Archive for Mathematical Logic

, Volume 54, Issue 7–8, pp 785–802 | Cite as

A logic of graded attributes

Article

Abstract

We present a logic for reasoning about attribute dependencies in data involving degrees such as a degree to which an object is red or a degree to which two objects are similar. The dependencies are of the form AB and can be interpreted in two ways: first, in data tables with entries representing degrees to which objects (rows) have attributes (columns); second, in database tables where each domain is equipped with a similarity relation. We assume that the degrees form a scale equipped with operations representing many-valued logical connectives. If 0 and 1 are the only degrees, the algebra of degrees becomes the two-element Boolean algebra and the two interpretations become well-known dependencies in Boolean data and functional dependencies of relational databases. In a setting with general scales, we obtain a new kind of dependencies with naturally arising degrees of validity, degrees of entailment, and related logical concepts. The deduction rules of the proposed logic are inspired by Armstrong rules and make it possible to infer dependencies to degrees—the degrees of provability. We provide a soundness and completeness theorem for such a setting asserting that degrees of entailment coincide with degrees of provability, prove the independence of deduction rules, and present further observations.

Keywords

Attribute implication Fuzzy logic Graded-style completeness Pavelka-style logic 

Mathematics Subject Classification

03B50 03B52 68T37 

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Copyright information

© Springer-Verlag Berlin Heidelberg 2015

Authors and Affiliations

  1. 1.Department of Computer SciencePalacký UniversityOlomoucCzech Republic

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