Fuzzy Description Logic Reasoning Using a Fixpoint Algorithm

We present ${\mathbf{FixIt}}{\mathbb{(ALC)}}$, a novel procedure for deciding knowledge base (KB) satisfiability in the Fuzzy Description Logic (FDL) ${\mathbb{ALC}}$. ${\mathbf{FixIt}}{\mathbb{(ALC)}}$ does not search for tree-structured models as in tableau-based proof procedures, but embodies a (greatest) fixpoint-computation of canonical models that are not necessarily tree-structured, based on a type-elimination process. Soundness, completeness and termination are proven and the runtime and space complexity are discussed. We give a precise characterization of the worst-case complexity of deciding KB satisfiability (as well as related terminological and assertional reasoning tasks) in ${\mathbb{ALC}}$ in the general case and show that our method yields a worst-case optimal decision procedure (under reasonable assumptions). To the best of our knowledge it is the first fixpoint-based decision procedure for FDLs, hence introducing a new class of inference procedures into FDL reasoning.

[1]  Sérgio Vale Aguiar Campos,et al.  Symbolic Model Checking , 1993, CAV.

[2]  Umberto Straccia,et al.  A fuzzy description logic for the semantic web , 2006, Fuzzy Logic and the Semantic Web.

[3]  Petr Hájek.  What does mathematical fuzzy logic offer to description logic? , 2006, Fuzzy Logic and the Semantic Web.

[4]  Frank Wolter,et al.  Semi-qualitative Reasoning about Distances: A Preliminary Report , 2000, JELIA.

[5]  Volker Haarslev,et al.  Uncertainty Reasoning for Ontologies with General TBoxes in Description Logic , 2008, URSW.

[6]  G. Stamou,et al.  Reasoning with Very Expressive Fuzzy Description Logics , 2007, J. Artif. Intell. Res..

[7]  Vaughan R. Pratt,et al.  A Near-Optimal Method for Reasoning about Action , 1980, J. Comput. Syst. Sci..

[8]  Gert Smolka,et al.  Attributive Concept Descriptions with Complements , 1991, Artif. Intell..

[9]  Umberto Straccia,et al.  Transforming Fuzzy Description Logics into Classical Description Logics , 2004, JELIA.

[10]  Dave Robertson,et al.  Probabilistic Dialogue Models for Dynamic Ontology Mapping , 2006, URSW.

[11]  Ulrike Sattler,et al.  BDD-based decision procedures for the modal logic K ★ , 2006, J. Appl. Non Class. Logics.

[12]  Umberto Straccia,et al.  Mixed Integer Programming, General Concept Inclusions and Fuzzy Description Logics , 2007, EUSFLAT Conf..

[13]  Yves Chiaramella,et al.  A Model for Multimedia Information Retrieval , 1996 .

[14]  Kenneth L. McMillan,et al.  Symbolic model checking , 1992 .

[15]  Baowen Xu,et al.  Discrete Tableau Algorithms for FSHI , 2006, Description Logics.

[16]  Klaus Schild,et al.  A Correspondence Theory for Terminological Logics: Preliminary Report , 1991, IJCAI.

[17]  Stephan Tobies,et al.  Complexity results and practical algorithms for logics in knowledge representation , 2001, ArXiv.

[18]  Umberto Straccia,et al.  General Concept Inclusions inFluzzy Description Logics , 2006, ECAI.

[19]  Elie Sanchez,et al.  Fuzzy Logic and the Semantic Web , 2005 .

[20]  Randal E. Bryant,et al.  Symbolic Boolean manipulation with ordered binary-decision diagrams , 1992, CSUR.

[21]  Umberto Straccia,et al.  Reasoning within Fuzzy Description Logics , 2011, J. Artif. Intell. Res..

[22]  Stijn Heymans,et al.  ON FIXPOINT-BASED DECISION PROCEDURES FOR FUZZY DESCRIPTION LOGICS I , 2008 .

[23]  Diego Calvanese,et al.  The Description Logic Handbook: Theory, Implementation, and Applications , 2003, Description Logic Handbook.

[24]  Petr Hájek,et al.  Making fuzzy description logic more general , 2005, Fuzzy Sets Syst..