Model Comparison Games for Horn Description Logics

Horn description logics are syntactically defined fragments of standard description logics that fall within the Horn fragment of first-order logic and for which ontology-mediated query answering is in PTIME for data complexity. They were independently introduced in modal logic to capture the intersection of Horn first-order logic with modal logic. In this paper, we introduce model comparison games for the basic Horn description logic $\pmb{horn}\mathcal{ALC}$ (corresponding to the basic Horn modal logic) and use them to obtain an Ehrenfeucht-Frafsse type definability result and a van Benthem style expressive completeness result for $\pmb{horn}\mathcal{ALC}$. We also establish a finite model theory version of the latter. The Ehrenfeucht-Frafsse type definability result is used to show that checking $\pmb{horn}\mathcal{ALC}$ indistinguishability of models is ExpTIME-complete, which is in sharp contrast to $\mathcal{ALC}$ indistinguishability (i.e., bisimulation equivalence) checkable in PTIME. In addition, we explore the behavior of Horn fragments of more expressive description and modal logics by defining a Horn guarded fragment of first-order logic and introducing model comparison games for it.

[1]  Eric Rosen,et al.  Modal Logic over Finite Structures , 1997, J. Log. Lang. Inf..

[2]  Carsten Lutz,et al.  The Data Complexity of Description Logic Ontologies , 2016, Log. Methods Comput. Sci..

[3]  Alfred Horn,et al.  On sentences which are true of direct unions of algebras , 1951, Journal of Symbolic Logic.

[4]  Martin Otto,et al.  The Freedoms of Guarded Bisimulation , 2011, CSL.

[5]  Andrea Calì,et al.  A general datalog-based framework for tractable query answering over ontologies , 2009, SEBD.

[6]  Marcelo Arenas,et al.  The Exact Complexity of the First-Order Logic Definability Problem , 2016, ACM Trans. Database Syst..

[7]  Balder ten Cate,et al.  The Product Homomorphism Problem and Applications , 2015, ICDT.

[8]  Balder ten Cate,et al.  Guarded Negation , 2011, Advances in Modal Logic.

[9]  Carsten Lutz,et al.  Deciding inseparability and conservative extensions in the description logic EL , 2010, J. Symb. Comput..

[10]  Anuj Dawar,et al.  Modal characterisation theorems over special classes of frames , 2005, 20th Annual IEEE Symposium on Logic in Computer Science (LICS' 05).

[11]  Magdalena Ortiz,et al.  Ontology-Mediated Query Answering with Data-Tractable Description Logics , 2015, Reasoning Web.

[12]  David Harel,et al.  On the Complexity of Verifying Concurrent Transition Systems , 1997, Inf. Comput..

[13]  Erich Grädel,et al.  On the Restraining Power of Guards , 1999, Journal of Symbolic Logic.

[14]  Ian Horrocks,et al.  An Introduction to Description Logic , 2017 .

[15]  I-Peng Lin,et al.  The Computational Complexity of Satisfiability of Temporal Horn Formulas in Propositional Linear-Time Temporal Logic , 1993, Inf. Process. Lett..

[16]  Carsten Lutz,et al.  Dichotomies in Ontology-Mediated Querying with the Guarded Fragment , 2017, PODS.

[17]  Liviu Badea,et al.  A Refinement Operator for Description Logics , 2000, ILP.

[18]  Georg Gottlob,et al.  Querying the Guarded Fragment , 2010, 2010 25th Annual IEEE Symposium on Logic in Computer Science.

[19]  Georg Gottlob,et al.  Query Answering in the Description Logic Horn- , 2008, JELIA.

[20]  Carsten Lutz,et al.  Query Inseparability for ALC Ontologies , 2019, Artif. Intell..

[21]  Robert Piro,et al.  Description Logic TBoxes: Model-Theoretic Characterizations and Rewritability , 2011, IJCAI.

[22]  Daniel Gorín,et al.  Simulations and Bisimulations for Coalgebraic Modal Logics , 2013, CALCO.

[23]  Boris Motik,et al.  Reasoning in Description Logics by a Reduction to Disjunctive Datalog , 2007, Journal of Automated Reasoning.

[24]  J. C. C. McKinsey,et al.  The decision problem for some classes of sentences without quantifiers , 1943, Journal of Symbolic Logic.

[25]  Barbara König,et al.  A van Benthem Theorem for Fuzzy Modal Logic , 2018, LICS.

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

[27]  Franz Baader,et al.  Pushing the EL Envelope , 2005, IJCAI.

[28]  M. de Rijke,et al.  Expressiveness of Concept Expressions in First-Order Description Logics , 1999, Artif. Intell..

[29]  William W. Cohen,et al.  Learning the Classic Description Logic: Theoretical and Experimental Results , 1994, KR.

[30]  Yevgeny Kazakov,et al.  Ontology Materialization by Abstraction Refinement in Horn SHOIF , 2017, Description Logics.

[31]  Diego Calvanese,et al.  Tractable Reasoning and Efficient Query Answering in Description Logics: The DL-Lite Family , 2007, Journal of Automated Reasoning.

[32]  Martin Otto,et al.  Highly Acyclic Groups, Hypergraph Covers and the Guarded Fragment , 2010, 2010 25th Annual IEEE Symposium on Logic in Computer Science.

[33]  Diego Calvanese,et al.  The DL-Lite Family and Relations , 2009, J. Artif. Intell. Res..

[34]  Valentin Goranko,et al.  Model theory of modal logic , 2007, Handbook of Modal Logic.

[35]  Boris Motik,et al.  Acyclicity Conditions and their Application to Query Answering in Description Logics , 2012, KR.

[36]  Chen C. Chang,et al.  Model Theory: Third Edition (Dover Books On Mathematics) By C.C. Chang;H. Jerome Keisler;Mathematics , 1966 .

[37]  Pablo Barceló,et al.  The complexity of reverse engineering problems for conjunctive queries , 2016, ICDT.

[38]  Carsten Lutz,et al.  First Order-Rewritability and Containment of Conjunctive Queries in Horn Description Logics , 2016, Description Logics.

[39]  Johan van Benthem,et al.  Modal Languages and Bounded Fragments of Predicate Logic , 1998, J. Philos. Log..

[40]  Sebastian Rudolph,et al.  Complexities of Horn Description Logics , 2013, TOCL.

[41]  D. Nardi,et al.  An Introduction to Description Logic , 2017 .

[42]  Davide Bresolin,et al.  On the Expressive Power of Sub-Propositional Fragments of Modal Logic , 2016, GandALF.

[43]  Luis Fariñas del Cerro,et al.  A Note of the Complexity of the Satisfiability of Modal Horn Clauses , 1987, J. Log. Program..

[44]  Tadeusz Litak,et al.  A Van Benthem/Rosen theorem for coalgebraic predicate logic , 2015, J. Log. Comput..

[45]  Jean Christoph Jung,et al.  Reverse Engineering Queries in Ontology-Enriched Systems: The Case of Expressive Horn Description Logic Ontologies , 2018, IJCAI.

[46]  Boris Motik,et al.  Data Complexity of Reasoning in Very Expressive Description Logics , 2005, IJCAI.

[47]  Martin Otto Modal and guarded characterisation theorems over finite transition systems , 2004, Ann. Pure Appl. Log..

[48]  Carsten Lutz,et al.  Horn-Rewritability vs PTime Query Evaluation in Ontology-Mediated Querying , 2018, IJCAI.

[49]  Joseph Michael Weinstein First order properties preserved by direct product , 1965 .

[50]  Holger Sturm Modal Horn Classes , 2000, Stud Logica.

[51]  Franz Baader,et al.  Query and Predicate Emptiness in Ontology-Based Data Access , 2016, J. Artif. Intell. Res..

[52]  Jens Lehmann,et al.  DL-Learner: Learning Concepts in Description Logics , 2009, J. Mach. Learn. Res..

[53]  Moshe Y. Vardi The complexity of relational query languages (Extended Abstract) , 1982, STOC '82.

[54]  G. Gottlob,et al.  Query Answering in the Description Logic Horn-SHIQ ⋆ , 2008 .

[55]  Sebastian Rudolph,et al.  Query Answering in the Horn Fragments of the Description Logics SHOIQ and SROIQ , 2011, IJCAI.

[56]  Martin Grohe,et al.  Learning first-order definable concepts over structures of small degree , 2017, 2017 32nd Annual ACM/IEEE Symposium on Logic in Computer Science (LICS).

[57]  Yevgeny Kazakov,et al.  Consequence-Driven Reasoning for Horn SHIQ Ontologies , 2009, IJCAI.

[58]  Linh Anh Nguyen On the Complexity of Fragments of Modal Logics , 2004, Advances in Modal Logic.

[59]  Andrea Calì,et al.  Taming the Infinite Chase: Query Answering under Expressive Relational Constraints , 2008, Description Logics.

[60]  Frank Wolter,et al.  Games for query inseparability of description logic knowledge bases , 2016, Artif. Intell..