21 research outputs found

    Distributivity of ordinal sum implications over overlap and grouping functions

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    summary:In 2015, a new class of fuzzy implications, called ordinal sum implications, was proposed by Su et al. They then discussed the distributivity of such ordinal sum implications with respect to t-norms and t-conorms. In this paper, we continue the study of distributivity of such ordinal sum implications over two newly-born classes of aggregation operators, namely overlap and grouping functions, respectively. The main results of this paper are characterizations of the overlap and/or grouping function solutions to the four usual distributive equations of ordinal sum fuzzy implications. And then sufficient and necessary conditions for ordinal sum implications distributing over overlap and grouping functions are given

    Fitting aggregation operators to data

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    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 /

    On the distributivity of T-power based implications

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    Due to the fact that Zadeh's quantifiers constitute the usual method to modify fuzzy propositions, the so-called family of T-power based implications was proposed. In this paper, the four basic distributive laws related to T-power based fuzzy implications and fuzzy logic operations (t-norms and t-conorms) are deeply studied. This study shows that two of the four distributive laws of the T-power based implications have a unique solution, while the other two have multiple solutions

    A Deep Study of Fuzzy Implications

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    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 nn 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))I(x,y)=I(x,I(x,y)), for all xx, y[0,1]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 II to define a fuzzy II-adjunction in F(Rn)\mathcal{F}(\mathbb{R}^{n}). And then we study the conditions under which a fuzzy dilation which is defined from a conjunction C\mathcal{C} on the unit interval and a fuzzy erosion which is defined from a fuzzy implication II^{'} to form a fuzzy II-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 C\mathcal{C} on the unit interval and the implication II or the implication II^{'} play important roles in such conditions

    Rotation-invariant t-norms

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    Generalized continuous and left-continuous t-norms arising from algebraic semantics for fuzzy logics

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    This paper focuses on the issue of how generalizations of continuous and left-continuous t-norms over linearly ordered sets should be from a logical point of view. Taking into account recent results in the scope of algebraic semantics for fuzzy logics over chains with a monoidal residuated operation, we advocate linearly ordered BL-algebras and MTL-algebras as adequate generalizations of continuous and left-continuous t-norms respectively. In both cases, the underlying basic structure is that of linearly ordered residuated lattices. Although the residuation property is equivalent to left-continuity in t-norms, continuous t-norms have received much more attention due to their simpler structure. We review their complete description in terms of ordinal sums and discuss the problem of describing the structure of their generalization to BL-chains. In particular we show the good behavior of BL-algebras over a finite or complete chain, and discuss the partial knowledge of rational BL-chains. Then we move to the general non-continuous case corresponding to left-continuous t-norms and MTL-chains. The unsolved problem of describing the structure of left-continuous t-norms is presented together with a fistful of construction-decomposition techniques that apply to some distinguished families of t-norms and, finally, we discuss the situation in the general study of MTL-chains as a natural generalization of left-continuous t-norms

    O równaniach funkcyjnych związanych z rozdzielnością implikacji rozmytych

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    In classical logic conjunction distributes over disjunction and disjunction distributes over conjunction. Moreover, implication is left-distributive over conjunction and disjunction: p ! (q ^ r) (p ! q) ^ (p ! r); p ! (q _ r) (p ! q) _ (p ! r): At the same time it is neither right-distributive over conjunction nor over disjunction. However, the following two equalities, that are kind of right-distributivity of implications, hold: (p ^ q) ! r (p ! r) _ (q ! r); (p _ q) ! r (p ! r) ^ (q ! r): We can rewrite the above four classical tautologies in fuzzy logic and obtain the following distributivity equations: I(x;C1(y; z)) = C2(I(x; y); I(x; z)); (D1) I(x;D1(y; z)) = D2(I(x; y); I(x; z)); (D2) I(C(x; y); z) = D(I(x; z); I(y; z)); (D3) I(D(x; y); z) = C(I(x; z); I(y; z)); (D4) that are satisfied for all x; y; z 2 [0; 1], where I is some generalization of classical implication, C, C1, C2 are some generalizations of classical conjunction and D, D1, D2 are some generalizations of classical disjunction. We can define and study those equations in any lattice L = (L;6L) instead of the unit interval [0; 1] with regular order „6” on the real line, as well. From the functional equation’s point of view J. Aczél was probably the one that studied rightdistributivity first. He characterized solutions of the functional equation (D3) in the case of C = D, among functions I there are bounded below and functions C that are continuous, increasing, associative and have a neutral element. Part of the results presented in this thesis may be seen as a generalization of J. Aczél’s theorem but with fewer assumptions on the functions F and G. As a generalization of classical implication we consider here a fuzzy implication and as a generalization of classical conjunction and disjunction - t-norms and t-conorms, respectively (or more general conjunctive and disjunctive uninorms). We study the distributivity equations (D1) - (D4) for such operators defined on different lattices with special focus on various functional equations that appear. In the first two sections necessary fuzzy logic concepts are introduced. The background and history of studies on distributivity of fuzzy implications are outlined, as well. In Sections 3, 4 and 5 new results are presented and among them solutions to the following functional equations (with different assumptions): f(m1(x + y)) = m2(f(x) + f(y)); x; y 2 [0; r1]; g(u1 + v1; u2 + v2) = g(u1; u2) + g(v1; v2); (u1; u2); (v1; v2) 2 L1; h(xc(y)) = h(x) + h(xy); x; y 2 (0;1); k(min(j(y); 1)) = min(k(x) + k(xy); 1); x 2 [0; 1]; y 2 (0; 1]; where: f : [0; r1] ! [0; r2], for some constants r1; r2 that may be finite or infinite, and for functions m2 that may be injective or not; g : L1 ! [1;1], for L1 = f(u1; u2) 2 [1;1]2 j u1 u2g (function g satisfies two-dimensional Cauchy equation extended to the infinities); h; c : (0;1) ! (0;1) and function h is continuous or is a bijection; k : [0; 1] ! [0; 1], g : (0; 1] ! [1;1) and function k is continuous. Most of the results in Sections 3, 4 and 5 are new and obtained by the author in collaboration with M. Baczynski, R. Ger, M. E. Kuczma or T. Szostok. Part of them have been already published either in scientific journals (see [5]) or in refereed papers in proceedings (see [4, 1, 2, 3])

    Fuzzy Description Logics with General Concept Inclusions

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    Description logics (DLs) are used to represent knowledge of an application domain and provide standard reasoning services to infer consequences of this knowledge. However, classical DLs are not suited to represent vagueness in the description of the knowledge. We consider a combination of DLs and Fuzzy Logics to address this task. In particular, we consider the t-norm-based semantics for fuzzy DLs introduced by Hájek in 2005. Since then, many tableau algorithms have been developed for reasoning in fuzzy DLs. Another popular approach is to reduce fuzzy ontologies to classical ones and use existing highly optimized classical reasoners to deal with them. However, a systematic study of the computational complexity of the different reasoning problems is so far missing from the literature on fuzzy DLs. Recently, some of the developed tableau algorithms have been shown to be incorrect in the presence of general concept inclusion axioms (GCIs). In some fuzzy DLs, reasoning with GCIs has even turned out to be undecidable. This work provides a rigorous analysis of the boundary between decidable and undecidable reasoning problems in t-norm-based fuzzy DLs, in particular for GCIs. Existing undecidability proofs are extended to cover large classes of fuzzy DLs, and decidability is shown for most of the remaining logics considered here. Additionally, the computational complexity of reasoning in fuzzy DLs with semantics based on finite lattices is analyzed. For most decidability results, tight complexity bounds can be derived

    DUALITIES AND REPRESENTATIONS FOR MANY-VALUED LOGICS IN THE HIERARCHY OF WEAK NILPOTENT MINIMUM.

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    In this thesis we study particular subclasses of WNM algebras. The variety of WNM algebras forms the algebraic semantics of the WNM logic, a propositional many-valued logic that generalizes some well-known case in the setting of triangular norms logics. WNM logic lies in the hierarchy of schematic extensions of MTL, which is proven to be the logic of all left-continuous triangular norms and their residua. In this work, I have extensively studied two extensions of WNM logic, namely RDP logic and NMG logic, from the point of view of algebraic and categorical logic. We develop spectral dualities between the varieties of algebras corresponding to RDP logic and NMG logic, and suitable defined combinatorial categories. Categorical dualities allow to give algorithmic construction of products in the dual categories obtaining computable descriptions of coproducts (which are notoriously hard to compute working only in the algebraic side) for the corresponding finite algebras. As a byproduct, representation theorems for finite algebras and free finitely generated algebras in the considered varieties are obtained. This latter characterization is especially useful to provide explicit construction of a number of objects relevant from the point of view of the logical interpretation of the varieties of algebras: normal forms, strongest deductive interpolants and most general unifiers

    North American Fuzzy Logic Processing Society (NAFIPS 1992), volume 2

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    This document contains papers presented at the NAFIPS '92 North American Fuzzy Information Processing Society Conference. More than 75 papers were presented at this Conference, which was sponsored by NAFIPS in cooperation with NASA, the Instituto Tecnologico de Morelia, the Indian Society for Fuzzy Mathematics and Information Processing (ISFUMIP), the Instituto Tecnologico de Estudios Superiores de Monterrey (ITESM), the International Fuzzy Systems Association (IFSA), the Japan Society for Fuzzy Theory and Systems, and the Microelectronics and Computer Technology Corporation (MCC). The fuzzy set theory has led to a large number of diverse applications. Recently, interesting applications have been developed which involve the integration of fuzzy systems with adaptive processes such a neural networks and genetic algorithms. NAFIPS '92 was directed toward the advancement, commercialization, and engineering development of these technologies
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