6,206 research outputs found
Retraction and Generalized Extension of Computing with Words
Fuzzy automata, whose input alphabet is a set of numbers or symbols, are a
formal model of computing with values. Motivated by Zadeh's paradigm of
computing with words rather than numbers, Ying proposed a kind of fuzzy
automata, whose input alphabet consists of all fuzzy subsets of a set of
symbols, as a formal model of computing with all words. In this paper, we
introduce a somewhat general formal model of computing with (some special)
words. The new features of the model are that the input alphabet only comprises
some (not necessarily all) fuzzy subsets of a set of symbols and the fuzzy
transition function can be specified arbitrarily. By employing the methodology
of fuzzy control, we establish a retraction principle from computing with words
to computing with values for handling crisp inputs and a generalized extension
principle from computing with words to computing with all words for handling
fuzzy inputs. These principles show that computing with values and computing
with all words can be respectively implemented by computing with words. Some
algebraic properties of retractions and generalized extensions are addressed as
well.Comment: 13 double column pages; 3 figures; to be published in the IEEE
Transactions on Fuzzy System
Covering rough sets based on neighborhoods: An approach without using neighborhoods
Rough set theory, a mathematical tool to deal with inexact or uncertain
knowledge in information systems, has originally described the indiscernibility
of elements by equivalence relations. Covering rough sets are a natural
extension of classical rough sets by relaxing the partitions arising from
equivalence relations to coverings. Recently, some topological concepts such as
neighborhood have been applied to covering rough sets. In this paper, we
further investigate the covering rough sets based on neighborhoods by
approximation operations. We show that the upper approximation based on
neighborhoods can be defined equivalently without using neighborhoods. To
analyze the coverings themselves, we introduce unary and composition operations
on coverings. A notion of homomorphismis provided to relate two covering
approximation spaces. We also examine the properties of approximations
preserved by the operations and homomorphisms, respectively.Comment: 13 pages; to appear in International Journal of Approximate Reasonin
A weak characterization of slow variables in stochastic dynamical systems
We present a novel characterization of slow variables for continuous Markov
processes that provably preserve the slow timescales. These slow variables are
known as reaction coordinates in molecular dynamical applications, where they
play a key role in system analysis and coarse graining. The defining
characteristics of these slow variables is that they parametrize a so-called
transition manifold, a low-dimensional manifold in a certain density function
space that emerges with progressive equilibration of the system's fast
variables. The existence of said manifold was previously predicted for certain
classes of metastable and slow-fast systems. However, in the original work, the
existence of the manifold hinges on the pointwise convergence of the system's
transition density functions towards it. We show in this work that a
convergence in average with respect to the system's stationary measure is
sufficient to yield reaction coordinates with the same key qualities. This
allows one to accurately predict the timescale preservation in systems where
the old theory is not applicable or would give overly pessimistic results.
Moreover, the new characterization is still constructive, in that it allows for
the algorithmic identification of a good slow variable. The improved
characterization, the error prediction and the variable construction are
demonstrated by a small metastable system
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