Computing a T-transitive lower approximation or opening of a proximity relation

Fuzzy Sets and Systems. IMPACT FACTOR: 1,181. Fuzzy Sets and Systems. IMPACT FACTOR: 1,181. Since transitivity is quite often violated even by decision makers that accept transitivity in their preferences as a condition for consistency, a standard approach to deal with intransitive preference elicitations is the search for a close enough transitive preference relation, assuming that such a violation is mainly due to decision maker estimation errors. In some way, the more number of elicitations, the more probable inconsistency is. This is mostly the case within a fuzzy framework, even when the number of alternatives or object to be classified is relatively small. In this paper we propose a fast method to compute a T-indistinguishability from a reflexive and symmetric fuzzy relation, being T any left-continuous t-norm. The computed approximation we propose will take O(n3) time complexity, where n is the number of elements under consideration, and is expected to produce a T-transitive opening. To the authors¿ knowledge, there are no other proposed algorithm that computes T-transitive lower approximations or openings while preserving the reflexivity and symmetry properties

An algorithm to compute the transitive closure, a transitive approximation and a transitive opening of a fuzzy proximity

A method to compute the transitive closure, a transitive opening and a transitive approximation of a reflexive and symmetric fuzzy relation is given. Other previous methods in literature compute just the transitive closure, some transitive approximations or some transitive openings. The proposed algorithm computes the three different similarities that approximate a proximity for the computational cost of computing just one. The shape of the binary partition tree for the three output similarities are the same.Peer ReviewedPostprint (published version

Fifty years of similarity relations: a survey of foundations and applications

On the occasion of the 50th anniversary of the publication of Zadeh's significant paper Similarity Relations and Fuzzy Orderings, an account of the development of similarity relations during this time will be given. Moreover, the main topics related to these fuzzy relations will be reviewed.Peer ReviewedPostprint (author's final draft

Knowledge Accumulation of Microbial Data Aiming at a Dynamic Taxonomic Framework

Deze thesis is een poging om precies dit onderzoeksgebied te overbruggen dat ligt tussen ruw gegeven en abstract concept, tussen praktijk en theorie, binnen het kader van de hedendaagse bacteriële taxonomie. Als gevolg hiervan is het een kruisbestuiving geworden tussen microbiologie, wiskunde en computerwetenschappen. De kunst om het landschap van de bacteriële diversiteit uit te tekenen, gebruikt als een metafoor voor het modelleren van de taxonomie, vereist het bepalen van een representatieve waaier aan reproduceerbare en vergelijkbare experimentele kenmerken van een verzameling bacteriën (microbiologie/taxonomie), het ontwerpen en implementeren van objectieve classificatiemethodes voor het groeperen van gegevens op een niet gecoördineerde manier (wiskunde/classificatie) en het consolideren van experimentele gegevens en hun verschillende onderverdelingen via een uniforme en weldoordachte aanpak (computerwetenschappen/kennisbeheer). Men kan zich gemakkelijk een globaal kennissysteem voor de geest halen dat de vellen vol experimentele gegevens die voortspruiten uit de microbiologische onderzoeksverrichtingen op een gestructureerde en geüniformiseerde manier kan absorberen. Een dergelijk kennisbeheersysteem zou een ongelofelijke vooruitgang betekenen voor de mogelijke toepassing van intelligente en goed gefundeerde methodes voor het ontginnen van de gegevens, ingezet als hulpmiddel om het afbakenen van objectieve en universele taxonomische consensusmodellen op een betere manier te stroomlijnen en te automatiseren. Bovendien kunnen dergelijke inferentiesystemen in staat worden geacht om ogenblikkelijk te reageren op een toevloed van nieuwe gegevens en interactief te communiceren met de buitenwereld indien noodzakelijke stukken voor het vervolledigen van de taxonomische puzzel zouden ontbreken. De geldigheid van nieuwe inzichten of hypothesen omtrent het leven en de evolutie van bacteriën zou onmiddellijk kunnen getoetst worden aan deze vergaarbakken vol kennis, mogelijks met een directe aanpassing van bestaande taxonomische modellen tot gevolg. Vooraleer de betrachtingen van een autodidactisch inferentiesysteem voor het uittekenen van het landschap van de bacteriële diversiteit kunnen gerealiseerd worden, moeten belangrijke technische en organisatorische hindernissen overwonnen worden. Dit vraagt het verleggen van de grenzen van een mondiale uitwisseling van gegevens, het nasporen en invullen van de hiaten in de waarnemingen, en het verkennen van de mogelijkheden van nieuwe technieken voor het ontginnen van gegevens, ten voordele van een beter inzicht in het leven en de evolutie van bacteriën. Spijts de nog vele onopgeloste kwesties, kunnen de ideeën die worden aangebracht in deze verhandeling als stimulans en leidraad dienen bij het integreren en exploiteren van microbiële gegevens, in plaats van het blijvend koesteren van een ijdele hoo

On FPGA implementations for bioinformatics, neural prosthetics and reinforcement learning problems.

Mak Sui Tung Terrence.Thesis (M.Phil.)--Chinese University of Hong Kong, 2005.Includes bibliographical references (leaves 132-142).Abstracts in English and Chinese.Abstract --- p.iList of Tables --- p.ivList of Figures --- p.vAcknowledgements --- p.ixChapter 1. --- Introduction --- p.1Chapter 1.1 --- Bioinformatics --- p.1Chapter 1.2 --- Neural Prosthetics --- p.4Chapter 1.3 --- Learning in Uncertainty --- p.5Chapter 1.4 --- The Field Programmable Gate Array (FPGAs) --- p.7Chapter 1.5 --- Scope of the Thesis --- p.10Chapter 2. --- A Hybrid GA-DP Approach for Searching Equivalence Sets --- p.14Chapter 2.1 --- Introduction --- p.16Chapter 2.2 --- Equivalence Set Criterion --- p.18Chapter 2.3 --- Genetic Algorithm and Dynamic Programming --- p.19Chapter 2.3.1 --- Genetic Algorithm Formulation --- p.20Chapter 2.3.2 --- Bounded Mutation --- p.21Chapter 2.3.3 --- Conditioned Crossover --- p.22Chapter 2.3.4 --- Implementation --- p.22Chapter 2.4 --- FPGAs Implementation of GA-DP --- p.24Chapter 2.4.1 --- System Overview --- p.25Chapter 2.4.2 --- Parallel Computation for Transitive Closure --- p.26Chapter 2.4.3 --- Genetic Operation Realization --- p.28Chapter 2.5 --- Discussion --- p.30Chapter 2.6 --- Limitation and Future Work --- p.33Chapter 2.7 --- Conclusion --- p.34Chapter 3. --- An FPGA-based Architecture for Maximum-Likelihood Phylogeny Evaluation --- p.35Chapter 3.1 --- Introduction --- p.36Chapter 3.2 --- Maximum-Likelihood Model --- p.39Chapter 3.3 --- Hardware Mapping for Pruning Algorithm --- p.41Chapter 3.3.1 --- Related Works --- p.41Chapter 3.3.2 --- Number Representation --- p.42Chapter 3.3.3 --- Binary Tree Representation --- p.43Chapter 3.3.4 --- Binary Tree Traversal --- p.45Chapter 3.3.5 --- Maximum-Likelihood Evaluation Algorithm --- p.46Chapter 3.4 --- System Architecture --- p.49Chapter 3.4.1 --- Transition Probability Unit --- p.50Chapter 3.4.2 --- State-Parallel Computation Unit --- p.51Chapter 3.4.3 --- Error Computation --- p.54Chapter 3.5 --- Discussion --- p.56Chapter 3.5.1 --- Hardware Resource Consumption --- p.56Chapter 3.5.2 --- Delay Evaluation --- p.57Chapter 3.6 --- Conclusion --- p.59Chapter 4. --- Field Programmable Gate Array Implementation of Neuronal Ion Channel Dynamics --- p.61Chapter 4.1 --- Introduction --- p.62Chapter 4.2 --- Background --- p.63Chapter 4.2.1 --- Analog VLSI Model for Hebbian Synapse --- p.63Chapter 4.2.2 --- A Unifying Model of Bi-directional Synaptic Plasticity --- p.64Chapter 4.2.3 --- Non-NMDA Receptor Channel Regulation --- p.65Chapter 4.3 --- FPGAs Implementation --- p.65Chapter 4.3.1 --- FPGA Design Flow --- p.65Chapter 4.3.2 --- Digital Model of NMD A and AMPA receptors --- p.65Chapter 4.3.3 --- Synapse Modification --- p.67Chapter 4.4 --- Results --- p.68Chapter 4.4.1 --- Simulation Results --- p.68Chapter 4.5 --- Discussion --- p.70Chapter 4.6 --- Conclusion --- p.71Chapter 5. --- Continuous-Time and Discrete-Time Inference Networks for Distributed Dynamic Programming --- p.72Chapter 5.1 --- Introduction --- p.74Chapter 5.2 --- Background --- p.77Chapter 5.2.1 --- Markov decision process (MDPs) --- p.78Chapter 5.2.2 --- Learning in the MDPs --- p.80Chapter 5.2.3 --- Bellman Optimal Criterion --- p.80Chapter 5.2.4 --- Value Iteration --- p.81Chapter 5.3 --- A Computational Framework for Continuous-Time Inference Network --- p.82Chapter 5.3.1 --- Binary Relation Inference Network --- p.83Chapter 5.3.2 --- Binary Relation Inference Network for MDPs --- p.85Chapter 5.3.3 --- Continuous-Time Inference Network for MDPs --- p.87Chapter 5.4 --- Convergence Consideration --- p.88Chapter 5.5 --- Numerical Simulation --- p.90Chapter 5.5.1 --- Example 1: Random Walk --- p.90Chapter 5.5.2 --- Example 2: Random Walk on a Grid --- p.94Chapter 5.5.3 --- Example 3: Stochastic Shortest Path Problem --- p.97Chapter 5.5.4 --- Relationships Between λ and γ --- p.99Chapter 5.6 --- Discrete-Time Inference Network --- p.100Chapter 5.6.1 --- Results --- p.101Chapter 5.7 --- Conclusion --- p.102Chapter 6. --- On Distributed g-Learning Network --- p.104Chapter 6.1 --- Introduction --- p.105Chapter 6.2 --- Distributed Q-Learniing Network --- p.108Chapter 6.2.1 --- Distributed Q-Learning Network --- p.109Chapter 6.2.2 --- Q-Learning Network Architecture --- p.111Chapter 6.3 --- Experimental Results --- p.114Chapter 6.3.1 --- Random Walk --- p.114Chapter 6.3.2 --- The Shortest Path Problem --- p.116Chapter 6.4 --- Discussion --- p.120Chapter 6.4.1 --- Related Work --- p.121Chapter 6.5 --- FPGAs Implementation --- p.122Chapter 6.5.1 --- Distributed Registering Approach --- p.123Chapter 6.5.2 --- Serial BRAM Storing Approach --- p.124Chapter 6.5.3 --- Comparison --- p.125Chapter 6.5.4 --- Discussion --- p.127Chapter 6.6 --- Conclusion --- p.128Chapter 7. --- Summary --- p.129Bibliography --- p.132AppendixChapter A. --- Simplified Floating-Point Arithmetic --- p.143Chapter B. --- "Logarithm, Exponential and Division Implementation" --- p.144Chapter B.1 --- Introduction --- p.144Chapter B.2 --- Approximation Scheme --- p.145Chapter B.2.1 --- Logarithm --- p.145Chapter B.2.2 --- Exponentiation --- p.147Chapter B.2.3 --- Division --- p.148Chapter C. --- Analog VLSI Implementation --- p.150Chapter C.1 --- Site Function --- p.150Chapter C.1.1 --- Multiplication Cell --- p.150Chapter C.2 --- The Unit Function --- p.153Chapter C.3 --- The Inference Network Computation --- p.154Chapter C.4 --- Layout --- p.157Chapter C.5 --- Fabrication --- p.159Chapter C.5.1 --- Testing and Characterization --- p.16

Fuzzy Sets, Fuzzy Logic and Their Applications

The present book contains 20 articles collected from amongst the 53 total submitted manuscripts for the Special Issue “Fuzzy Sets, Fuzzy Loigic and Their Applications” of the MDPI journal Mathematics. The articles, which appear in the book in the series in which they were accepted, published in Volumes 7 (2019) and 8 (2020) of the journal, cover a wide range of topics connected to the theory and applications of fuzzy systems and their extensions and generalizations. This range includes, among others, management of the uncertainty in a fuzzy environment; fuzzy assessment methods of human-machine performance; fuzzy graphs; fuzzy topological and convergence spaces; bipolar fuzzy relations; type-2 fuzzy; and intuitionistic, interval-valued, complex, picture, and Pythagorean fuzzy sets, soft sets and algebras, etc. The applications presented are oriented to finance, fuzzy analytic hierarchy, green supply chain industries, smart health practice, and hotel selection. This wide range of topics makes the book interesting for all those working in the wider area of Fuzzy sets and systems and of fuzzy logic and for those who have the proper mathematical background who wish to become familiar with recent advances in fuzzy mathematics, which has entered to almost all sectors of human life and activity

Uncertainty and indistinguishability. Application to modelling with words.

El concepte d'igualtat és fonamental en qualsevol teoria donat que és una noció essencial a l'hora de discernir entre els elements objecte del seu estudi i possibilitar la definició de mecanismes de classificació.Quan totes les propietats són perfectament precises (absència d'incertesa), hom obtè la igualtat clàssica a on dos objectes són considerats iguals si i només si comparteixen el mateix conjunt de propietats. Però, què passa quan considerem l'aparició d'incertesa, com en el cas a on els objectes compleixen una determinada propietat només fins a un cert grau?. Llavors, donat que alguns objectes seran més similars entre si que d'altres, sorgeix la necessitat de una noció gradual del concepte d'igualtat.Aquestes consideracions refermen la idea de que certs contextos requereixen una definició més flexible, que superi la rigidesa de la noció clàssica d'igualtat. Els operadors de T-indistingibilitat semblen bons candidats per aquest nou tipus d'igualtat que cerquem.D'altra banda, La Teoria de l'Evidència de Dempster-Shafer, com a marc pel tractament d'evidències, defineix implícitament una noció d'indistingibilitat entre els elements del domini de discurs basada en la seva compatibilitat relativa amb l'evidència considerada. El capítol segon analitza diferents mètodes per definir l'operador de T-indistingibilitat associat a una evidència donada.En el capítol tercer, després de presentar un exhaustiu estat de l'art en mesures d'incertesa, ens centrem en la qüestió del còmput de l'entropia quan sobre els elements del domini s'ha definit una relació d'indistingibilitat. Llavors, l'entropia hauria de ser mesurada no en funció de l'ocurrència d'events diferents, sinó d'acord amb la variabilitat percebuda per un observador equipat amb la relació d'indistingibilitat considerada. Aquesta interpretació suggereix el "paradigma de l'observador" que ens porta a la introducció del concepte d'entropia observacional.La incertesa és un fenomen present al món real. El desenvolupament de tècniques que en permetin el tractament és doncs, una necessitat. La 'computació amb paraules' ('computing with words') pretén assolir aquest objectiu mitjançant un formalisme basat en etiquetes lingüístiques, en contrast amb els mètodes numèrics tradicionals. L'ús d'aquestes etiquetes millora la comprensibilitat del llenguatge de representació delconeixement, a l'hora que requereix una adaptació de les tècniques inductives tradicionals.En el quart capítol s'introdueix un nou tipus d'arbre de decisió que incorpora les indistingibilitats entre elements del domini a l'hora de calcular la impuresa dels nodes. Hem anomenat arbres de decisió observacionals a aquests nou tipus, donat que es basen en la incorporació de l'entropia observacional en la funció heurística de selecció d'atributs. A més, presentem un algorisme capaç d'induir regles lingüístiques mitjançant un tractament adient de la incertesa present a les etiquetes lingüístiques o a les dades mateixes. La definició de l'algorisme s'acompanya d'una comparació formal amb altres algorismes estàndards.The concept of equality is a fundamental notion in any theory since it is essential to the ability of discerning the objects to whom it concerns, ability which in turn is a requirement for any classification mechanism that might be defined. When all the properties involved are entirely precise, what we obtain is the classical equality, where two individuals are considered equal if and only if they share the same set of properties. What happens, however, when imprecision arises as in the case of properties which are fulfilled only up to a degree? Then, because certain individuals will be more similar than others, the need for a gradual notion of equality arises.These considerations show that certain contexts that are pervaded with uncertainty require a more flexible concept of equality that goes beyond the rigidity of the classic concept of equality. T-indistinguishability operators seem to be good candidates for this more flexible and general version of the concept of equality that we are searching for.On the other hand, Dempster-Shafer Theory of Evidence, as a framework for representing and managing general evidences, implicitly conveys the notion of indistinguishability between the elements of the domain of discourse based on their relative compatibility with the evidence at hand. In chapter two we are concerned with providing definitions for the T-indistinguishability operator associated to a given body of evidence.In chapter three, after providing a comprehensive summary of the state of the art on measures of uncertainty, we tackle the problem of computing entropy when an indistinguishability relation has been defined over the elements of the domain. Entropy should then be measured not according to the occurrence of different events, but according to the variability perceived by an observer equipped with indistinguishability abilities as defined by the indistinguishability relation considered. This idea naturally leads to the introduction of the concept of observational entropy.Real data is often pervaded with uncertainty so that devising techniques intended to induce knowledge in the presence of uncertainty seems entirely advisable.The paradigm of computing with words follows this line in order to provide a computation formalism based on linguistic labels in contrast to traditional numerical-based methods.The use of linguistic labels enriches the understandability of the representation language, although it also requires adapting the classical inductive learning procedures to cope with such labels.In chapter four, a novel approach to building decision trees is introduced, addressing the case when uncertainty arises as a consequence of considering a more realistic setting in which decision maker's discernment abilities are taken into account when computing node's impurity measures. This novel paradigm results in what have been called --observational decision trees' since the main idea stems from the notion of observational entropy in order to incorporate indistinguishability concerns. In addition, we present an algorithm intended to induce linguistic rules from data by properly managing the uncertainty present either in the set of describing labels or in the data itself. A formal comparison with standard algorithms is also provided