Parameterizing the semantics of fuzzy attribute implications by systems of isotone Galois connections

TL;DR

This paper introduces a general framework for the semantics of fuzzy attribute implications using systems of isotone Galois connections, unifying previous approaches and providing formal semantics, entailment notions, and data-driven bases.

Contribution

It formalizes a broad parameterization of fuzzy attribute implications, including previous methods, and offers semantic, axiomatic, and data-based characterizations.

Findings

Unified semantics for fuzzy attribute implications.

Complete axiomatization of semantic entailment.

Characterization of implication bases from data.

Abstract

We study the semantics of fuzzy if-then rules called fuzzy attribute implications parameterized by systems of isotone Galois connections. The rules express dependencies between fuzzy attributes in object-attribute incidence data. The proposed parameterizations are general and include as special cases the parameterizations by linguistic hedges used in earlier approaches. We formalize the general parameterizations, propose bivalent and graded notions of semantic entailment of fuzzy attribute implications, show their characterization in terms of least models and complete axiomatization, and provide characterization of bases of fuzzy attribute implications derived from data.