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Introduction to Probabilistic Ontologies

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Reasoning Web. Declarative Artificial Intelligence (Reasoning Web 2020)

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Abstract

There is no doubt about it; an accurate representation of a real knowledge domain must be able to capture uncertainty. As the best known formalism for handling uncertainty, probability theory is often called upon this task, giving rise to probabilistic ontologies. Unfortunately, things are not as simple as they might appear, and different choices made can deeply affect the semantics and computational properties of probabilistic ontology languages. In this tutorial, we explore the main design choices available, and the situations in which they may be meaningful or not. We then dive deeper into a specific family of probabilistic ontology languages which can express logical and probabilistic dependencies between axioms.

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Notes

  1. 1.

    As is standard in probability theory, we use concatenation to denote intersection. That is, XY stands for the event \(X\cap Y\).

  2. 2.

    Again, as usual in probability theory, \(\overline{X}\) denotes the event \(\varOmega \setminus X\); that is, the negation of X.

  3. 3.

    Alternatively, read the second statement as “the probability of rain tomorrow is 0.3”.

  4. 4.

    Note that by definition of ontology languages, any subset of an ontology is also an ontology; hence this construction works without issues.

  5. 5.

    The actual semantics of open-world databases is more complex than this, but we wanted to provide just a basic intuition.

  6. 6.

    We will come back to this issue in Sect. 5.5.

  7. 7.

    Recall that the extensional description of a joint probability distribution requires exponential space on the number of variables involved.

  8. 8.

    It should go without saying, but this is only an example and should in no way be considered medical advice of any kind.

References

  1. Artale, A., Calvanese, D., Kontchakov, R., Zakharyaschev, M.: The DL-Lite family and relations. J. Artif. Intell. Res. 36, 1–69 (2009). https://doi.org/10.1613/jair.2820

    Article  MathSciNet  MATH  Google Scholar 

  2. Baader, F., Calvanese, D., McGuinness, D., Nardi, D., Patel-Schneider, P. (eds.): The Description Logic Handbook: Theory, Implementation, and Applications, 2nd edn. University Press, Cambridge (2007)

    MATH  Google Scholar 

  3. Baader, F., Hladik, J., Peñaloza, R.: Automata can show PSpace results for description logics. Inf. Comput. 206(9–10), 1045–1056 (2008). https://doi.org/10.1016/j.ic.2008.03.006

    Article  MathSciNet  MATH  Google Scholar 

  4. Bellomarini, L., Benedetto, D., Gottlob, G., Sallinger, E.: Vadalog: a modern architecture for automated reasoning with large knowledge graphs. Inf. Syst., 101528 (2020). https://doi.org/10.1016/j.is.2020.101528

  5. Bianchi, F., Rossiello, G., Costabello, L., Palmonari, M., Minervini, P.: Knowledge graph embeddings and explainable AI. CoRR abs/2004.14843 (2020). https://arxiv.org/abs/2004.14843

  6. Biere, A., Heule, M.J.H., van Maaren, H., Walsh, T. (eds.): Handbook of Satisfiability. Frontiers in Artificial Intelligence and Applications, vol. 185. IOS Press, Amsterdam (2009)

    MATH  Google Scholar 

  7. Bobillo, F., Straccia, U.: Reasoning within fuzzy OWL 2 EL revisited. Fuzzy Sets Syst. 351, 1–40 (2018). https://doi.org/10.1016/j.fss.2018.03.011

    Article  MathSciNet  MATH  Google Scholar 

  8. Borgwardt, S.: Fuzzy description logics with general concept inclusions. Ph.D. thesis, Technische Universität Dresden, Germany (2014)

    Google Scholar 

  9. Borgwardt, S., Ceylan, İ.İ., Lukasiewicz, T.: Ontology-mediated query answering over log-linear probabilistic data. In: Proceedings of the 33rd AAAI Conference on Artificial Intelligence, AAAI 2019, pp. 2711–2718. AAAI Press (2019). https://doi.org/10.1609/aaai.v33i01.33012711

  10. Borgwardt, S., Peñaloza, R.: The complexity of lattice-based fuzzy description logics. J. Data Semant. 2(1), 1–19 (2013). https://doi.org/10.1007/s13740-012-0013-x

    Article  Google Scholar 

  11. Borgwardt, S., Peñaloza, R.: Fuzzy description logics – a survey. In: Moral, S., Pivert, O., Sánchez, D., Marín, N. (eds.) SUM 2017. LNCS (LNAI), vol. 10564, pp. 31–45. Springer, Cham (2017). https://doi.org/10.1007/978-3-319-67582-4_3

    Chapter  Google Scholar 

  12. Botha, L., Meyer, T., Peñaloza, R.: A Bayesian extension of the description logic \(\cal{ALC}\). In: Calimeri, F., Leone, N., Manna, M. (eds.) JELIA 2019. LNCS (LNAI), vol. 11468, pp. 339–354. Springer, Cham (2019). https://doi.org/10.1007/978-3-030-19570-0_22

    Chapter  Google Scholar 

  13. Brewka, G.: Nonmonotonic Reasoning - Logical Foundations of Commonsense. Cambridge Tracts in Theoretical Computer Science, vol. 12. Cambridge University Press, Cambridge (1991)

    MATH  Google Scholar 

  14. Calì, A., Gottlob, G., Lukasiewicz, T., Pieris, A.: Datalog+/-: a family of languages for ontology querying. In: de Moor, O., Gottlob, G., Furche, T., Sellers, A. (eds.) Datalog 2.0 2010. LNCS, vol. 6702, pp. 351–368. Springer, Heidelberg (2011). https://doi.org/10.1007/978-3-642-24206-9_20

    Chapter  Google Scholar 

  15. Calvanese, D., De Giacomo, G., Lembo, D., Lenzerini, M., Rosati, R.: Data complexity of query answering in description logics. Artif. Intell. 195, 335–360 (2013). https://doi.org/10.1016/j.artint.2012.10.003

    Article  MathSciNet  MATH  Google Scholar 

  16. Cano, A., Cozman, F.G., Lukasiewicz, T.: Reasoning with imprecise probabilities. Int. J. Approx. Reason. 44(3), 197–199 (2007). https://doi.org/10.1016/j.ijar.2006.09.001

    Article  MathSciNet  MATH  Google Scholar 

  17. Carvalho, R.N., Laskey, K.B., Costa, P.C.G.: PR-OWL - a language for defining probabilistic ontologies. Int. J. Approx. Reason. 91, 56–79 (2017). https://doi.org/10.1016/j.ijar.2017.08.011

    Article  MathSciNet  MATH  Google Scholar 

  18. Ceylan, İ.İ., Darwiche, A., den Broeck, G.V.: Open-world probabilistic databases. In: Baral, C., Delgrande, J.P., Wolter, F. (eds.) Proceedings of the 15th International Conference on Principles of Knowledge Representation and Reasoning, KR 2016, pp. 339–348. AAAI Press (2016). http://www.aaai.org/ocs/index.php/KR/KR16/paper/view/12908

  19. Ceylan, İİ., Lukasiewicz, T.: A tutorial on query answering and reasoning over probabilistic knowledge bases. In: d’Amato, C., Theobald, M. (eds.) Reasoning Web 2018. LNCS, vol. 11078, pp. 35–77. Springer, Cham (2018). https://doi.org/10.1007/978-3-030-00338-8_3

    Chapter  Google Scholar 

  20. Ceylan, İİ., Peñaloza, R.: The Bayesian ontology language BEL. J. Autom. Reason. 58(1), 67–95 (2017). https://doi.org/10.1007/s10817-016-9386-0

    Article  MathSciNet  MATH  Google Scholar 

  21. Concannon, L.G., Kaufman, M.S., Herring, S.A.: Counseling athletes on the risk of chronic traumatic encephalopathy. Sports Health 6(5), 396–401 (2014). https://doi.org/10.1177/1941738114530958

    Article  Google Scholar 

  22. Cooper, G.F.: The computational complexity of probabilistic inference using Bayesian belief networks. Artif. Intell. 42(2–3), 393–405 (1990). https://doi.org/10.1016/0004-3702(90)90060-D

    Article  MathSciNet  MATH  Google Scholar 

  23. Cozman, F.G.: Graphical models for imprecise probabilities. Int. J. Approx. Reason. 39(2–3), 167–184 (2005). https://doi.org/10.1016/j.ijar.2004.10.003

    Article  MathSciNet  MATH  Google Scholar 

  24. Cuenca Grau, B., Horrocks, I., Motik, B., Parsia, B., Patel-Schneider, P., Sattler, U.: OWL 2: The next step for OWL. J. Web Semant. 6, 309–322 (2008)

    Article  Google Scholar 

  25. Cyganiak, R., Wood, D., Lanthaler, M. (eds.): RDF 1.1 Concepts and Abstract Syntax. W3C Recommendation (25 February 2014). http://www.w3.org/TR/rdf11-concepts/

  26. Dalvi, N.N., Suciu, D.: The dichotomy of probabilistic inference for unions of conjunctive queries. J. ACM 59(6), 30:1–30:87 (2012). https://doi.org/10.1145/2395116.2395119

    Article  MathSciNet  MATH  Google Scholar 

  27. Darwiche, A.: Modeling and Reasoning with Bayesian Networks. Cambridge University Press, Cambridge (2009). http://www.cambridge.org/uk/catalogue/catalogue.asp?isbn=9780521884389

    Book  Google Scholar 

  28. Dechter, R.: Bucket elimination: a unifying framework for reasoning. Artif. Intell. 113(1–2), 41–85 (1999). https://doi.org/10.1016/S0004-3702(99)00059-4

    Article  MathSciNet  MATH  Google Scholar 

  29. Diestel, R.: Graph Theory. Graduate Texts in Mathematics, vol. 173, 5th edn. Springer, Heidelberg (2017). https://doi.org/10.1007/978-3-662-53622-3

    Book  MATH  Google Scholar 

  30. Dubois, D., Prade, H.: Possibility Theory - An Approach to Computerized Processing of Uncertainty. Springer, Boston (1988). https://doi.org/10.1007/978-1-4684-5287-7

    Book  MATH  Google Scholar 

  31. Gallo, G., Longo, G., Pallottino, S., Nguyen, S.: Directed hypergraphs and applications. Discret. Appl. Math. 42(2), 177–201 (1993). https://doi.org/10.1016/0166-218X(93)90045-P

    Article  MathSciNet  MATH  Google Scholar 

  32. Gottwald, S.: Many-valued and fuzzy logics. In: Kacprzyk, J., Pedrycz, W. (eds.) Springer Handbook of Computational Intelligence, pp. 7–29. Springer, Heidelberg (2015). https://doi.org/10.1007/978-3-662-43505-2_2

    Chapter  Google Scholar 

  33. Gutiérrez-Basulto, V., Jung, J.C., Lutz, C., Schröder, L.: A closer look at the probabilistic description logic prob-EL. In: Burgard, W., Roth, D. (eds.) Proceedings of the 25th AAAI Conference on Artificial Intelligence, AAAI 2011. AAAI Press (2011). http://www.aaai.org/ocs/index.php/AAAI/AAAI11/paper/view/3702

  34. Gutiérrez-Basulto, V., Jung, J.C., Lutz, C., Schröder, L.: Probabilistic description logics for subjective uncertainty. J. Artif. Intell. Res. 58, 1–66 (2017). https://doi.org/10.1613/jair.5222

    Article  MathSciNet  MATH  Google Scholar 

  35. Hájek, P.: Metamathematics of Fuzzy Logic. Trends in Logic, vol. 4. Springer, Dordrecht (1998). https://doi.org/10.1007/978-94-011-5300-3

    Book  MATH  Google Scholar 

  36. Halpern, J.Y.: An analysis of first-order logics of probability. In: Sridharan, N.S. (ed.) Proceedings of the 11th International Joint Conference on Artificial Intelligence, IJCAI 1989, pp. 1375–1381. Morgan Kaufmann (1989). http://ijcai.org/Proceedings/89-2/Papers/084.pdf

  37. Halpern, J.Y.: An analysis of first-order logics of probability. Artif. Intell. 46(3), 311–350 (1990). https://doi.org/10.1016/0004-3702(90)90019-V

    Article  MathSciNet  MATH  Google Scholar 

  38. Halpern, J.Y.: Reasoning about Uncertainty. MIT Press, Cambridge (2005)

    MATH  Google Scholar 

  39. Hogan, A., et al.: Knowledge graphs. CoRR abs/2003.02320 (2020). https://arxiv.org/abs/2003.02320

  40. Horrocks, I., Kutz, O., Sattler, U.: The even more irresistible \(\cal{SROIQ}\). In: Doherty, P., Mylopoulos, J., Welty, C.A. (eds.) Proceedings of the 10th International Conference on Principles of Knowledge Representation and Reasoning, KR 2006, pp. 57–67. AAAI Press (2006)

    Google Scholar 

  41. Jaeger, M.: Probabilistic reasoning in terminological logics. In: Doyle, J., Sandewall, E., Torasso, P. (eds.) Proceedings of the 4th International Conference on Principles of Knowledge Representation and Reasoning, KR 1994, pp. 305–316. Morgan Kaufmann (1994)

    Google Scholar 

  42. Klinov, P., Parsia, B.: Pronto: a practical probabilistic description logic reasoner. In: Bobillo, F., et al. (eds.) UniDL/URSW 2008-2010. LNCS (LNAI), vol. 7123, pp. 59–79. Springer, Heidelberg (2013). https://doi.org/10.1007/978-3-642-35975-0_4

    Chapter  Google Scholar 

  43. Koller, D., Levy, A.Y., Pfeffer, A.: P-CLASSIC: a tractable probablistic description logic. In: Kuipers, B., Webber, B.L. (eds.) Proceedings of the 14th National Conference on Artificial Intelligence, AAAI 1997, pp. 390–397. AAAI Press (1997). http://www.aaai.org/Library/AAAI/1997/aaai97-060.php

  44. Kyburg Jr., H.E., Teng, C.M.: Uncertain Inference. Cambridge University Press, Cambridge (2001). https://doi.org/10.1017/CBO9780511612947

    Book  MATH  Google Scholar 

  45. Laskey, K.B., da Costa, P.C.G.: Of starships and klingons: Bayesian logic for the 23rd century. In: Proceedings of the 21st Conference in Uncertainty in Artificial Intelligence, UAI 2005, pp. 346–353. AUAI Press (2005)

    Google Scholar 

  46. Littman, M.L., Majercik, S.M., Pitassi, T.: Stochastic Boolean satisfiability. J. Autom. Reason. 27(3), 251–296 (2001). https://doi.org/10.1023/A:1017584715408

    Article  MathSciNet  MATH  Google Scholar 

  47. Lukasiewicz, T.: Expressive probabilistic description logics. Artif. Intell. 172(6–7), 852–883 (2008). https://doi.org/10.1016/j.artint.2007.10.017

    Article  MathSciNet  MATH  Google Scholar 

  48. Lukasiewicz, T., Straccia, U.: Managing uncertainty and vagueness in description logics for the semantic web. J. Web Semant. 6(4), 291–308 (2008)

    Article  Google Scholar 

  49. Lutz, C., Schröder, L.: Probabilistic description logics for subjective uncertainty. In: Lin, F., Sattler, U., Truszczynski, M. (eds.) Proceedings of the 12th International Conference on Principles of Knowledge Representation and Reasoning, KR 2010. AAAI Press (2010). http://aaai.org/ocs/index.php/KR/KR2010/paper/view/1243

  50. Maggi, F.M., Montali, M., Peñaloza, R.: Temporal logics over finite traces with uncertainty. In: Proceedings of the 34th AAAI Conference on Artificial Intelligence, AAAI 2020, pp. 10218–10225. AAAI Press (2020)

    Google Scholar 

  51. Maggi, F.M., Montali, M., Peñaloza, R., Alman, A.: Extending temporal business constraints with uncertainty. In: Proceedings of BPM-20 (2020, to appear)

    Google Scholar 

  52. Maggi, F.M., Montali, M., Westergaard, M., van der Aalst, W.M.P.: Monitoring business constraints with linear temporal logic: an approach based on colored automata. In: Rinderle-Ma, S., Toumani, F., Wolf, K. (eds.) BPM 2011. LNCS, vol. 6896, pp. 132–147. Springer, Heidelberg (2011). https://doi.org/10.1007/978-3-642-23059-2_13

    Chapter  Google Scholar 

  53. Mintz, M., Bills, S., Snow, R., Jurafsky, D.: Distant supervision for relation extraction without labeled data. In: Su, K., Su, J., Wiebe, J. (eds.), Proceedings of the 47th Annual Meeting of the Association for Computational Linguistics, ACL 2009, pp. 1003–1011. The Association for Computer Linguistics (2009). https://www.aclweb.org/anthology/P09-1113/

  54. Motik, B., Cuenca Grau, B., Horrocks, I., Wu, Z., Fokoue, A., Lutz, C. (eds.): OWL 2 Web Ontology Language: Profiles. W3C Recommendation (27 October 2009). http://www.w3.org/TR/owl2-profiles/

  55. Niepert, M., Noessner, J., Stuckenschmidt, H.: Log-linear description logics. In: Walsh, T. (ed.) Proceedings of the 22nd International Joint Conference on Artificial Intelligence, IJCAI 2011, pp. 2153–2158. IJCAI/AAAI (2011). https://doi.org/10.5591/978-1-57735-516-8/IJCAI11-359

  56. Nilsson, N.J.: Probabilistic logic. Artif. Intell. 28(1), 71–87 (1986). https://doi.org/10.1016/0004-3702(86)90031-7

    Article  MathSciNet  MATH  Google Scholar 

  57. Pearl, J.: Probabilistic Reasoning in Intelligent Systems - Networks of Plausible Inference. Morgan Kaufmann Series in Representation and Reasoning. Morgan Kaufmann, Burlington (1989)

    MATH  Google Scholar 

  58. Peñaloza, R.: Inconsistency-tolerant instance checking in tractable description logics. In: Costantini, S., Franconi, E., Van Woensel, W., Kontchakov, R., Sadri, F., Roman, D. (eds.) RuleML+RR 2017. LNCS, vol. 10364, pp. 215–229. Springer, Cham (2017). https://doi.org/10.1007/978-3-319-61252-2_15

    Chapter  Google Scholar 

  59. Peñaloza, R.: Making decisions with knowledge base repairs. In: Torra, V., Narukawa, Y., Pasi, G., Viviani, M. (eds.) MDAI 2019. LNCS (LNAI), vol. 11676, pp. 259–271. Springer, Cham (2019). https://doi.org/10.1007/978-3-030-26773-5_23

    Chapter  Google Scholar 

  60. Peñaloza, R.: Axiom pinpointing. CoRR abs/2003.08298 (2020). https://arxiv.org/abs/2003.08298

  61. Peñaloza, R., Sertkaya, B.: Understanding the complexity of axiom pinpointing in lightweight description logics. Artif. Intell. 250, 80–104 (2017). https://doi.org/10.1016/j.artint.2017.06.002

    Article  MathSciNet  MATH  Google Scholar 

  62. Pesic, M., Schonenberg, H., van der Aalst, W.M.P.: DECLARE: full support for loosely-structured processes. In: Proceedings of the 11th IEEE International Enterprise Distributed Object Computing Conference, EDOC 2007, pp. 287–300. IEEE Computer Society (2007). https://doi.org/10.1109/EDOC.2007.14

  63. Raedt, L.D., Kimmig, A., Toivonen, H.: Problog: a probabilistic prolog and its application in link discovery. In: Veloso, M.M. (ed.) Proceedings of the 20th International Joint Conference on Artificial Intelligence, IJCAI 2007, pp. 2462–2467. IJCAI (2007). http://ijcai.org/Proceedings/07/Papers/396.pdf

  64. Riguzzi, F., Bellodi, E., Lamma, E., Zese, R.: Probabilistic description logics under the distribution semantics. Semant. Web 6(5), 477–501 (2015). https://doi.org/10.3233/SW-140154

    Article  MATH  Google Scholar 

  65. Roth, D.: On the hardness of approximate reasoning. Artif. Intell. 82(1–2), 273–302 (1996). https://doi.org/10.1016/0004-3702(94)00092-1

    Article  MathSciNet  Google Scholar 

  66. Schild, K.: A correspondence theory for terminological logics: preliminary report. In: Mylopoulos, J., Reiter, R. (eds.) Proceedings of the 12th International Joint Conference on Artificial Intelligence, IJCAI 1991, pp. 466–471. Morgan Kaufmann (1991)

    Google Scholar 

  67. Straccia, U.: Fuzzy semantic web languages and beyond. In: Benferhat, S., Tabia, K., Ali, M. (eds.) IEA/AIE 2017. LNCS (LNAI), vol. 10350, pp. 3–8. Springer, Cham (2017). https://doi.org/10.1007/978-3-319-60042-0_1

    Chapter  Google Scholar 

  68. Suciu, D., Olteanu, D., Ré, C., Koch, C.: Probabilistic Databases. Synthesis Lectures on Data Management. Morgan & Claypool Publishers, San Rafael (2011). https://doi.org/10.2200/S00362ED1V01Y201105DTM016

    Book  Google Scholar 

  69. Tijms, H.: Understanding Probability, 3rd edn. Cambridge University Press, Cambridge (2012). https://doi.org/10.1017/CBO9781139206990

    Book  MATH  Google Scholar 

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  1. IKR3 Lab, University of Milano-Bicocca, Milan, Italy

    Rafael Peñaloza

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  1. Department of Mathematics and Computer Science, University of Calabria, Rende, Italy

    Marco Manna

  2. School of Informatics, University of Edinburgh, Edinburgh, UK

    Andreas Pieris

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Peñaloza, R. (2020). Introduction to Probabilistic Ontologies. In: Manna, M., Pieris, A. (eds) Reasoning Web. Declarative Artificial Intelligence. Reasoning Web 2020. Lecture Notes in Computer Science(), vol 12258. Springer, Cham. https://doi.org/10.1007/978-3-030-60067-9_1

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