Implication functions in interval-valued fuzzy set theory

Interval-valued fuzzy set theory is an extension of fuzzy set theory in which the real, but unknown, membership degree is approximated by a closed interval of possible membership degrees. Since implications on the unit interval play an important role in fuzzy set theory, several authors have extended this notion to interval-valued fuzzy set theory. This chapter gives an overview of the results pertaining to implications in interval-valued fuzzy set theory. In particular, we describe several possibilities to represent such implications using implications on the unit interval, we give a characterization of the implications in interval-valued fuzzy set theory which satisfy the Smets-Magrez axioms, we discuss the solutions of a particular distributivity equation involving strict t-norms, we extend monoidal logic to the interval-valued fuzzy case and we give a soundness and completeness theorem which is similar to the one existing for monoidal logic, and finally we discuss some other constructions of implications in interval-valued fuzzy set theory

Interval-valued and intuitionistic fuzzy mathematical morphologies as special cases of L-fuzzy mathematical morphology

Mathematical morphology (MM) offers a wide range of tools for image processing and computer vision. MM was originally conceived for the processing of binary images and later extended to gray-scale morphology. Extensions of classical binary morphology to gray-scale morphology include approaches based on fuzzy set theory that give rise to fuzzy mathematical morphology (FMM). From a mathematical point of view, FMM relies on the fact that the class of all fuzzy sets over a certain universe forms a complete lattice. Recall that complete lattices provide for the most general framework in which MM can be conducted. The concept of L-fuzzy set generalizes not only the concept of fuzzy set but also the concepts of interval-valued fuzzy set and Atanassov’s intuitionistic fuzzy set. In addition, the class of L-fuzzy sets forms a complete lattice whenever the underlying set L constitutes a complete lattice. Based on these observations, we develop a general approach towards L-fuzzy mathematical morphology in this paper. Our focus is in particular on the construction of connectives for interval-valued and intuitionistic fuzzy mathematical morphologies that arise as special, isomorphic cases of L-fuzzy MM. As an application of these ideas, we generate a combination of some well-known medical image reconstruction techniques in terms of interval-valued fuzzy image processing

A Novel Method of the Generalized Interval-Valued Fuzzy Rough Approximation Operators

Rough set theory is a suitable tool for dealing with the imprecision, uncertainty, incompleteness, and vagueness of knowledge. In this paper, new lower and upper approximation operators for generalized fuzzy rough sets are constructed, and their definitions are expanded to the interval-valued environment. Furthermore, the properties of this type of rough sets are analyzed. These operators are shown to be equivalent to the generalized interval fuzzy rough approximation operators introduced by Dubois, which are determined by any interval-valued fuzzy binary relation expressed in a generalized approximation space. Main properties of these operators are discussed under different interval-valued fuzzy binary relations, and the illustrative examples are given to demonstrate the main features of the proposed operators

Aggregation operators on partially ordered sets and their categorical foundations

summary:In spite of increasing studies and investigations in the field of aggregation operators, there are two fundamental problems remaining unsolved: aggregation of $L$-fuzzy set-theoretic notions and their justification. In order to solve these problems, we will formulate aggregation operators and their special types on partially ordered sets with universal bounds, and introduce their categories. Furthermore, we will show that there exists a strong connection between the category of aggregation operators on partially ordered sets with universal bounds (Agop) and the category of partially ordered groupoids with universal bounds (Pogpu). Moreover, the subcategories of Agop consisting of associative aggregation operators, symmetric and associative aggregation operators and associative aggregation operators with neutral elements are, respectively, isomorphic to the subcategories of Pogpu formed by partially ordered semigroups, commutative partially ordered semigroups and partially ordered monoids in the sense of Birkhoff. As a justification of the present notions and results, some relevant examples for aggregations operators on partially ordered sets are given. Particularly, aggregation process in probabilistic metric spaces is also considered

Foundations of fuzzy answer set programming

Answer set programming (ASP) is a declarative language that is tailored towards combinatorial search problems. Although ASP has been applied to many problems, such as planning, configuration and verification of software, and database repair, it is less suitable for describing continuous problems. In this thesis we therefore studied fuzzy answer set programming (FASP). FASP is a language that combines ASP with ideas from fuzzy logic -- a class of many-valued logics that are able to describe continuous problems. We study the following topics: 1. An important issue when modeling continuous optimization problems is how to cope with overconstrained problems. In many cases we can opt to allow imperfect solutions, i.e. solutions that do not satisfy all constraints, but are sufficiently acceptable. However, the question which one of these imperfect solutions is most suitable then arises. Current approaches to fuzzy answer set programming solve this problem by attaching weights to the rules of the program. However, it is often not clear how these weights should be chosen and moreover weights do not allow to order different solutions. We improve upon this technique by using aggregators, which eliminate the aforementioned problems. This allows a richer modeling language and bridges the gap between FASP and other techniques such as valued constraint satisfaction problems. 2. The wishes of users and implementers of a programming language are often in direct conflict with each other. Users prefer a rich language that is easy to model in, whereas implementers prefer a small language that is easy to implement. We reconcile these differences by identifying a core language for FASP, called core FASP (CFASP), that only consists of non-constraint rules with monotonically increasing functions and negators in the body. We show that CFASP is capable of simulating constraint rules, monotonically decreasing functions, aggregators, S-implicators and classical negation. Moreover we remark that the simulations of constraints and classical negation bear a great resemblance to their simulations in classical ASP, which provides further insight into the relationship between ASP and FASP. 3. As a first step towards the creation of an implementation method for FASP we research whether it is possible to translate a FASP program to a fuzzy SAT problem. We introduce the concept of the completion of a FASP program and show that for programs without loops the models of the completion coincide with the answer sets. Furthermore we show that if a program has loops, we can translate the program to a fuzzy SAT problem by generalizing the concept of loop formulas. We illustrate this on a continuous version of the k-center problem. Such a translation is important because it allows us to solve FASP programs by means of solvers for fuzzy SAT. Under the appropriate conditions it is for example possible to solve FASP programs by means of off-the-shelf solvers for mixed integer programming (MIP)

Fuzzy techniques for noise removal in image sequences and interval-valued fuzzy mathematical morphology

Image sequences play an important role in today's world. They provide us a lot of information. Videos are for example used for traffic observations, surveillance systems, autonomous navigation and so on. Due to bad acquisition, transmission or recording, the sequences are however usually corrupted by noise, which hampers the functioning of many image processing techniques. A preprocessing module to filter the images often becomes necessary. After an introduction to fuzzy set theory and image processing, in the first main part of the thesis, several fuzzy logic based video filters are proposed: one filter for grayscale video sequences corrupted by additive Gaussian noise and two color extensions of it and two grayscale filters and one color filter for sequences affected by the random valued impulse noise type. In the second main part of the thesis, interval-valued fuzzy mathematical morphology is studied. Mathematical morphology is a theory intended for the analysis of spatial structures that has found application in e.g. edge detection, object recognition, pattern recognition, image segmentation, image magnification… In the thesis, an overview is given of the evolution from binary mathematical morphology over the different grayscale morphology theories to interval-valued fuzzy mathematical morphology and the interval-valued image model. Additionally, the basic properties of the interval-valued fuzzy morphological operators are investigated. Next, also the decomposition of the interval-valued fuzzy morphological operators is investigated. We investigate the relationship between the cut of the result of such operator applied on an interval-valued image and structuring element and the result of the corresponding binary operator applied on the cut of the image and structuring element. These results are first of all interesting because they provide a link between interval-valued fuzzy mathematical morphology and binary mathematical morphology, but such conversion into binary operators also reduces the computation. Finally, also the reverse problem is tackled, i.e., the construction of interval-valued morphological operators from the binary ones. Using the results from a more general study in which the construction of an interval-valued fuzzy set from a nested family of crisp sets is constructed, increasing binary operators (e.g. the binary dilation) are extended to interval-valued fuzzy operators