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Reasoner = Logical Calculus + Rule Engine


We propose using rule languages to encode complex reasoning algorithms in a declarative way. This approach—which follows the classical slogan “Algorithm = Logic + Control”—promises to turn high-level specifications of logical calculi as systems of inference rules into declarative rule-based models that can be executed on state-of-the-art rule engines. Simple rule languages suffice for simple logics, and we review our results on using Datalog rules to reason in the description logic \(\mathcal {EL}\). For more expressive logics, a suitably expressive yet implementable rule language often seems to be missing. To fill this gap, we consider an extension of Datalog with sets, Datalog(S), that can be executed by modern existential-rule reasoners, and we use it to present a rule-based reasoning calculus for the expressive description logic \(\mathcal {ALC}\).

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  1. 1.

    For example, the SAT solver of Howe and King uses the operational block directive of SICStus Prolog to implement “watched literals” [20]; this is an example of traditional (albeit elegant) computer programming where controlling execution details is an important part of the implementation.

  2. 2.

    A fixed Datalog rule set can be evaluated in P w.r.t. the number of facts, but there are polynomial-time algorithms that cannot be captured by Datalog [1].

  3. 3.

  4. 4., formerly VLog4j

  5. 5.

    A subtle but important difference is that their Datalog\(^{\textsc {S}}\) is based on a sort signature that defines which sets may be considered. Data complexity thus becomes P-complete [30, Thm 2], while it is ExpTime-complete for our Datalog(S).


  1. 1.

    Abiteboul S, Hull R, Vianu V (1994) Foundations of databases. Addison Wesley, Boston

    Google Scholar 

  2. 2.

    Ahmetaj S, Ortiz M, Simkus M (2018) Rewriting guarded existential rules into small datalog programs. In: Kimelfeld B, Amsterdamer Y (eds) Proceedings of 21st International Conference on database theory (ICDT’18), LIPIcs, vol 98, Schloss Dagstuhl - Leibniz-Zentrum fuer Informatik, pp 4:1–4:24

  3. 3.

    Aref M, ten Cate B, Green TJ, Kimelfeld B, Olteanu D, Pasalic E, Veldhuizen TL, Washburn G (2015) Design and implementation of the LogicBlox system. In: Sellis TK, Davidson SB, Ives ZG (eds) Proceedings of 2015 ACM SIGMOD International Conference on management of data. ACM, pp 1371–1382

  4. 4.

    Baader F, Brandt S, Lutz C (2005) Pushing the \(\cal{EL}\) envelope. In: Kaelbling L, Saffiotti A (eds) Proceedings of 19th International Joint Conference on artificial intelligence (IJCAI’05), Professional Book Center, pp 364– 369.

  5. 5.

    Baget J, Leclère M, Mugnier M, Rocher S, Sipieter C (2015) Graal: a toolkit for query answering with existential rules. In: Bassiliades N, Gottlob G, Sadri F, Paschke A, Roman D (eds) Proceedings of 9th International web rule symposium (RuleML’15), LNCS, vol 9202, Springer, pp 328–344.

  6. 6.

    Bellomarini L, Sallinger E, Gottlob G (2018) The vadalog system: Datalog-based reasoning for knowledge graphs. PVLDB 11(9):975–987

    Google Scholar 

  7. 7.

    Benedikt M, Konstantinidis G, Mecca G, Motik B, Papotti P, Santoro D, Tsamoura E (2017) Benchmarking the chase. In: Sallinger E, den Bussche JV, Geerts F (eds) Proceedings of 36th symposium on principles of database systems (PODS’17), ACM, pp 37–52.

  8. 8.

    Benedikt M, Leblay J, Tsamoura E (2014) PDQ: proof-driven query answering over web-based data. PVLDB 7(13):1553–1556

    Google Scholar 

  9. 9.

    Bienvenu M, ten Cate B, Lutz C, Wolter F (2014) Ontology-based data access: a study through disjunctive datalog, CSP, and MMSNP. ACM Trans Database Syst 39(4):33:1–33:44

    MathSciNet  Article  Google Scholar 

  10. 10.

    Carral D, Dragoste I, González L, Jacobs C, Krötzsch M, Urbani J (2019) VLog: a rule engine for knowledge graphs. In: Ghidini C (eds) Proceedings of 18th International semantic web conference (ISWC’19, Part II), LNCS, Springer, vol 11779, pp 19–35.

  11. 11.

    Carral D, Dragoste I, Krötzsch M (2017) Restricted chase (non)termination for existential rules with disjunctions. In: Sierra C (eda) Proceedings of 26th International Joint Conference on artificial intelligence (IJCAI’17), pp 922–928.

  12. 12.

    Carral D, Dragoste I, Krötzsch M (2018) The combined approach to query answering in Horn-\(\cal{ALCHOIQ}\). In: Thielscher M, Toni F, Wolter F (eds) Proceedings of 16th International Conference on principles of knowledge representation and reasoning (KR’18), AAAI Press, pp 339–348.

  13. 13.

    Carral D, Dragoste I, Krötzsch M, Lewe C (2019) Chasing sets: how to use existential rules for expressive reasoning. In: Kraus S (eds) Proceedings of 28th International Joint Conference on artificial intelligence (IJCAI’19), pp 1624–1631.

  14. 14.

    Carral D, González L, Koopmann P (2019) From Horn-SRIQ to Datalog: a data-independent transformation that preserves assertion entailment. In: Proceedings of 33rd AAAI Conference on artificial intelligence (AAAI’19), AAAI Press, pp 2736–2743.

  15. 15.

    Cuenca Grau B, Horrocks I, Krötzsch M, Kupke C, Magka D, Motik B, Wang Z (2013) Acyclicity notions for existential rules and their application to query answering in ontologies. J Artif Intell Res 47:741–808

    MathSciNet  Article  Google Scholar 

  16. 16.

    Dantsin E, Eiter T, Gottlob G, Voronkov A (2001) Complexity and expressive power of logic programming. ACM Comput Surv 33(3):374–425

    Article  Google Scholar 

  17. 17.

    Egly U, Gaggl SA, Woltran S (2010) Answer-set programming encodings for argumentation frameworks. Argum Comput 1(2):147–177

    Article  Google Scholar 

  18. 18.

    Eiter T, Ortiz M, Simkus M, Tran T, Xiao G (2012) Query rewriting for Horn-\(\cal{SHIQ}\) plus rules. In: Hoffmann J, Selman B (eds) Proceedings of 26th AAAI Conference on artificial intelligence (AAAI’12). AAAI Press.

  19. 19.

    Geerts F, Mecca G, Papotti P, Santoro D (2014) That’s all folks!. LLUNATIC goes open source. PVLDB 7(13):1565–1568

    Google Scholar 

  20. 20.

    Howe JM, King A (2012) A pearl on SAT and SMT solving in Prolog. Theor Comput Sci 435:43–55

    MathSciNet  Article  Google Scholar 

  21. 21.

    Hustadt U, Motik B, Sattler U (2007) Reasoning in description logics by a reduction to disjunctive Datalog. J Autom Reason 39(3):351–384

    MathSciNet  Article  Google Scholar 

  22. 22.

    Kowalski RA (1979) Algorithm = logic + control. Commun ACM 22(7):424–436

    Article  Google Scholar 

  23. 23.

    Krötzsch M (2010) Efficient inferencing for OWL EL. In: Janhunen T, Niemelä I (eds) Proceedings of 12th European Conference on logics in artificial intelligence (JELIA’10), LNAI, vol 6341, Springer, pp 234–246.

  24. 24.

    Krötzsch M (2011) Efficient rule-based inferencing for OWL EL. In: Proceedings of 22nd International Joint Conference on artificial intelligence (IJCAI’11), AAAI Press/IJCAI, pp 2668–2673.

  25. 25.

    Krötzsch M, Rudolph S, Hitzler P (2013) Complexities of Horn description logics. ACM Trans Comput Logic 14(1):2:1–2:36

    MathSciNet  Article  Google Scholar 

  26. 26.

    Krötzsch M, Simančík F, Horrocks I (2014) Description logics. IEEE Intell Syst 29:12–19

    Article  Google Scholar 

  27. 27.

    Krötzsch M, Marx M, Rudolph S (2019) The power of theterminating chase. In: Barceló P, Calautti M (eds) Proceedings of 22nd International Conference on database theory (ICDT’19), LIPIcs, vol 127, Schloss Dagstuhl - Leibniz-Zentrum fuerInformatik, pp 3:1–3:17.

  28. 28.

    Motik B, Cuenca Grau B, Horrocks I, Wu Z, Fokoue A, Lutz C (eds) (2009) OWL 2 web ontology language: profiles. W3C Recommendation. Available at Accessed 27 Oct 2009.

  29. 29.

    Nenov Y, Piro R, Motik B, Horrocks I, Wu Z, Banerjee J (2015) RDFox: a highly-scalable RDF store. In: Arenas M (eds) Proceedings of 14th International semantic web conference (ISWC’15), Part II, LNCS, vol 9367, Springer, pp 3–20.

  30. 30.

    Ortiz M, Rudolph S, Simkus M (2010) Worst-case optimal reasoning for the Horn-DL fragments of OWL 1 and 2. In: Lin F, Sattler U, Truszczynski M (eds) Proceedings of 12th International Conference on principles of knowledge representation and reasoning (KR’10), AAAI Press, pp 269–279.

  31. 31.

    Otten J, Bibel W (2003) leanCoP: lean connection-based theorem proving. J Symb Comput 36(1–2):139–161

    MathSciNet  Article  Google Scholar 

  32. 32.

    Rudolph S, Krötzsch M, Hitzler P (2012) Type-elimination-based reasoning for the description logic \(\cal{SHIQ}b_s\) using decision diagrams and disjunctive datalog. Logical Methods Comput Sci 8(1):1–38

    MathSciNet  Article  Google Scholar 

  33. 33.

    Simančík F, Kazakov Y, Horrocks I (2011) Consequence-based reasoning beyond Horn ontologies. In: Proceedings of 22nd International Joint Conference on artificial intelligence (IJCAI’11), AAAI Press/IJCAI, pp 1093–1098.

  34. 34.

    Urbani J, Jacobs C, Krötzsch M (2016) Column-oriented Datalog materialization for large knowledge graphs. In: Schuurmans D, Wellman MP (eds) Proceedings of 30th AAAI Conference on artificial intelligence (AAAI’16), AAAI Press, pp 258–264.

  35. 35.

    Urbani J, Krötzsch M, Jacobs CJH, Dragoste I, Carral D (2018) Efficient model construction for Horn logic with VLog: system description. In: Galmiche D, Schulz S, Sebastiani R (eds) Proceedings of 9th International Joint Conference on automated reasoning (IJCAR’18), LNCS, vol 10900, Springer, pp 680–688.

  36. 36.

    Xiao G, Heymans S, Eiter T (2010) DReW: a reasoner for Datalog-rewritable description logics and dl-programs. In: Eiter T, Ghali AE, Fernández S, Heymans S, Krennwallner T, Lévy F (eds) Proceedings of 1st International workshop on business models, business rules and ontologies (BuRO’10), ONTORULE Project, pp 1–14.

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This work is partly supported by Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) in projects number 389792660 (TRR 248, Center for Perspicuous Systems) and KR 4381/1-1 (Emmy Noether grant DIAMOND), and by the Bundesministerium für Bildung und Forschung (BMBF, Federal Ministry of Education and Research) in the Center for Scalable Data Analytics and Artificial Intelligence (ScaDS.AI).

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Correspondence to Markus Krötzsch.

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Carral, D., Dragoste, I. & Krötzsch, M. Reasoner = Logical Calculus + Rule Engine. Künstl Intell 34, 453–463 (2020).

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