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Computing the lattice of all fixpoints of a fuzzy closure operator

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Abstract
We present a fast bottom-up algorithm to compute all fixpoints of a fuzzy closure operator in a finite set over a finite chain of truth degrees, along with the partial order on the set of all fixpoints. Fuzzy closure operators appear in several areas of fuzzy logic and its applications, including formal concept analysis (FCA) that we use as a reference area of application in this paper. Several problems in FCA, such as computing all formal concepts from data with graded attributes or computing non-redundant bases of all attribute dependencies, can be reduced to the problem of computing fixpoints of particular fuzzy closure operators associated with the input data. The development of a general algorithm that is applicable, in particular, to these problems is the ultimate purpose of this paper. We present the algorithm, its theoretical foundations, and experimental evaluation.
Keywords
fuzzy closure operator, fuzzy logic, fixpoint, Algorithm, DECOMPOSITION, SUBALGEBRAS, FUNDAMENTALS, SYSTEMS

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Chicago
Belohlavek, Radim, Bernard De Baets, Jan Outrata, and Vilem Vychodil. 2010. “Computing the Lattice of All Fixpoints of a Fuzzy Closure Operator.” Ieee Transactions on Fuzzy Systems 18 (3): 546–557.
APA
Belohlavek, R., De Baets, B., Outrata, J., & Vychodil, V. (2010). Computing the lattice of all fixpoints of a fuzzy closure operator. IEEE TRANSACTIONS ON FUZZY SYSTEMS, 18(3), 546–557.
Vancouver
1.
Belohlavek R, De Baets B, Outrata J, Vychodil V. Computing the lattice of all fixpoints of a fuzzy closure operator. IEEE TRANSACTIONS ON FUZZY SYSTEMS. 2010;18(3):546–57.
MLA
Belohlavek, Radim, Bernard De Baets, Jan Outrata, et al. “Computing the Lattice of All Fixpoints of a Fuzzy Closure Operator.” IEEE TRANSACTIONS ON FUZZY SYSTEMS 18.3 (2010): 546–557. Print.
@article{1085748,
  abstract     = {We present a fast bottom-up algorithm to compute all fixpoints of a fuzzy closure operator in a finite set over a finite chain of truth degrees, along with the partial order on the set of all fixpoints. Fuzzy closure operators appear in several areas of fuzzy logic and its applications, including formal concept analysis (FCA) that we use as a reference area of application in this paper. Several problems in FCA, such as computing all formal concepts from data with graded attributes or computing non-redundant bases of all attribute dependencies, can be reduced to the problem of computing fixpoints of particular fuzzy closure operators associated with the input data. The development of a general algorithm that is applicable, in particular, to these problems is the ultimate purpose of this paper. We present the algorithm, its theoretical foundations, and experimental evaluation.},
  author       = {Belohlavek, Radim and De Baets, Bernard and Outrata, Jan and Vychodil, Vilem},
  issn         = {1063-6706},
  journal      = {IEEE TRANSACTIONS ON FUZZY SYSTEMS},
  keyword      = {fuzzy closure operator,fuzzy logic,fixpoint,Algorithm,DECOMPOSITION,SUBALGEBRAS,FUNDAMENTALS,SYSTEMS},
  language     = {eng},
  number       = {3},
  pages        = {546--557},
  title        = {Computing the lattice of all fixpoints of a fuzzy closure operator},
  url          = {http://dx.doi.org/10.1109/TFUZZ.2010.2041006},
  volume       = {18},
  year         = {2010},
}

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