Decidability and Complexity of ALCOIF with Transitive Closure (and More)

We prove that satisfiability and finite satisfiability in the description logic ALCOIFreg are NExpTime-complete when every regular role expression of the form α∗ contains either no functional role or only a single role name (and possibly its inverse). Notably, this encompasses the extension of ALCOIF with transitive closure of roles and the modal logic of linear orders and successor, with converse.

[1]  Richard E. Ladner,et al.  Propositional Dynamic Logic of Regular Programs , 1979, J. Comput. Syst. Sci..

[2]  Maurizio Lenzerini,et al.  TBox and ABox Reasoning in Expressive Description Logics , 1996, KR.

[3]  Helmut Veith,et al.  Extending ALCQIO with Trees , 2015, 2015 30th Annual ACM/IEEE Symposium on Logic in Computer Science.

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

[5]  Christoph Haase,et al.  A survival guide to presburger arithmetic , 2018, SIGL.

[6]  Thomas Schwentick,et al.  On the Complexity of Equational Horn Clauses , 2005, CADE.

[7]  Olivier Curé,et al.  A Decision Procedure for SHOIQ with Transitive Closure of Roles , 2013, SEMWEB.

[8]  Myriam Lamolle,et al.  Decidability of Description Logics with Transitive Closure of Roles in Concept and Role Inclusion Axioms , 2010, Description Logics.

[9]  Franz Baader.  Augmenting Concept Languages by Transitive Closure of Roles: An Alternative to Terminological Cycles , 1991, IJCAI.

[10]  Emanuel Kieronski,et al.  Decidability Issues for Two-Variable Logics with Several Linear Orders , 2011, CSL.

[11]  Moshe Y. Vardi.  Reasoning about The Past with Two-Way Automata , 1998, ICALP.

[12]  Erich Grädel,et al.  Two-variable logic with counting is decidable , 1997, Proceedings of Twelfth Annual IEEE Symposium on Logic in Computer Science.

[13]  Maurizio Lenzerini,et al.  Boosting the Correspondence between Description Logics and Propositional Dynamic Logics , 1994, AAAI.

[14]  Piero A. Bonatti,et al.  On the undecidability of logics with converse, nominals, recursion and counting , 2004, Artif. Intell..

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

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

[17]  Ian Pratt-Hartmann.  Complexity of the Two-Variable Fragment with Counting Quantifiers , 2005, J. Log. Lang. Inf..

[18]  Martin Otto,et al.  Small substructures and decidability issues for first-order logic with two variables , 2005, 20th Annual IEEE Symposium on Logic in Computer Science (LICS' 05).

[19]  Leszek Pacholski,et al.  Complexity Results for First-Order Two-Variable Logic with Counting , 2000, SIAM J. Comput..

[20]  Martin Otto,et al.  Two variable first-order logic over ordered domains , 2001, Journal of Symbolic Logic.

[21]  Ian Pratt-Hartmann,et al.  The two‐variable fragment with counting and equivalence , 2015, Math. Log. Q..

[22]  Thomas Zeume,et al.  Order-Invariance of Two-Variable Logic is Decidable , 2016, 2016 31st Annual ACM/IEEE Symposium on Logic in Computer Science (LICS).

[23]  Lidia Tendera,et al.  Two-variable logics with counting and semantic constraints , 2018, SIGL.

[24]  H. Jerome Keisler,et al.  Model theory for infinitary logic : logic with countable conjunctions and finite quantifiers , 1971 .

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

[26]  Martin Otto,et al.  Undecidability Results on Two-Variable Logics , 1997, STACS.

[27]  Sebastian Rudolph,et al.  Undecidability Results for Database-Inspired Reasoning Problems in Very Expressive Description Logics , 2016, KR.

[28]  Ian Pratt-Hartmann,et al.  Complexity of the Guarded Two-variable Fragment with Counting Quantifiers , 2006, J. Log. Comput..

[29]  Giora Slutzki,et al.  Alternating Tree Automata , 1983, Theor. Comput. Sci..

[30]  Aniello Murano,et al.  The Complexity of Enriched µ-Calculi , 2006, ICALP.

[31]  Giuseppe De Giacomo.  Decidability of Class Based Knowledge Representation Formalisms , 2009 .

[32]  Thomas Schwentick,et al.  Two-Variable Logic with Two Order Relations - (Extended Abstract) , 2010, CSL.