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On Proofs and Rule of Multiplication in Fuzzy Attribute Logic

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Foundations of Fuzzy Logic and Soft Computing (IFSA 2007)

Part of the book series: Lecture Notes in Computer Science ((LNAI,volume 4529))

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Abstract

The paper develops fuzzy attribute logic, i.e. a logic for reasoning about formulas of the form \(A\Rightarrow B\) where A and B are fuzzy sets of attributes. A formula \(A\Rightarrow B\) represents a dependency which is true in a data table with fuzzy attributes iff each object having all attributes from A has also all attributes from B, membership degrees in A and B playing a role of thresholds. We study axiomatic systems of fuzzy attribute logic which result by adding a single deduction rule, called a rule of multiplication, to an ordinary system of deduction rules complete w.r.t. bivalent semantics, i.e. to well-known Armstrong axioms. In this paper, we concentrate on the rule of multiplication and its role in fuzzy attribute logic. We show some advantageous properties of the rule of multiplication. In addition, we show that these properties enable us to reduce selected problems concerning proofs in fuzzy attribute logic to the corresponding problems in the ordinary case. As an example, we discuss the problem of normalization of proofs and present, in the setting of fuzzy attribute logic, a counterpart to a well-known theorem from database theory saying that each proof can be transformed to a so-called RAP-sequence.

Supported by grant No. 1ET101370417 of GA AV ČR, by grant No. 201/05/0079 of the Czech Science Foundation, and by institutional support, research plan MSM 6198959214.

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Author information

Authors and Affiliations

  1. Dept. Systems Science and Industrial Engineering, Binghamton University—SUNY, Binghamton, NY 13902, USA

    Radim Belohlavek

  2. Dept. Computer Science, Palacky University, Olomouc, Tomkova 40, CZ-779 00 Olomouc, Czech Republic

    Radim Belohlavek & Vilem Vychodil

Authors
  1. Radim Belohlavek
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  2. Vilem Vychodil
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Editor information

Patricia Melin Oscar Castillo Luis T. Aguilar Janusz Kacprzyk Witold Pedrycz

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Belohlavek, R., Vychodil, V. (2007). On Proofs and Rule of Multiplication in Fuzzy Attribute Logic. In: Melin, P., Castillo, O., Aguilar, L.T., Kacprzyk, J., Pedrycz, W. (eds) Foundations of Fuzzy Logic and Soft Computing. IFSA 2007. Lecture Notes in Computer Science(), vol 4529. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-72950-1_47

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  • DOI: https://doi.org/10.1007/978-3-540-72950-1_47

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-72917-4

  • Online ISBN: 978-3-540-72950-1

  • eBook Packages: Computer ScienceComputer Science (R0)

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