A version of the Stone-Weierstrass theorem in fuzzy analysis

Let C ( K , E 1 ) be the space of continuous functions defined between a compact Hausdorff space K and the space of fuzzy numbers E 1 endowed with the supremum metric. We provide a set of sufficient conditions on a subspace of C ( K , E 1 ) in order that it be dense. We also obtain a similar result for interpolating families of C ( K , E 1 ) . As a corollary of the above results we prove that certain fuzzy-number-valued neural networks can approximate any continuous fuzzy-number-valued function defined on a compact subspace of R

Completeness, metrizability and compactness in spaces of fuzzy-number-valued functions

Fuzzy-number-valued functions, that is, functions defined on a topological space taking values in the space of fuzzy numbers, play a central role in the development of Fuzzy Analysis. In this paper we study completeness, metrizability and compactness of spaces of continuous fuzzy-number-valued functions

Gromov–Hausdorff convergence of non-Archimedean fuzzy metric spaces

We introduce the notion of the Gromov–Hausdorff fuzzy distance between two non-Archimedean fuzzy metric spaces (in the sense of Kramosil and Michalek). Basic properties involving convergence and the fuzzy version of the completeness theorem are presented. We show that the topological properties induced by the classic Gromov–Hausdorff distance on metric spaces can be deduced from our approach

Feebly compact paratopological groups and real-valued functions

We present several examples of feebly compact Hausdorff paratopological groups (i.e., groups with continuous multiplication) which provide answers to a number of questions posed in the literature. It turns out that a 2-pseudocompact, feebly compact Hausdorff paratopological group G can fail to be a topological group. Our group G has the Baire property, is Fréchet–Urysohn, but it is not precompact. It is well known that every infinite pseudocompact topological group contains a countable non-closed subset. We construct an infinite feebly compact Hausdorff paratopological group G all countable subsets of which are closed. Another peculiarity of the group G is that it contains a nonempty open subsemigroup C such that C−1 is closed and discrete, i.e., the inversion in G is extremely discontinuous. We also prove that for every continuous real-valued function g on a feebly compact paratopological group G , one can find a continuous homomorphism φ of G onto a second countable Hausdorff topological group H and a continuous real-valued function h on H such that g=h∘φ . In particular, every feebly compact paratopological group is R3 -factorizable. This generalizes a theorem of Comfort and Ross established in 1966 for real-valued functions on pseudocompact topological groups

Sequentially compact subsets and monotone functions: An application to fuzzy theory

Let (X,<,τO) be a first countable compact linearly ordered topological space. If (Y,D) is a uniform sequentially compact linearly ordered space with weight less than the splitting number s, then we characterize the sequentially compact subsets of the space M(X,Y) of all monotone functions from X into Y endowed with the topology of the uniform convergence induced by the uniformity D. In particular, our results are applied to identify the compact subsets of M([0,1],Y) for a wide class of linearly ordered topological spaces, including Y=R. This allows us to provide a characterization of the compact subsets of an extended version of the fuzzy number space (with the supremum metric) where the reals are replaced by certain linearly ordered topological spaces, which corrects some characterizations which appear in the literature. Since fuzzy analysis is based on the notion of fuzzy number just as much as classical analysis is based on the concept of real number, our results open new possibilities of research in this field

Bilinear isometries on spaces of vector-valued continuous functions

Let X, Y, Z be compact Hausdorff spaces and let E1, E2, E3 be Banach spaces. If T:C(X,E1)×C(Y,E2)→C(Z,E3) is a bilinear isometry which is stable on constants and E3 is strictly convex, then there exist a nonempty subset Z0 of Z, a surjective continuous mapping h:Z0→X×Y and a continuous function ω:Z0→Bil(E1×E2,E3) such that T(f,g)(z)=ω(z)(f(πX(h(z))),g(πY(h(z)))) for all z∈Z0 and every pair (f,g)∈C(X,E1)×C(Y,E2). This result generalizes the main theorems in Cambern (1978) [2] and Moreno and Rodríguez (2005) [7]

Pointwise convergence topology and function spaces in fuzzy analysis

We study the space of all continuous fuzzy-valued functions from a space $X$ into the space of fuzzy numbers (\mathbb{E}\sp{1},d\sb{\infty}) endowed with the pointwise convergence topology. Our results generalize the classical ones for continuous real-valued functions. The field of applications of this approach seems to be large, since the classical case allows many known devices to be fitted to general topology, functional analysis, coding theory, Boolean rings, etc

RTCS: a reactive with tags classifier system

In this work, a new Classifier System is proposed (CS). The system, a Reactive with Tags Classifier System (RTCS), is able to take into account environmental situations in intermediate decisions. CSs are special production systems, where conditions and actions are codified in order to learn new rules by means of Genetic Algorithms (GA). The RTCS has been designed to generate sequences of actions like the traditional classifier systems, but RTCS also has the capability of chaining rules among different time instants and reacting to new environmental situations, considering the last environmental situation to take a decision. In addition to the capability to react and generate sequences of actions, the design of a new rule codification allows the evolution of groups of specialized rules. This new codification is based on the inclusion of several bits, named tags, in conditions and actions, which evolve by means of GA. RTCS has been tested in robotic navigation. Results show the suitability of this approximation to the navigation problem and the coherence of tag values in rules classification.Publicad

Learning sequences of rules using classifier systems with tags

IEEE International Conference on Systems, Man, and Cybernetics. Tokyo, 12-15 October 1999.The objective of this paper was to obtain an encoding structure that would allow the genetic evolution of rules in such a manner that the number of rules and relationship in a classifier system (CS) would be learnt in the evolution process. For this purpose, an area that allows the definition of rule groups has been entered into the condition and message part of the encoded rules. This area is called internal tag. This term was coined because the system has some similarities with natural processes that take place in certain animal species, where the existence of tags allows them to communicate and recognize each other. Such CS is called a tag classifier system (TCS). The TCS has been tested in the game of draughts and compared with the classical CS. The results show an improving of the CS performance

Applying classifier systems to learn the reactions in mobile robots

The navigation problem involves how to reach a goal avoiding obstacles in dynamic environments. This problem can be faced considering reactions and sequences of actions. Classifier systems (CSs) have proven their ability of continuous learning, however, they have some problems in reactive systems. A modified CS, namely a reactive classifier system (RCS), is proposed to overcome those problems. Two special mechanisms are included in the RCS: the non-existence of internal cycles inside the CS (no internal cycles) and the fusion of environmental message with the messages posted to the message list in the previous instant (generation list through fusion). These mechanisms allow the learning of both reactions and sequences of actions. This learning process involves two main tasks: first, discriminate between rules and, second, the discovery of new rules to obtain a successful operation in dynamic environments. DiVerent experiments have been carried out using a mini-robot Khepera to find a generalized solution. The results show the ability of the system for continuous learning and adaptation to new situations.Publicad