Algorithms for reasoning in very expressive description logics under infinitely valued Gödel semantics

Fuzzy description logics (FDLs) are knowledge representation formalisms capable of dealing with imprecise knowledge by allowing intermediate membership degrees in the interpretation of concepts and roles. One option for dealing with these intermediate degrees is to use the so-called Gdel semantics, under which conjunction is interpreted by the minimum of the degrees of the conjuncts. Despite its apparent simplicity, developing reasoning techniques for expressive FDLs under this semantics is a hard task.In this paper, we introduce two new algorithms for reasoning in very expressive FDLs under Gdel semantics. They combine the ideas of a previous automata-based algorithm for Gdel FDLs with the known crispification and tableau approaches for FDL reasoning. The results are the two first practical algorithms capable of reasoning in infinitely valued FDLs supporting general concept inclusions. A novel tableau algorithm for reasoning in very expressive fuzzy description logics.A new crispification-based approach that can handle all sublogics of G-SROIQ with the forest-model property.Closing undecidability gaps for expressive fuzzy description logics.

[1]  Ian Horrocks,et al.  A Tableaux Decision Procedure for SHOIQ , 2005, IJCAI.

[2]  Umberto Straccia,et al.  Joining Gödel and Zadeh Fuzzy Logics in Fuzzy Description Logics , 2012, Int. J. Uncertain. Fuzziness Knowl. Based Syst..

[3]  Yevgeny Kazakov,et al.  SRIQ and SROIQ are Harder than SHOIQ , 2008, Description Logics.

[4]  Boris Motik,et al.  HermiT: An OWL 2 Reasoner , 2014, Journal of Automated Reasoning.

[5]  Ian Horrocks,et al.  Practical Reasoning for Very Expressive Description Logics , 2000, Log. J. IGPL.

[6]  Dusan Guller.  On the Satisfiability and Validity Problems in the Propositional Gödel Logic , 2010, IJCCI.

[7]  Rafael Peñaloza,et al.  On the Undecidability of Fuzzy Description Logics with GCIs and Product T-norm , 2011, FroCoS.

[8]  Giorgos Stoilos,et al.  A Framework for Reasoning with Expressive Continuous Fuzzy Description Logics , 2009, Description Logics.

[9]  García-Cerdaña.  Fuzzy Description Logic , 2017 .

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

[11]  Christopher Tresp,et al.  A Description Logic for Vague Knowledge , 1998, ECAI.

[12]  Rafael Peñaloza,et al.  Reasoning in Expressive Description Logics under Infinitely Valued Gödel Semantics , 2015, FroCos.

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

[14]  Radko Mesiar,et al.  Triangular Norms , 2000, Trends in Logic.

[15]  Umberto Straccia,et al.  Reasoning with the finitely many-valued Lukasiewicz fuzzy Description Logic SROIQ , 2011, Inf. Sci..

[16]  Fernando Bobillo,et al.  DeLorean: A reasoner for fuzzy OWL 2 , 2012, Expert Syst. Appl..

[17]  Ian Horrocks,et al.  The Even More Irresistible SROIQ , 2006, KR.

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

[19]  Petr Hájek,et al.  Handbook of mathematical fuzzy logic , 2011 .

[20]  Stefanos D. Kollias,et al.  Reasoning with qualified cardinality restrictions in fuzzy Description Logics , 2008, 2008 IEEE International Conference on Fuzzy Systems (IEEE World Congress on Computational Intelligence).

[21]  Rafael Peñaloza,et al.  Reasoning in Fuzzy Description Logics using Automata , 2016, Fuzzy Sets Syst..

[22]  Diego Calvanese,et al.  Regular Path Queries in Expressive Description Logics with Nominals , 2009, IJCAI.

[23]  Umberto Straccia,et al.  Finite Fuzzy Description Logics: A Crisp Representation for Finite Fuzzy ALCH , 2010, URSW.

[24]  Ian Horrocks,et al.  Decidability of SHIQ with Complex Role Inclusion Axioms , 2003, IJCAI.

[25]  Rafael Peñaloza,et al.  The limits of decidability in fuzzy description logics with general concept inclusions , 2015, Artif. Intell..

[26]  Fernando Bobillo,et al.  A Crisp Representation for Fuzzy SHOIN with Fuzzy Nominals and General Concept Inclusions , 2006, URSW.

[27]  Robert LIN,et al.  NOTE ON FUZZY SETS , 2014 .

[28]  Giorgos Stoilos,et al.  Reasoning with fuzzy extensions of OWL and OWL 2 , 2013, Knowledge and Information Systems.

[29]  Umberto Straccia,et al.  Finite Fuzzy Description Logics and Crisp Representations , 2010, URSW.

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

[31]  Umberto Straccia,et al.  The fuzzy ontology reasoner fuzzyDL , 2016, Knowl. Based Syst..

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

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

[34]  Umberto Straccia,et al.  Fuzzy description logics with general t-norms and datatypes , 2009, Fuzzy Sets Syst..

[35]  Umberto Straccia,et al.  Fuzzy description logics under Gödel semantics , 2009, Int. J. Approx. Reason..

[36]  Umberto Straccia,et al.  Description Logics over Lattices , 2006, Int. J. Uncertain. Fuzziness Knowl. Based Syst..

[37]  M. Droste,et al.  Handbook of Weighted Automata , 2009 .

[38]  Rafael Peñaloza,et al.  Decidable Gödel Description Logics without the Finitely-Valued Model Property , 2014, KR.

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

[40]  Umberto Straccia,et al.  Managing uncertainty and vagueness in description logics for the Semantic Web , 2008, J. Web Semant..

[41]  Dirk Pattinson,et al.  Syntactic Labelled Tableaux for Lukasiewicz Fuzzy ALC , 2013, IJCAI.

[42]  I. Horrocks,et al.  A Tableau Decision Procedure for $\mathcal{SHOIQ}$ , 2007, Journal of Automated Reasoning.

[43]  Volker Haarslev,et al.  A formal framework for description logics with uncertainty , 2009, Int. J. Approx. Reason..

[44]  Stephan Tobies,et al.  The Complexity of Reasoning with Cardinality Restrictions and Nominals in Expressive Description Logics , 2011, ArXiv.

[45]  Fernando Bobillo,et al.  Optimizing the Crisp Representation of the Fuzzy Description Logic SROIQ , 2007, URSW.

[46]  Umberto Straccia,et al.  On the (un)decidability of fuzzy description logics under Łukasiewicz t-norm , 2013, Inf. Sci..

[47]  Umberto Straccia,et al.  A Fuzzy Description Logic with Product T-norm , 2007, 2007 IEEE International Fuzzy Systems Conference.

[48]  Fernando Bobillo,et al.  A Crisp Representation for Fuzzy with Fuzzy Nominals and General Concept Inclusions. , 2008 .

[49]  Birte Glimm,et al.  Konclude: System description , 2014, J. Web Semant..

[50]  Rafael Peñaloza,et al.  Answering Fuzzy Conjunctive Queries Over Finitely Valued Fuzzy Ontologies , 2015, Journal on Data Semantics.

[51]  Petr Hájek,et al.  Metamathematics of Fuzzy Logic , 1998, Trends in Logic.

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

[53]  Birte Glimm,et al.  Extended Caching, Backjumping and Merging for Expressive Description Logics , 2012, IJCAR.