Computing the lattice of all fixpoints of a fuzzy closure operator

  • Authors:

  • Radim Belohlavek;Bernard De Baets;Jan Outrata;Vilem Vychodil

  • Affiliations:

  • Department of Computer Science, Palacky University, Olomouc, Czech Republic and Department of Systems Science and Industrial Engineering, T. J. Watson School of Engineering and Applied Science, Bi ...;Department of Applied Mathematics, Biometrics, and Process Control, Ghent University, Ghent, Belgium;Department of Computer Science, Palacky University, Olomouc, Czech Republic;Department of Computer Science, Palacky University, Olomouc, Czech Republic and Department of Systems Science and Industrial Engineering, T. J. Watson School of Engineering and Applied Science, Bi ...

  • Venue:
  • IEEE Transactions on Fuzzy Systems - Special section on computing with words
  • Year:
  • 2010

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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.