On the Failure of the Finite Model Property in some Fuzzy Description Logics

TL;DR

This paper demonstrates that fuzzy Description Logics like ALC under Lukasiewicz and Product logics do not have the Finite Model Property when arbitrary knowledge bases are allowed, impacting reasoning algorithms.

Contribution

It shows the failure of the Finite Model Property in fuzzy ALC logics with arbitrary knowledge bases, highlighting limitations in existing reasoning methods.

Findings

FMP fails for fuzzy ALC under Lukasiewicz and Product logics.

Finite and witnessed satisfiability differ for arbitrary knowledge bases.

Implications for reasoning algorithms in fuzzy DLs.

Abstract

Fuzzy Description Logics (DLs) are a family of logics which allow the representation of (and the reasoning with) structured knowledge affected by vagueness. Although most of the not very expressive crisp DLs, such as ALC, enjoy the Finite Model Property (FMP), this is not the case once we move into the fuzzy case. In this paper we show that if we allow arbitrary knowledge bases, then the fuzzy DLs ALC under Lukasiewicz and Product fuzzy logics do not verify the FMP even if we restrict to witnessed models; in other words, finite satisfiability and witnessed satisfiability are different for arbitrary knowledge bases. The aim of this paper is to point out the failure of FMP because it affects several algorithms published in the literature for reasoning under fuzzy ALC.