Graded consequence relations and fuzzy closure operator

Gerla, Giangiacomo

doi:10.1080/11663081.1996.10510892

Cited by 31 publications

(18 citation statements)

References 9 publications

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“…Note that for L ¼ f0; 1g, L K -closure operators coincide with (classical) closure operators. Note also that for L ¼ ½0; 1, L f1g -closure operators are precisely fuzzy closure operators studied by Gerla [9,15,16].…”

Section: Fuzzy Closure Operatorsmentioning

confidence: 97%

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Fuzzy closure operators II: induced relations, representation, and examples

Bělohlávek¹

2002

Soft Computing

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Closure operators (and related structures) are investigated from the point of view of fuzzy set theory. The paper is a follow up to [7] where fundamental notions and result have been established. The present approach generalizes the existing approaches in two ways: first, complete residuated lattices are used as the structures of truth values (leaving the unite interval [0,1] with minimum and other t-norms particular cases); second, the monotony condition is formulated so that it can reflect also partial subsethood (not only full subsethood as in other approaches). In this paper, we study relations induced by fuzzy closure operators (fuzzy quasiorders and similarities); factorization of closure systems by similarities and by so-called decrease of logical precision; representation of fuzzy closure operators by (crisp) closure operators; relation to consequence relations; and natural examples illustrating the notions and results. IntroductionThis is a follow up to my paper [7]. In [7], closure operators and related structures have been considered from the point of view of fuzzy approach (graded truth approach; with complete residuated lattices taken for the structures of truth values). The aim of this paper is to present further results on fuzzy closure operators.The organization and the content of the paper are as follows: Sect. 2 recalls the notions and main results of [7]. In Sect. 3, we study some induced (fuzzy) relations: fuzzy quasiorder and equivalence (similarity). We show a way to factorize the complete lattice of closed (w.r.t. to a given fuzzy closure operator) fuzzy sets by an a-cut of a naturally defined similarity relation, parameter a having the role of controlling the coarsity of the factorization.Another way to factorize the lattice of closed fuzzy sets is by a so-called decrease of logical precision. The factorization processes have natural applications if the structure of closed sets has some natural interpretation and one needs to simplify the structure (as an example, we demonstrate the results on factorization of so-called fuzzy concept lattices). In Sect. 4, we present a natural representation of fuzzy closure operators by (classical) closure operators. Section 5 presents some examples of fuzzy closure operators. In Sect. 6, fuzzy closure operators and consequence relations are briefly discussed. Fuzzy closure operatorsClosure operators (and closure systems) play a significant role in both pure and applied mathematics. In the framework of fuzzy set theory, several particular examples of closure operators and systems have been considered (e.g. so-called fuzzy subalgebras, fuzzy congruences, fuzzy topology etc.). Recently, fuzzy closure operators and fuzzy closure systems themselves have been studied, see e.

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Section: Fuzzy Closure Operatorsmentioning

confidence: 97%

“…Recently, fuzzy closure operators and fuzzy closure systems themselves have been studied, see e.g. [8,9,15,16]. As a matter of fact, a fuzzy set A is usually defined as a mapping from a universe set X into the real interval ½0; 1 in the above mentioned works.…”

Section: Fuzzy Closure Operatorsmentioning

confidence: 99%

Fuzzy closure operators II: induced relations, representation, and examples

Bělohlávek¹

2002

Soft Computing

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“…In the framework of fuzzy set theory, several particular examples of closure operators and systems have been considered (e.g., so-called fuzzy subalgebras, fuzzy congruences, fuzzy topology, etc.). Recently, fuzzy closure operators and fuzzy closure systems themselves (i.e., operators which map fuzzy sets to fuzzy sets and the corresponding systems of closed fuzzy sets) have been studied by Gerla et al; see, e.g., [3,4,6,7]. As a matter of fact, a fuzzy set A is usually defined as a mapping from a universe set X into the real interval [0, 1] in the above mentioned works.…”

Section: Introductionmentioning

confidence: 99%

Fuzzy Closure Operators

Bělohlávek¹

2001

Journal of Mathematical Analysis and Applications

150

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“…non truth-functional) way to represent and reasoning with probability or other uncertainty measures. This is the case for instance of the approach developed by Gerla [1994b]. Roughly speaking, Gerla devises a probability logic by defining a suitable fuzzy consequence operator C, in the sense of Pavelka (see Section 3.8), on fuzzy sets v of the set B of classical formulas (modulo classical equivalence) in a given language, where the membership degree v(p) of a proposition p is understood as lower bound on its probability.…”

Section: Fuzzy Logic Theories To Reason Under Uncertaintymentioning

confidence: 99%

“…Links between fuzzy closure operators and graded consequence relations were examined by Gerla [1996] and by Castro Trillas and Cubillo [1994]. In particular Castro et al point out that several methods of approximate reasoning used in Artificial Intelligence, such as Polya's models of plausible reasoning [Polya, 1954] or Nilsson's probabilistic logic [Nilsson, 1974], are not covered by the formalism of graded consequence relations, and they introduce a new concept of consequence relations, called fuzzy consequence relations which, unlike Chakraborty's graded consequence relation, apply over fuzzy sets of formulas.…”

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confidence: 99%