391 research outputs found
On Fuzzy Concepts
In this paper we try to combine two approaches. One is the theory of knowledge graphs in which concepts are represented by graphs. The other is the axiomatic theory of fuzzy sets (AFS).
The discussion will focus on the idea of fuzzy concept. It will be argued that the fuzziness of a concept in natural language is mainly due to the difference in interpretation that people give to a certain word. As different interpretations lead to different knowledge graphs, the notion of fuzzy concept should be describable in terms of sets of graphs. This leads to a natural introduction of membership values for elements of graphs. Using these membership values we apply AFS theory as well as an alternative approach to calculate fuzzy decision trees, that can be used to determine the most relevant elements of a concept
On the relation between the base of an EI algebra and word graphs
This paper is an attempt to investigate the possibilities to link algebraic fuzzy set theory with the theory of word graphs. In both theories concepts are studied and concepts can be set in correspondence. This enables to use algebraic results in the context of word graph theory
On the Notion of Proposition in Classical and Quantum Mechanics
The term proposition usually denotes in quantum mechanics (QM) an element of
(standard) quantum logic (QL). Within the orthodox interpretation of QM the
propositions of QL cannot be associated with sentences of a language stating
properties of individual samples of a physical system, since properties are
nonobjective in QM. This makes the interpretation of propositions
problematical. The difficulty can be removed by adopting the objective
interpretation of QM proposed by one of the authors (semantic realism, or SR,
interpretation). In this case, a unified perspective can be adopted for QM and
classical mechanics (CM), and a simple first order predicate calculus L(x) with
Tarskian semantics can be constructed such that one can associate a physical
proposition (i.e., a set of physical states) with every sentence of L(x). The
set of all physical propositions is partially ordered and contains a
subset of testable physical propositions whose order structure
depends on the criteria of testability established by the physical theory. In
particular, turns out to be a Boolean lattice in CM, while it can
be identified with QL in QM. Hence the propositions of QL can be associated
with sentences of L(x), or also with the sentences of a suitable quantum
language , and the structure of QL characterizes the notion of
testability in QM. One can then show that the notion of quantum truth does not
conflict with the classical notion of truth within this perspective.
Furthermore, the interpretation of QL propounded here proves to be equivalent
to a previous pragmatic interpretation worked out by one of the authors, and
can be embodied within a more general perspective which considers states as
first order predicates of a broader language with a Kripkean semantics.Comment: 22 pages. To appear in "The Foundations of Quantum Mechanics:
Historical Analysis and Open Questions-Cesena 2004", C. Garola, A. Rossi and
S. Sozzo Eds., World Scientific, Singapore, 200
Quantum Chains of Hopf Algebras with Quantum Double Cosymmetry
Given a finite dimensional C^*-Hopf algebra H and its dual H^ we construct
the infinite crossed product A=... x H x H^ x H ... and study its
superselection sectors in the framework of algebraic quantum field theory. A is
the observable algebra of a generalized quantum spin chain with H-order and
H^-disorder symmetries, where by a duality transformation the role of order and
disorder may also appear interchanged. If H=\CC G is a group algebra then A
becomes an ordinary G-spin model. We classify all DHR-sectors of A --- relative
to some Haag dual vacuum representation --- and prove that their symmetry is
described by the Drinfeld double D(H). To achieve this we construct localized
coactions \rho: A \to (A \otimes D(H)) and use a certain compressibility
property to prove that they are universal amplimorphisms on A. In this way the
double D(H) can be recovered from the observable algebra A as a universal
cosymmetrty.Comment: Latex, 48 pages, no figures, extended version of hep-th/9507174, but
without the field algebra construction, contains full proofs of the slightly
shortened article published in Commun.Math.Phys., the revision only concerns
some misprints and an update of the literatur
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