On Proofs and Rule of Multiplication in Fuzzy Attribute Logic

  • Authors:

  • Radim Belohlavek;Vilem Vychodil

  • Affiliations:

  • Dept. Systems Science and Industrial Engineering, Binghamton University--SUNY, Binghamton, NY 13902, USA and Dept. Computer Science, Palacky University, Olomouc, Tomkova 40, CZ-779 00 Olomouc, Cze ...;Dept. Computer Science, Palacky University, Olomouc, Tomkova 40, CZ-779 00 Olomouc, Czech Republic

  • Venue:
  • IFSA '07 Proceedings of the 12th international Fuzzy Systems Association world congress on Foundations of Fuzzy Logic and Soft Computing
  • Year:
  • 2007

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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 Aand Bare 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 Ahas also all attributes from B, membership degrees in Aand Bplaying 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.