebras, and a normal form theorem for Lukasiewicz logic. Today is actively committed to apply ideas from algebraic geometry in the MV-algebra and in the study of probability which admit infinitesimal values.
He is author/coauthor of more than 150 scientific works, published on international journals of logic, algebra and computer science.