On Yager and Hamacher t-Norms and Fuzzy Metric Spaces

Recently, Gregori et al. have discussed (Fuzzy Sets Syst 2011;161:2193 2205) the so-called strong fuzzy metrics when looking for a class of completable fuzzy metric spaces in the sense of George and Veeramani and state the question of finding a non-strong fuzzy metric space for a continuous t-norm different from the minimum. Later on, Gutíerrez-García and Romaguera solved this question (Fuzzy Sets Syst 2011;162:91 93) by means of two examples for the product and the Lukasiewicz t-norm, respectively. In this direction, they posed to find further examples of nonstrong fuzzy metrics for continuous t-norms that are greater than the product but different from minimum. In this paper, we found an example of this kind. On the other hand, Tirado established (Fixed Point Theory 2012;13:273 283) a fixed-point theorem in fuzzy metric spaces, which was successfully used to prove the existence and uniqueness of solution for the recurrence equation associated with the probabilistic divide and conquer algorithms. Here, we generalize this result by using a class of continuous t-norms known as ω-Yager t-norms.The second author acknowledges the support of the Ministry of Economy and Competitiveness of Spain under grant MTM2012-37894-C02-01 and the support of Universitat Politecnica de Valencia under grant PAID-06-12-SP20120471.Castro Company, F.; Tirado Peláez, P. (2014). On Yager and Hamacher t-Norms and Fuzzy Metric Spaces. International Journal of Intelligent Systems. 29:1173-1180. https://doi.org/10.1002/int.21688S1173118029Sherwood, H. (1966). On the completion of probabilistic metric spaces. Zeitschrift f�r Wahrscheinlichkeitstheorie und Verwandte Gebiete, 6(1), 62-64. doi:10.1007/bf00531809Gregori, V. (2002). On completion of fuzzy metric spaces. Fuzzy Sets and Systems, 130(3), 399-404. doi:10.1016/s0165-0114(02)00115-xGregori, V., Morillas, S., & Sapena, A. (2010). On a class of completable fuzzy metric spaces. Fuzzy Sets and Systems, 161(16), 2193-2205. doi:10.1016/j.fss.2010.03.013Gutiérrez García, J., & Romaguera, S. (2011). Examples of non-strong fuzzy metrics. Fuzzy Sets and Systems, 162(1), 91-93. doi:10.1016/j.fss.2010.09.017Yager, R. R. (1980). On a general class of fuzzy connectives. Fuzzy Sets and Systems, 4(3), 235-242. doi:10.1016/0165-0114(80)90013-5Castro-Company, F., & Tirado, P. (2012). Some classes of t-norms and fuzzy metric spaces. doi:10.1063/1.4756272George, A., & Veeramani, P. (1994). On some results in fuzzy metric spaces. Fuzzy Sets and Systems, 64(3), 395-399. doi:10.1016/0165-0114(94)90162-7Hadžić, O., & Pap, E. (2001). Fixed Point Theory in Probabilistic Metric Spaces. doi:10.1007/978-94-017-1560-7Klement, E. P., Mesiar, R., & Pap, E. (2000). Triangular Norms. Trends in Logic. doi:10.1007/978-94-015-9540-

Fuzzy Quasi-Metric Spaces: Bicompletion, Contractions on Product Spaces, and Applications to Access Predictions

Desde que L.A. Zadeh presentó la teoría de conjuntos difusos en 1965, esta se ha usado en una amplia serie de áreas de las matemáticas y se ha aplicado en una gran variedad de escenarios de la vida real. Estos escenarios cubren procesos complejos sin modelo matemático sencillo tales como dispositivos de control industrial, reconocimiento de patrones o sistemas que gestionen información imprecisa o altamente impredecible. La topología difusa es un importante ejemplo de uso de la teoría de L.A. Zadeh. Durante años, los autores de este campo han buscado obtener la definición de un espacio métrico difuso para medir la distancia entre elementos según grados de proximidad. El presente trabajo trata acerca de la bicompletación de espacios casi-métricos difusos en el sentido de Kramosil y Michalek. Sherwood probó que todo espacio métrico difuso admite completación que es única excepto por isometría basándose en propiedades de la métrica de Lévy. Probamos aquí que todo espacio casi-métrico difuso tiene bicompletación usando directamente el supremo de conjuntos en [0,1] y límites inferiores de secuencias en [0,1] en lugar de usar la métrica de Lévy. Aprovechamos tanto la bicompletitud y bicompletación de espacios casi-métricos difusos como las propiedades de los espacios métricos difusos y difusos intuicionistas para presentar varias aplicaciones a problemas del campo de la informática. Así estudiamos la existencia y unicidad de solución para las ecuaciones de recurrencia asociadas a ciertos algoritmos formados por dos procedimientos recursivos. Para analizar su complejidad aplicamos el principio de contracción de Banach tanto en un producto de casi-métricas no-Arquimedianas en el dominio de las palabras como en la casi-métrica producto de dos espacios de complejidad casi-métricos de Schellekens. Estudiamos también una aplicación de espacios métricos difusos a sistemas de información basados en localidad de accesos.Castro Company, F. (2010). Fuzzy Quasi-Metric Spaces: Bicompletion, Contractions on Product Spaces, and Applications to Access Predictions [Tesis doctoral no publicada]. Universitat Politècnica de València. https://doi.org/10.4995/Thesis/10251/8420Palanci

Comparison of different T-norm operators in classification problems

Fuzzy rule based classification systems are one of the most popular fuzzy modeling systems used in pattern classification problems. This paper investigates the effect of applying nine different T-norms in fuzzy rule based classification systems. In the recent researches, fuzzy versions of confidence and support merits from the field of data mining have been widely used for both rules selecting and weighting in the construction of fuzzy rule based classification systems. For calculating these merits the product has been usually used as a T-norm. In this paper different T-norms have been used for calculating the confidence and support measures. Therefore, the calculations in rule selection and rule weighting steps (in the process of constructing the fuzzy rule based classification systems) are modified by employing these T-norms. Consequently, these changes in calculation results in altering the overall accuracy of rule based classification systems. Experimental results obtained on some well-known data sets show that the best performance is produced by employing the Aczel-Alsina operator in terms of the classification accuracy, the second best operator is Dubois-Prade and the third best operator is Dombi. In experiments, we have used 12 data sets with numerical attributes from the University of California, Irvine machine learning repository (UCI).Comment: 6 pages, 1 figure, 4 tables; International Journal of Fuzzy Logic Systems (IJFLS) Vol.2, No.3, July 201

Power of Continuous Triangular Norms with Application to Intuitionistic Fuzzy Information Aggregation

The paper aims to investigate the power operation of continuous triangular norms (t-norms) and develop some intuitionistic fuzzy information aggregation methods. It is proved that a continuous t-norm is power stable if and only if every point is a power stable point, and if and only if it is the minimum t-norm, or it is strict, or it is an ordinal sum of strict t-norms. Moreover, the representation theorem of continuous t-norms is used to obtain the computational formula for the power of continuous t-norms. Based on the power operation of t-norms, four fundamental operations induced by a continuous t-norm for the intuitionistic fuzzy (IF) sets are introduced. Furthermore, various aggregation operators, namely the IF weighted average (IFWA), IF weighted geometric (IFWG), and IF mean weighted average and geometric (IFMWAG) operators, are defined, and their properties are analyzed. Finally, a new decision-making algorithm is designed based on the IFMWAG operator, which can remove the hindrance of indiscernibility on the boundaries of some classical aggregation operators. The practical applicability, comparative analysis, and advantages of the study with other decision-making methods are furnished to ascertain the efficacy of the designed method

A Fuzzy Rule Based Approach to Geographic Classification of Virgin Olive Oil Using T-Operators

Olive oil is an important agricultural food product. Especially, protected designation of origin (PDO) and protected geographic indications (PGI) are useful to protect the intellectual property rights of the consumers and producers. For this reason, the importance of the geographic classification increases to trace geographical indications. This chapter suggests a geographical classification system for the virgin olive oils. This system is formed on chemical parameters. These parameters include fuzziness. Novel proposed system constructs the rules by using fuzzy decision tree algorithm. It produces rules over fuzzy ID3 algorithm. It uses fuzzy entropy on the fuzzified data. The reasoning procedure depends on weighted rule-based system and is adapted into the fuzzy reasoning handled with different T-operators. Fuzzification is performed with fuzzy c-means algorithm for the olive oil data set. The cluster numbers of each variable are selected based on partition coefficient validity criteria. The model is examined by using different decision tree approaches (C4.5 and standard version fuzzy ID3 algorithm) and FID3 reasoning method with eight different T-operators. Also, the conclusions are supported by statistical analysis. Experimental results support that the weights have important manner on fuzzy reasoning method for the geographic classification system

A Banach contraction principle in fuzzy metric spaces defined by means of t-conorms

[EN] Fixed point theory in fuzzy metric spaces has grown to become an intensive field of research. The difficulty of demonstrating a fixed point theorem in such kind of spaces makes the authors to demand extra conditions on the space other than completeness. In this paper, we introduce a new version of the celebrated Banach contracion principle in the context of fuzzy metric spaces. It is defined by means of t-conorms and constitutes an adaptation to the fuzzy context of the mentioned contracion principle more "faithful" than the ones already defined in the literature. In addition, such a notion allows us to prove a fixed point theorem without requiring any additional condition on the space apart from completeness. Our main result (Theorem 1) generalizes another one proved by Castro-Company and Tirado. Besides, the celebrated Banach fixed point theorem is obtained as a corollary of Theorem 1.Juan-José Miñana acknowledges financial support from FEDER/Ministerio de Ciencia, Innovación y Universidades-Agencia Estatal de Investigación/¿Proyecto PGC2018-095709-B-C21. This work is also partially supported by Programa Operatiu FEDER 2014-2020 de les Illes Balears, by project PROCOE/4/2017 (Direcció General d¿Innovació i Recerca, Govern de les Illes Balears) and by projects ROBINS and BUGWRIGHT2. These two latest projects have received funding from the European Union¿s Horizon 2020 research and innovation programme under grant agreements No 779776 and No 871260, respectively. This publication reflects only the authors views and the European Union is not liable for any use that may be made of the information contained therein. Valentín Gregori acknowledges the support of Generalitat Valenciana under grant AICO-2020-136.Gregori Gregori, V.; Miñana, J. (2021). A Banach contraction principle in fuzzy metric spaces defined by means of t-conorms. Revista de la Real Academia de Ciencias Exactas Físicas y Naturales Serie A Matemáticas. 115(3):1-11. https://doi.org/10.1007/s13398-021-01068-6S111115

Evolutionary Design for Computational Visual Attention

A new framework for simulating the visual attention system in primates is introduced. The proposed architecture is an abstraction of existing approaches influenced by the work of Koch and Ullman, and Tompa. Each stage of the attentional hierarchy is chosen with consideration for both psychophysics and mathematical optimality. A set of attentional operators are derived that act on basic image channels of intensity, hue and orientation to produce maps representing perceptual importance of each image pixel. The development of such operators is realized within the context of a genetic optimization. The model includes the notion of an information domain where feature maps are transformed to a domain that more closely corresponds to the human visual system. A careful analysis of various issues including feature extraction, density estimation and data fusion is presented within the context of the visual attention problem