On the failure of the finite model property in some Fuzzy Description Logics

Authors:
Fernando Bobillo;Félix Bou;Umberto Straccia
Affiliations:
Department of Computer Science and Systems Engineering, University of Zaragoza, Spain;Department of Probability, Logic and Statistics, University of Barcelona, Spain;Istituto di Scienza e Tecnologie dell'Informazione, Consiglio Nazionale delle Ricerche (ISTI - CNR), Pisa, Italy
Venue:
Fuzzy Sets and Systems
Year:
2011

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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 DLs.