# On Proofs and Rule of Multiplication in Fuzzy Attribute Logic

• Vilem Vychodil
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 4529)

## 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.

## Keywords

Fuzzy Logic Association Rule Fuzzy Rule Residuated Lattice Formal Concept Analysis
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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