450 research outputs found
Informational Paradigm, management of uncertainty and theoretical formalisms in the clustering framework: A review
Fifty years have gone by since the publication of the first paper on clustering based on fuzzy sets theory. In 1965, L.A. Zadeh had published “Fuzzy Sets” [335]. After only one year, the first effects of this seminal paper began to emerge, with the pioneering paper on clustering by Bellman, Kalaba, Zadeh [33], in which they proposed a prototypal of clustering algorithm based on the fuzzy sets theory
Decidability and Universality in Symbolic Dynamical Systems
Many different definitions of computational universality for various types of
dynamical systems have flourished since Turing's work. We propose a general
definition of universality that applies to arbitrary discrete time symbolic
dynamical systems. Universality of a system is defined as undecidability of a
model-checking problem. For Turing machines, counter machines and tag systems,
our definition coincides with the classical one. It yields, however, a new
definition for cellular automata and subshifts. Our definition is robust with
respect to initial condition, which is a desirable feature for physical
realizability.
We derive necessary conditions for undecidability and universality. For
instance, a universal system must have a sensitive point and a proper
subsystem. We conjecture that universal systems have infinite number of
subsystems. We also discuss the thesis according to which computation should
occur at the `edge of chaos' and we exhibit a universal chaotic system.Comment: 23 pages; a shorter version is submitted to conference MCU 2004 v2:
minor orthographic changes v3: section 5.2 (collatz functions) mathematically
improved v4: orthographic corrections, one reference added v5:27 pages.
Important modifications. The formalism is strengthened: temporal logic
replaced by finite automata. New results. Submitte
Evolution of Voronoi-based Fuzzy Controllers
A fuzzy controller is usually designed by formulating the knowledge of a human expert into a set of linguistic variables and fuzzy rules. One of the most successful methods to automate the fuzzy controllers development process are evolutionary algorithms. In this work, we propose a so-called ``approximative'' representation for fuzzy systems, where the antecedent of the rules are determined by a multivariate membership function defined in terms of Voronoi regions. Such representation guarantees the -completeness property and provides a synergistic relation between the rules. An evolutionary algorithm based on this representation can evolve all the components of the fuzzy system, and due to the properties of the representation, the algorithm (1) can benefit from the use of geometric genetic operators, (2) does not need genetic repair algorithms, (3) guarantees the completeness property and (4) can implement previous knowledge in a simple way by using adaptive a priori rules. The proposed representation is evaluated on an obstacle avoidance problem with a simulated mobile robot
Accelerated Fuzzy C-Means Clustering Based on New Affinity Filtering and Membership Scaling
Fuzzy C-Means (FCM) is a widely used clustering method. However, FCM and its
many accelerated variants have low efficiency in the mid-to-late stage of the
clustering process. In this stage, all samples are involved in the update of
their non-affinity centers, and the fuzzy membership grades of the most of
samples, whose assignment is unchanged, are still updated by calculating the
samples-centers distances. All those lead to the algorithms converging slowly.
In this paper, a new affinity filtering technique is developed to recognize a
complete set of the non-affinity centers for each sample with low computations.
Then, a new membership scaling technique is suggested to set the membership
grades between each sample and its non-affinity centers to 0 and maintain the
fuzzy membership grades for others. By integrating those two techniques, FCM
based on new affinity filtering and membership scaling (AMFCM) is proposed to
accelerate the whole convergence process of FCM. Many experimental results
performed on synthetic and real-world data sets have shown the feasibility and
efficiency of the proposed algorithm. Compared with the state-of-the-art
algorithms, AMFCM is significantly faster and more effective. For example,
AMFCM reduces the number of the iteration of FCM by 80% on average
Rough sets, their extensions and applications
Rough set theory provides a useful mathematical foundation for developing automated computational systems that can help understand and make use of imperfect knowledge. Despite its recency, the theory and its extensions have been widely applied to many problems, including decision analysis, data-mining, intelligent control and pattern recognition. This paper presents an outline of the basic concepts of rough sets and their major extensions, covering variable precision, tolerance and fuzzy rough sets. It also shows the diversity of successful applications these theories have entailed, ranging from financial and business, through biological and medicine, to physical, art, and meteorological
Shadowing, asymptotic shadowing and s-limit shadowing
We study three notions of shadowing: classical shadowing, limit (or
asymptotic) shadowing, and s-limit shadowing. We show that classical and
s-limit shadowing coincide for tent maps and, more generally, for piecewise
linear interval maps with constant slopes, and are further equivalent to the
linking property introduced by Chen in 1991.
We also construct a system which exhibits shadowing but not limit shadowing,
and we study how shadowing properties transfer to maximal transitive subsystems
and inverse limits (sometimes called natural extensions).
Where practicable, we show that our results are best possible by means of
examples.Comment: 28 pages, 4 figure
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