A mathematical study of fuzzy logic; an algebraic approach

Fuzzy set theory and Fuzzy logic is studied from a mathematical point of view. The main goal is to investigatecommon mathematical structures in various fuzzy logical inference systems and to establish a general mathematical basis for fuzzy logic when considered as multi-valued logic. The study is composed of six distinct publications. The first paper deals with Mattila'sLPC+Ch Calculus. THis fuzzy inference system is an attempt to introduce linguistic objects to mathematical logic without defining these objects mathematically.LPC+Ch Calculus is analyzed from algebraic point of view and it is demonstratedthat suitable factorization of the set of well formed formulae (in fact, Lindenbaum algebra) leads to a structure called ET-algebra and introduced in the beginning of the paper. On its basis, all the theorems presented by Mattila and many others can be proved in a simple way which is demonstrated in the Lemmas 1 and 2and Propositions 1-3. The conclusion critically discusses some other issues of LPC+Ch Calculus, specially that no formal semantics for it is given.In the second paper the characterization of solvability of the relational equation RoX=T, where R, X, T are fuzzy relations, X the unknown one, and o the minimum-induced composition by Sanchez, is extended to compositions induced by more general products in the general value lattice. Moreover, the procedure also applies to systemsof equations. In the third publication common features in various fuzzy logicalsystems are investigated. It turns out that adjoint couples and residuated lattices are very often present, though not always explicitly expressed. Some minor new results are also proved.The fourth study concerns Novak's paper, in which Novak introduced first-order fuzzy logic and proved, among other things, the semantico-syntactical completeness of this logic. He also demonstrated that the algebra of his logic is a generalized residuated lattice. In proving that the examination of Novak's logic can be reduced to the examination of locally finite MV-algebras.In the fifth paper a multi-valued sentential logic with values of truth in an injective MV-algebra is introduced and the axiomatizability of this logic is proved. The paper developes some ideas of Goguen and generalizes the results of Pavelka on the unit interval. Our proof for the completeness is purely algebraic. A corollary of the Completeness Theorem is that fuzzy logic on the unit interval is semantically complete if, and only if the algebra of the valuesof truth is a complete MV-algebra. The Compactness Theorem holds in our well-defined fuzzy sentential logic, while the Deduction Theorem and the Finiteness Theorem do not. Because of its generality and good-behaviour, MV-valued logic can be regarded as a mathematical basis of fuzzy reasoning. The last paper is a continuation of the fifth study. The semantics and syntax of fuzzy predicate logic with values of truth in ana injective MV-algerba are introduced, and a list of universally valid sentences is established. The system is proved to be semanticallycomplete. This proof is based on an idea utilizing some elementary properties of injective MV-algebras and MV-homomorphisms, and is purely algebraic.

Video activity extraction and reporting with incremental unsupervised learning

The present work presents a new method for activity extraction and reporting from video based on the aggregation of fuzzy relations. Trajectory clustering is first employed mainly to discover the points of entry and exit of mobiles appearing in the scene. In a second step, proximity relations between resulting clusters of detected mobiles and contextual elements from the scene are modeled employing fuzzy relations. These can then be aggregated employing typical soft-computing algebra. A clustering algorithm based on the transitive closure calculation of the fuzzy relations allows building the structure of the scene and characterises the ongoing different activities of the scene. Discovered activity zones can be reported as activity maps with different granularities thanks to the analysis of the transitive closure matrix. Taking advantage of the soft relation properties, activity zones and related activities can be labeled in a more human-like language. We present results obtained on real videos corresponding to apron monitoring in the Toulouse airport in France.

Traffic flow breakdown prediction using feature reduction through Rough-Neuro fuzzy Networks

The prediction of the traffic behavior could help to make decision about the routing process, as well as enables gains on effectiveness and productivity on the physical distribution. This need motivated the search for technological improvements in the Routing performance in metropolitan areas. The purpose of this paper is to present computational evidences that Artificial Neural Network ANN could be use to predict the traffic behavior in a metropolitan area such So Paulo (around 16 million inhabitants). The proposed methodology involves the application of Rough-Fuzzy Sets to define inference morphology for insertion of the behavior of Dynamic Routing into a structured rule basis, without human expert aid. The dynamics of the traffic parameters are described through membership functions. Rough Sets Theory identifies the attributes that are important, and suggest Fuzzy relations to be inserted on a Rough Neuro Fuzzy Network (RNFN) type Multilayer Perceptron (MLP) and type Radial Basis Function (RBF), in order to get an optimal surface response. To measure the performance of the proposed RNFN, the responses of the unreduced rule basis are compared with the reduced rule one. The results show that by making use of the Feature Reduction through RNFN, it is possible to reduce the need for human expert in the construction of the Fuzzy inference mechanism in such flow process like traffic breakdown. © 2011 IEEE.

Teoria de conjuntos fuzzy e aplicações

Pós-graduação em Matemática Universitária - IGCE

Coverage and invariability in fuzzy systems

This paper presents a new complex system systemic. Here, we are working in a fuzzy environment, so we have to adapt all the previous concepts and results that were obtained in a non-fuzzy environment, for this fuzzy case. The direct and indirect influences between variables will provide the basis for obtaining fuzzy and/or non-fuzzy relationships, so that the concepts of coverage and invariability between sets of variables will appear naturally. These two concepts and their interconnections will be analyzed from the viewpoint of algebraic properties of inclusion, union and intersection (fuzzy and non-fuzzy), and also for the loop concept, which, as we shall see, will be of special importance.

Invariability, orbits and fuzzy attractors

In this paper, we present a generalization of a new systemic approach to abstract fuzzy systems. Using a fuzzy relations structure will retain the information provided by degrees of membership. In addition, to better suit the situation to be modelled, it is advisable to use T-norm or T-conorm distinct from the minimum and maximum, respectively. This gain in generality is due to the completeness of the work on a higher level of abstraction. You cannot always reproduce the results obtained previously, and also sometimes different definitions with different views are obtained. In any case this approach proves to be much more effective when modelling reality.

A Fuzzy Decision Strategy for Topic Identification and Dynamic Selection of Language Models

International audience

Linguistic evaluation method for the environmental assessment of infrastructure projects

A soft linguistic evaluation method is proposed for the environmental assessment of physical infrastructure projects based on fuzzy relations. Infrastructure projects are characterized in terms of linguistic expressions of 'performance' with respect to factors or impacts and the 'importance' of those factors/impacts. A simple example is developed to illustrate the method in the context of three road infrastructure projects assessed against five factors/impacts. In addition, a means to include hard or crisp factors is presented and illustrated with respect to a sixth factor.

Soft Computing Methods in Metallurgy

In metallurgic plants a high quality metal production is always required. Nowadays soft computing applications are more often used for automation of manufacturing process and quality control instead of mechanical techniques. In this thesis an overview of soft computing methods presents. As an example of soft computing application, an effective model of fuzzy expert system for the automotive quality control of steel degassing process was developed. The purpose of this work is to describe the fuzzy relations as quality hypersurfaces by varying number of linguistic variables and fuzzy sets.

Évaluation d'une expérience de thérapie de groupe avec des épouses d'alcooliques

[…] À travers ces trois années passées auprès des alcooliques et leurs conjoints, nous avons constaté que la très grande majorité de ces individus avait un dénominateur commun: l'isolement. En effet, à la longue, l'alcoolisme fausse et brise les relations de l'alcoolique avec sa famille, son entourage, ses amis, son milieu de travail, son milieu de loisirs. Il y a bien les "amis" de taverne, mais de l'aveu même des alcooliques, il n'existe pas là de relations humaines interpersonnelles satisfaisantes, mais ce sont plutôt des relations d'intérêt, car ce qui réunit d'abord et avant tout ces gens, c'est leur besoin de boire. Et petit à petit, la famille de l'alcoolique se retrouve elle aussi isolée socialement.[…] Nous avons voulu, dans la présente étude, évaluer les résultats d'une expérience de thérapie de groupe avec des épouses d'alcooliques qui s'est déroulée à la clinique externe d'alcoologie du Centre Hospitalier St-Vincent de Paul. De façon plus spécifique, nous avons voulu, 1) mesurer les changements survenus au niveau des connaissances et des attitudes des épouses qui auraient participé à la thérapie de groupe, 2) voir jusqu'où les changements survenus étaient associés à certaines caractéristiques personnelles des participantes, et 3) voir s'il y avait une relation entre les changements survenus au niveau des connaissances et les changements survenus au niveau des attitudes. L'intérêt de la présente étude était double: d'une part, elle nous permettait d'évaluer une expérience de thérapie de groupe avec des épouses d'alcooliques selon les critères d'une recherche scientifique et de situer la thérapie de groupe en terme d'efficacité à l'intérieur des services offerts par la clinique où nous travaillions. D'autre part, étant donné la rareté de telles études dans ce domaine de l'alcoolisme (voir annexe A), la présente étude pouvait certes mieux faire connaître la thérapie de groupe avec des épouses d'alcooliques, illustrer la valeur de celle-ci et enfin, fournir des données susceptibles de contribuer au développement de ce mode d'intervention. […]

Para lá dos jornalistas: influencer relations, uma nova era de Media Relations

Dissertação apresentada à Escola Superior de Comunicação Social como parte dos requisitos para obtenção de grau de mestre em Gestão Estratégica das Relações Públicas.

Intra-relations of African canaries, genus Serinus [by] Austin L. Rand.

v.51:no.9(1968)