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Characterizing Boundedness in Chase Variants
Published online by Cambridge University Press: 04 June 2020
Abstract
Existential rules are a positive fragment of first-order logic that generalizes function-free Horn rules by allowing existentially quantified variables in rule heads. This family of languages has recently attracted significant interest in the context of ontology-mediated query answering. Forward chaining, also known as the chase, is a fundamental tool for computing universal models of knowledge bases, which consist of existential rules and facts. Several chase variants have been defined, which differ on the way they handle redundancies. A set of existential rules is bounded if it ensures the existence of a bound on the depth of the chase, independently from any set of facts. Deciding if a set of rules is bounded is an undecidable problem for all chase variants. Nevertheless, when computing universal models, knowing that a set of rules is bounded for some chase variant does not help much in practice if the bound remains unknown or even very large. Hence, we investigate the decidability of the k-boundedness problem, which asks whether the depth of the chase for a given set of rules is bounded by an integer k. We identify a general property which, when satisfied by a chase variant, leads to the decidability of k-boundedness. We then show that the main chase variants satisfy this property, namely the oblivious, semi-oblivious (aka Skolem), and restricted chase, as well as their breadth-first versions.
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References
Abiteboul, S. 1989. Boundedness is undecidable for datalog programs with a single recursive rule. Information Processing Letters 32, 6, 281–287.CrossRefGoogle Scholar
Abiteboul, S., Hull, R. and Vianu, V. 1995. Foundations of Databases: The Logical Level, 1st ed. Addison-Wesley Longman Publishing Co., Inc., Boston, MA, USA.Google Scholar
Baader, F., Brandt, S. and Lutz, C. 2005. Pushing the EL envelope. In Proceedings of the 19th International Joint Conference on Artificial Intelligence, IJCAI’05. Morgan Kaufmann Publishers Inc., San Francisco, CA, USA, 364–369.Google Scholar
Baget, J., Garreau, F., Mugnier, M. and Rocher, S. 2014. Extending acyclicity notions for existential rules. In ECAI 2014 - 21st European Conference on Artificial Intelligence, 18-22 August 2014, Prague, Czech Republic – Including Prestigious Applications of Intelligent Systems (PAIS 2014), 39–44.Google Scholar
Baget, J., Leclère, M. and Mugnier, M. 2010. Walking the decidability line for rules with existential variables. In Proceedings of the Twelfth International Conference on Principles of Knowledge Representation and Reasoning, KR’10. AAAI Press, 466–476.Google Scholar
Baget, J., Leclère, M., Mugnier, M. and Salvat, E. 2009. Extending decidable cases for rules with existential variables. In Proceedings of the 21st International Jont Conference on Artifical Intelligence, IJCAI’09. Morgan Kaufmann Publishers Inc., San Francisco, CA, USA, 677–682.Google Scholar
Baget, J., Mugnier, M., Rudolph, S. and Thomazo, M. 2011. Walking the complexity lines for generalized guarded existential rules. In IJCAI 2011, Proceedings of the 22nd International Joint Conference on Artificial Intelligence, Barcelona, Catalonia, Spain, July 16–22, 2011, 712–717.Google Scholar
Baget, J.-F., Leclère, M., Mugnier, M.-L. and Salvat, E. 2011. On rules with existential variables: Walking the decidability line. Artificial Intelligence 175, 9–10, 1620–1654.CrossRefGoogle Scholar
Beeri, C. and Vardi, M. Y. 1981. The implication problem for data dependencies. In International Colloquium on Automata, Languages, and Programming. Springer, 73–85.Google Scholar
Beeri, C. and Vardi, M. Y. 1984. A proof procedure for data dependencies. Journal of the Association for Computing Machinery 31, 4 (September), 718-741.CrossRefGoogle Scholar
Bourhis, P., Leclère, M., Mugnier, M., Tison, S., Ulliana, F. and Gallois, L. 2019. Oblivious and semi-oblivious boundedness for existential rules. In Proceedings of the Twenty-Eighth International Joint Conference on Artificial Intelligence, IJCAI 2019, Macao, China, August 10–16, 2019, 1581–1587.Google Scholar
Calì, A., Gottlob, G. and Kifer, M. 2008. Taming the infinite chase: Query answering under expressive relational constraints. In Proceedings of the Eleventh International Conference on Principles of Knowledge Representation and Reasoning, KR’08. AAAI Press, 70-80.Google Scholar
Calì, A., Gottlob, G. and Lukasiewicz, T. 2009. A general datalog-based framework for tractable query answering over ontologies. In Proceedings of the Twenty-Eighth ACM SIGMOD-SIGACT-SIGART Symposium on Principles of Database Systems, PODS’09. Association for Computing Machinery, New York, NY, USA, 77-86.Google Scholar
Calvanese, D., De Giacomo, G., Lembo, D., Lenzerini, M. and Rosati, R. 2005. DL-Lite: Tractable description logics for ontologies. In Proceedings of the 20th National Conference on Artificial Intelligence – Volume 2, AAAI’05. AAAI Press, 602–607.Google Scholar
Calvanese, D., De Giacomo, G., Lembo, D., Lenzerini, M. and Rosati, R. 2007. Tractable reasoning and efficient query answering in description logics: The DL-Lite family. Journal of Automated Reasoning (JAR) 39, 3, 385–429.CrossRefGoogle Scholar
Carral, D., Dragoste, I. and Krötzsch, M. 2017. Restricted chase (non)termination for existential rules with disjunctions. In Proceedings of the Twenty-Sixth International Joint Conference on Artificial Intelligence, IJCAI 2017, Melbourne, Australia, August 19–25, 2017, 922–928.Google Scholar
Chandra, A. K. and Merlin, P. M. 1977. Optimal implementation of conjunctive queries in relational data bases. In Proceedings of the 9th Annual ACM Symposium on Theory of Computing, May 4–6, 1977, Boulder, Colorado, USA, 77–90.Google Scholar
Cosmadakis, S. S., Gaifman, H., Kanellakis, P. C. and Vardi, M. Y. 1988. Decidable optimization problems for database logic programs (preliminary report). In ACM Symposium on Theory of Computing, 477–490.Google Scholar
Cuenca Grau, B., Horrocks, I., Krötzsch, M., Kupke, C., Magka, D., Motik, B. and Wang, Z. 2013. Acyclicity notions for existential rules and their application to query answering in ontologies. Journal of Artificial Intelligence Research 47, 1 (May), 741-808.CrossRefGoogle Scholar
Delivorias, S., Leclère, M., Mugnier, M. and Ulliana, F. 2018. On the k-boundedness for existential rules. In Computing Research Repository (initially published in Rules and Reasoning – Second International Joint Conference, RuleML+RR 2018, Luxembourg, September 18–21, 2018, Proceedings) abs/1810.09304.Google Scholar
Deutsch, A., Nash, A. and Remmel, J. B. 2008. The chase revisited. In Proceedings of the Twenty-Seventh ACM SIGMOD-SIGACT-SIGART Symposium on Principles of Database Systems, PODS ’08. Association for Computing Machinery, New York, NY, USA, 149-158.Google Scholar
Fagin, R., Kolaitis, P. G., Miller, R. J. and Popa, L. 2005. Data exchange: Semantics and query answering. Theoretical Computer Science 336, 1 (May), 89–124.CrossRefGoogle Scholar
Gaifman, H., Mairson, H. G., Sagiv, Y. and Vardi, M. Y. 1993. Undecidable optimization problems for database logic programs. Journal of the Association for Computing Machinery 40, 3 (July), 683–713.CrossRefGoogle Scholar
Gogacz, T. and Marcinkowski, J. 2014. All-instances termination of chase is undecidable. In International Colloquium on Automata, Languages, and Programming 2014 Proceedings, Part II, 293-304.Google Scholar
Gottlob, G., Orsi, G., Pieris, A. and Simkus, M. 2012. Datalog and its extensions for semantic web databases. In Reasoning Web, Eiter, T. and Krennwallner, T., Eds. Springer Berlin Heidelberg, Berlin, Heidelberg, 54–77.Google Scholar
Grahne, G. and Onet, A. 2018. Anatomy of the chase. Fundamenta Informaticae 157, 3, 221–270.CrossRefGoogle Scholar
Guessarian, I. and Peixoto, M. V. 1994. About boundedness for some datalog and DATALOGneg programs. Journal of Logic and Computation 4, 4, 375–403.CrossRefGoogle Scholar
Hernich, A. and Schweikardt, N. 2007. CWA-solutions for data exchange settings with target dependencies. In Proceedings of the Twenty-Sixth ACM SIGACT-SIGMOD-SIGART Symposium on Principles of Database Systems, June 11–13, 2007, Beijing, China, 113–122.Google Scholar
Hillebrand, G. G., Kanellakis, P. C., Mairson, H. G. and Vardi, M. Y. 1995. Undecidable boundedness problems for datalog programs. Journal of Logic Programming 25, 2, 163–190.CrossRefGoogle Scholar
Krötzsch, M. and Rudolph, S. 2011. Extending decidable existential rules by joining acyclicity and guardedness. In Proceedings of the Twenty-Second International Joint Conference on Artificial Intelligence – Volume Volume Two. IJCAI’11. AAAI Press, 963–968.Google Scholar
Krötzsch, M., Rudolph, S. and Hitzler, P. 2007. Complexity boundaries for Horn description logics. In Proceedings of the 22nd National Conference on Artificial Intelligence – Volume 1, AAAI’07. AAAI Press, 452–457.Google Scholar
Leclère, M., Mugnier, M. and Ulliana, F. 2016. On bounded positive existential rules. In Proceedings of the 29th International Workshop on Description Logics.Google Scholar
Leone, N., Manna, M., Terracina, G. and Veltri, P. 2012. Efficiently computable Datalog∃ programs. In Proceedings of the Thirteenth International Conference on Principles of Knowledge Representation and Reasoning, KR’12. AAAI Press, 13–23.Google Scholar
Marcinkowski, J. 1999. Achilles, turtle, and undecidable boundedness problems for small datalog programs. Society for Industrial and Applied Mathematics Journal on Computing 29, 1, 231–257.Google Scholar
Marnette, B. 2009. Generalized schema-mappings: From termination to tractability. In Proceedings of the Twenty-Eighth ACM SIGMOD-SIGACT-SIGART Symposium on Principles of Database Systems, PODS’09. Association for Computing Machinery, New York, NY, USA, 13–22.Google Scholar
Mugnier, M. and Thomazo, M. 2014. An introduction to ontology-based query answering with existential rules. In Reasoning Web 2014, Koubarakis, M., Stamou, G., Stoilos, G., Horrocks, I., Kolaitis, P., Lausen, G. and Weikum, G., Eds. Springer International Publishing, Cham, 245–278.Google Scholar
Rocher, S. 2016. Querying existential rule knowledge bases: Decidability and complexity (Interrogation de Bases de Connaissances avec Règles Existentielles : Décidabilité et Complexité). Ph.D. thesis, University of Montpellier, France.Google Scholar
Thomazo, M., Baget, J., Mugnier, M. and Rudolph, S. 2012. A generic querying algorithm for greedy sets of existential rules. In Principles of Knowledge Representation and Reasoning: Proceedings of the Thirteenth International Conference, KR 2012, Rome, Italy, June 10–14, 2012.Google Scholar