Hostname: page-component-546b4f848f-w58md Total loading time: 0 Render date: 2023-06-02T11:44:23.427Z Has data issue: false Feature Flags: { "useRatesEcommerce": true } hasContentIssue false

Efficient TBox Reasoning with Value Restrictions using the ${\cal F}{{\cal L}_0}$wer Reasoner

Published online by Cambridge University Press:  15 October 2021

Technische Universitat Dresden, Dresden, Germany (e-mails:,,,,
Technische Universitat Dresden, Dresden, Germany (e-mails:,,,,
Technische Universitat Dresden, Dresden, Germany (e-mails:,,,,
Technische Universitat Dresden, Dresden, Germany (e-mails:,,,,
Technische Universitat Dresden, Dresden, Germany (e-mails:,,,,
Rights & Permissions[Opens in a new window]


HTML view is not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

The inexpressive Description Logic (DL) ${\cal F}{{\cal L}_0}$ , which has conjunction and value restriction as its only concept constructors, had fallen into disrepute when it turned out that reasoning in ${\cal F}{{\cal L}_0}$ w.r.t. general TBoxes is ExpTime-complete, that is, as hard as in the considerably more expressive logic ${\cal A}{\cal L}{\cal C}$ . In this paper, we rehabilitate ${\cal F}{{\cal L}_0}$ by presenting a dedicated subsumption algorithm for ${\cal F}{{\cal L}_0}$ , which is much simpler than the tableau-based algorithms employed by highly optimized DL reasoners. Our experiments show that the performance of our novel algorithm, as prototypically implemented in our ${\cal F}{{\cal L}_0}$ wer reasoner, compares very well with that of the highly optimized reasoners. ${\cal F}{{\cal L}_0}$ wer can also deal with ontologies written in the extension ${\cal F}{{\cal L}_ \bot }$ of ${\cal F}{{\cal L}_0}$ with the top and the bottom concept by employing a polynomial-time reduction, shown in this paper, which eliminates top and bottom. We also investigate the complexity of reasoning in DLs related to the Horn-fragments of ${\cal F}{{\cal L}_0}$ and ${\cal F}{{\cal L}_ \bot }$ .

Original Article
Creative Commons
Creative Common License - CCCreative Common License - BY
This is an Open Access article, distributed under the terms of the Creative Commons Attribution licence (, which permits unrestricted re-use, distribution, and reproduction in any medium, provided the original work is properly cited.
© The Author(s), 2021. Published by Cambridge University Press



This paper is under consideration in Theory and Practice of Logic Programming (TPLP).


Baader, F. 1990. Terminological cycles in KL-ONE-based knowledge representation languages. In Proc. of the 8th Nat. Conf. on Artificial Intelligence (AAAI’90), Boston, MA, USA, 621–626.Google Scholar
Baader, F., Brandt, S. and Lutz, C. 2005. Pushing the ${\cal E}{\cal L}$ envelope. In Proc. of the 19th Int. Joint Conf. on Artificial Intelligence (IJCAI 2005), Kaelbling, L. P. and Saffiotti, A., Eds. Edinburgh, UK. Morgan Kaufmann, Los Altos, 364–369.Google Scholar
Baader, F., Calvanese, D., McGuinness, D., Nardi, D. and Patel-Schneider, P. F., Eds. 2003. The Description Logic Handbook: Theory, Implementation, and Applications. Cambridge University Press, New York.Google Scholar
Baader, F., Fernandez Gil, O. and Marantidis, P. 2018. Matching in the description logic ${\cal F}{{\cal L}_0}$ with respect to general TBoxes. In LPAR-22. 22nd International Conference on Logic for Programming, Artificial Intelligence and Reasoning, Barthe, G., Sutcliffe, G. and Veanes, M., Eds. vol. 57 of EPiC Series in Computing. EasyChair, 76–94.Google Scholar
Baader, F., Fernandez Gil, O. and Pensel, M. 2018a. Standard and non-standard inferences in the description logic ${\cal F}{{\cal L}_0}$ using tree automata. In GCAI-2018, 4th Global Conference on Artificial Intelligence, Lee, D. D., Steen, A. and Walsh, T., Eds. vol. 55 of EPiC Series in Computing. EasyChair, 1–14.Google Scholar
Baader, F., Horrocks, I., Lutz, C. and Sattler, U. 2017. An Introduction to Description Logic. Cambridge University Press.CrossRefGoogle Scholar
Baader, F., Marantidis, P. and Okhotin, A. 2016. Approximate unification in the description logic ${\cal F}{{\cal L}_0}$ . In Logics in Artificial Intelligence - 15th European Conference, JELIA 2016, Proceedings, Michael, L. and Kakas, A. C., Eds. vol. 10021 of Lecture Notes in Computer Science, 49–63.Google Scholar
Baader, F., Marantidis, P. and Pensel, M. 2018b. The data complexity of answering instance queries in ${\cal F}{{\cal L}_0}$ . In Companion of the The Web Conference WWW, Champin, P., Gandon, F. L., Lalmas, M. and Ipeirotis, P. G., Eds. ACM, 1603–1607.Google Scholar
Baader, F. and Sattler, U. 2001. An overview of tableau algorithms for description logics. Studia Logica 69, 1, 540.CrossRefGoogle Scholar
Baader, F. and Théron, C. 2020. Role-value maps and general concept inclusions in the minimal description logic with value restrictions – or revisiting old skeletons in the DL cupboard. KI – Journal für Künstliche Intelligenz 34, 3, 291301.CrossRefGoogle Scholar
Brachman, R. J., McGuinness, D. L., Patel-Schneider, P. F., Alperin Resnick, L. and Borgida, A. 1991. Living with CLASSIC: When and how to use a KL-ONE-like language. In Principles of Semantic Networks, Sowa, J. F., Eds. Morgan Kaufmann, Los Altos, 401–456.Google Scholar
Brachman, R. J. and Schmolze, J. G. 1985. An overview of the KL-ONE knowledge representation system. Cognitive Science 9, 2, 171216.CrossRefGoogle Scholar
Brandt, S. 2004. Polynomial time reasoning in a description logic with existential restrictions, GCI axioms, and—what else? In Proc. of the 16th Eur. Conf. on Artificial Intelligence (ECAI 2004), de Mántaras, R. L. and Saitta, L., Eds., pp. 298–302.Google Scholar
Cuenca Grau, B., Horrocks, I., Motik, B., Parsia, B., Patel-Schneider, P. F. and Sattler, U. 2008. OWL 2: The next step for OWL. Journal of Web Semantics 6, 4, 309322.CrossRefGoogle Scholar
Forgy, C. 1982. Rete: A fast algorithm for the many patterns/many objects match problem. Artificial Intelligence 19, 1, 1737.CrossRefGoogle Scholar
Hoehndorf, R., Schofield, P. N. and Gkoutos, G. V. 2015. The role of ontologies in biological and biomedical research: A functional perspective. Briefings in Bioinformatics 16, 6, 10691080.CrossRefGoogle ScholarPubMed
Hofmann, M. 2005. Proof-theoretic approach to description-logic. In Proc. of the 20th IEEE Symp. on Logic in Computer Science (LICS 2005), Panangaden, P., Ed. IEEE Computer Society Press, 229–237.Google Scholar
Horridge, M. and Bechhofer, S. 2011. The OWL API: A Java API for OWL ontologies. Semantic Web 2, 1, 1121.CrossRefGoogle Scholar
Horrocks, I., Patel-Schneider, P. F. and van Harmelen, F. 2003. From SHIQ and RDF to OWL: The making of a web ontology language. Journal of Web Semantics 1, 1, 726.CrossRefGoogle Scholar
Kazakov, Y. 2009. Consequence-driven reasoning for Horn ${\cal S}{\cal H}{\cal I}{\cal Q}$ ontologies. In Proc. of the 21st Int. Joint Conf. on Artificial Intelligence (IJCAI 2009), C. Boutilier, Ed. IJCAI/AAAI, 2040–2045.Google Scholar
Kazakov, Y. and de Nivelle, H. 2003. Subsumption of concepts in ${\cal F}{{\cal L}_0}$ for (cyclic) terminologies with respect to descriptive semantics is PSPACE-complete. In Proc. of the 2003 Description Logic Workshop (DL 2003). CEUR Electronic Workshop Proceedings, Scholar
Krötzsch, M., Rudolph, S. and Hitzler, P. 2007. Complexity boundaries for Horn description logics. In Proceedings of the Twenty-Second AAAI Conference on Artificial Intelligence, July 22–26, 2007, Vancouver, British Columbia, Canada. AAAI Press, 452457.Google Scholar
Krötzsch, M., Rudolph, S. and Hitzler, P. 2013. Complexities of Horn description logics. ACM Transactions on Computational Logic 14, 1, 2:12:36.CrossRefGoogle Scholar
Matentzoglu, N., Bail, S. and Parsia, B. 2013. A snapshot of the OWL Web. In The Semantic Web - ISWC 2013 - 12th International Semantic Web Conference, Sydney, NSW, Australia, October 21–25, 2013, Proceedings, Part I, Alani, H., Kagal, L., Fokoue, A., Groth, P. T., Biemann, C., Parreira, J. X., Aroyo, L., Noy, N. F., Welty, C. and Janowicz, K., Eds. vol. 8218 of Lecture Notes in Computer Science. Springer, 331–346.Google Scholar
Mays, E., Dionne, R. and Weida, R. 1991. K-REP system overview. SIGART Bulletin 2, 3.CrossRefGoogle Scholar
Michel, F., Turhan, A.-Y. and Zarriess, B. 2019. Efficient TBox reasoning with value restrictions—introducing the ${\cal F}{{\cal L}_0}$ wer reasoner. In Proceedings of the 3rd International Joint Conference on Rules and Reasoning (RuleML+RR 2019), Fodor, P. and Montali, M., Eds. LNCS, Bolzano, Italy. Springer.Google Scholar
Nebel, B. 1990. Terminological reasoning is inherently intractable. Artificial Intelligence 43, 235249.CrossRefGoogle Scholar
Parsia, B., Matentzoglu, N., Gonçalves, R. S., Glimm, B. and Steigmiller, A. 2017. The owl reasoner evaluation (ore) 2015 competition report. Journal of Automated Reasoning 59, 4, 455482.CrossRefGoogle ScholarPubMed
Peltason, C. 1991. The BACK system — an overview. SIGART Bulletin 2, 3, 114119.CrossRefGoogle Scholar
Romero, A. A., Cuenca Grau, B. and Horrocks, I. 2012. More: Modular combination of OWL reasoners for ontology classification. In International Semantic Web Conference (1), vol. 7649 of Lecture Notes in Computer Science. Springer, 1–16.Google Scholar
Schild, K. 1991. A correspondence theory for terminological logics: Preliminary report. In Proc. of the 12th Int. Joint Conf. on Artificial Intelligence (IJCAI’91), 466–471.Google Scholar
Simančík, F., Kazakov, Y. and Horrocks, I. 2011. Consequence-based reasoning beyond Horn ontologies. In IJCAI 2011, Proceedings of the 22nd International Joint Conference on Artificial Intelligence, Walsh, T., Ed. IJCAI/AAAI, 1093–1098.Google Scholar
Woods, W. A. and Schmolze, J. G. 1992. The KL-ONE family. In Semantic Networks in Artificial Intelligence, F. W. Lehmann, Ed., pp. 133–178. Pergamon Press. Published as a special issue of Computers & Mathematics with Applications 23, 2–9.Google Scholar