The limits of decidability in fuzzy description logics with general concept inclusions

Fuzzy description logics (DLs) can be used to represent and reason with vague knowledge. This family of logical formalisms is very diverse, each member being characterized by a specific choice of constructors, axioms, and triangular norms, which are used to specify the semantics. Unfortunately, it has recently been shown that the consistency problem in many fuzzy DLs with general concept inclusion axioms is undecidable. In this paper, we present a proof framework that allows us to extend these results to cover large classes of fuzzy DLs. On the other hand, we also provide matching decidability results for most of the remaining logics. As a result, we obtain a near-universal classification of fuzzy DLs according to the decidability of their consistency problem.

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

[2]  Christian Eitzinger,et al.  Triangular Norms , 2001, Künstliche Intell..

[3]  Fernando Bobillo,et al.  Reasoning in Fuzzy OWL 2 with DeLorean , 2010, URSW.

[4]  Àngel García-Cerdaña,et al.  Fuzzy Description Logics and t-norm based fuzzy logics , 2010, Int. J. Approx. Reason..

[5]  Jan Hladik.  A Tableau System for the Description Logic SHIO , 2004, IJCAR Doctoral Programme.

[6]  Rafael Peñaloza,et al.  Non-Gödel Negation Makes Unwitnessed Consistency Undecidable , 2012, Description Logics.

[7]  Marco Cerami,et al.  From classical Description Logic to n-graded Fuzzy Description Logic , 2010, International Conference on Fuzzy Systems.

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

[9]  Rafael Peñaloza,et al.  Consistency reasoning in lattice-based fuzzy Description Logics , 2014, Int. J. Approx. Reason..

[10]  Rafael Peñaloza,et al.  How Fuzzy Is My Fuzzy Description Logic? , 2012, IJCAR.

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

[12]  Ian Horrocks,et al.  Keys, Nominals, and Concrete Domains , 2003, IJCAI.

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

[14]  Umberto Straccia,et al.  On Qualified Cardinality Restrictions in Fuzzy Description Logics under Ëukasiewicz semantics , 2008 .

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

[16]  Rafael Peñaloza,et al.  GCIs Make Reasoning in Fuzzy DL with the Product T-norm Undecidable , 2011, Description Logics.

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

[18]  Umberto Straccia,et al.  On the Failure of the Finite Model Property in some Fuzzy Description Logics , 2010, Fuzzy Sets Syst..

[19]  Petr Hájek,et al.  Residuated fuzzy logics with an involutive negation , 2000, Arch. Math. Log..

[20]  Rafael Peñaloza,et al.  Are fuzzy description logics with general concept inclusion axioms decidable? , 2011, 2011 IEEE International Conference on Fuzzy Systems (FUZZ-IEEE 2011).

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

[22]  Stefan Borgwardt,et al.  Fuzzy DLs over Finite Lattices with Nominals , 2014, Description Logics.

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

[24]  Marco Cerami,et al.  Decidability of a Description Logic over Infinite-Valued Product Logic , 2010, KR.

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

[26]  I. Horrocks,et al.  A PSPACE-algorithm for deciding ALCNIR+-satisfiability , 1998 .

[27]  C. Tresp,et al.  Extending Description Logics to Vague Knowledge in Medicine , 2000 .

[28]  Umberto Straccia,et al.  Combining Fuzzy Logic and Semantic Web to Enable Situation-Awareness in Service Recommendation , 2010, DEXA.

[29]  Nguyen Hoang Nga,et al.  THE FUZZY DESCRIPTION LOGIC ALCFLH , 2005 .

[30]  Grant DA-AR.  A Fuzzy-Set-Theoretic Interpretation of Linguistic Hedges , 2015 .

[31]  Emil L. Post.  A variant of a recursively unsolvable problem , 1946 .

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

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

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

[35]  Diego Calvanese,et al.  The Description Logic Handbook , 2007 .

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

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

[38]  Stefanos D. Kollias,et al.  Tractable reasoning with vague knowledge using fuzzy $\mathcal{EL}^{++}$ , 2012, Journal of Intelligent Information Systems.

[39]  P ? ? ? ? ? ? ? % ? ? ? ? , 1991 .

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

[41]  Rafael Peñaloza,et al.  Chair for Automata Theory LTCS – Report Towards a Tableau Algorithm for Fuzzy ALC with Product T-norm , 2011 .

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

[43]  P. Mostert,et al.  On the Structure of Semigroups on a Compact Manifold With Boundary , 1957 .

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

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

[46]  Rafael Peñaloza,et al.  Gödel Description Logics with General Models , 2014, Description Logics.

[47]  John Yen,et al.  Generalizing Term Subsumption Languages to Fuzzy Logic , 1991, IJCAI.

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

[49]  Yevgeny Kazakov,et al.  RIQ and SROIQ Are Harder than SHOIQ , 2008, KR.

[50]  Rafael Peñaloza,et al.  Gödel Negation Makes Unwitnessed Consistency Crisp , 2012, Description Logics.

[51]  Umberto Straccia,et al.  fuzzyDL: An expressive fuzzy description logic reasoner , 2008, 2008 IEEE International Conference on Fuzzy Systems (IEEE World Congress on Computational Intelligence).

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

[53]  Boris Motik,et al.  Hypertableau Reasoning for Description Logics , 2009, J. Artif. Intell. Res..

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

[55]  Umberto Straccia,et al.  Description Logics with Fuzzy Concrete Domains , 2005, UAI.

[56]  Paulo J. G. Lisboa,et al.  Fuzzy systems in medicine , 2000 .

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

[58]  Umberto Straccia,et al.  Aggregation operators for fuzzy ontologies , 2013, Appl. Soft Comput..

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

[60]  Rafael Peñaloza,et al.  The Complexity of Lattice-Based Fuzzy Description Logics , 2012, Journal on Data Semantics.

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

[62]  Laurence A. Wolsey,et al.  Mixed Integer Programming , 2008, Wiley Encyclopedia of Computer Science and Engineering.

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

[64]  Rafael Peñaloza,et al.  Undecidability of Fuzzy Description Logics , 2012, KR.

[65]  Rafael Peñaloza,et al.  Positive Subsumption in Fuzzy EL with General t-Norms , 2013, IJCAI.