Extending \(\mathscr {E\!L}^{++}\) with Linear Constraints on the Probability of Axioms

  • Marcelo FingerEmail author
Part of the Lecture Notes in Computer Science book series (LNCS, volume 11560)


One of the main reasons to employ a description logic such as \(\mathscr {E\!L}^{++}\) is the fact that it has efficient, polynomial-time algorithmic properties such as deciding consistency and inferring subsumption. However, simply by adding negation of concepts to it, we obtain the expressivity of description logics whose decision procedure is ExpTime-complete. Similar complexity explosion occurs if we add probability assignments on concepts. To lower the resulting complexity, we instead concentrate on assigning probabilities to Axioms/GCIs. We show that the consistency detection problem for such a probabilistic description logic is NP-complete, and present a linear algebraic deterministic algorithm to solve it, using the column generation technique. We also examine and provide algorithms for the probabilistic extension problem, which consists of inferring the minimum and maximum probabilities for a new axiom, given a consistent probabilistic knowledge base.


  1. Baader, F., Brandt, S., Lutz, C.: Pushing the EL envelope. In: Proceedings of IJCAI 2005, San Francisco, CA, USA, pp. 364–369. Morgan Kaufmann Publishers Inc. (2005a)Google Scholar
  2. Baader, F., Brandt, S., Lutz, C.: Pushing the EL envelope. Technical report LTCS-Report LTCS-05-01 (2005b)Google Scholar
  3. Baader, F., Horrocks, I., Lutz, C., Sattler, U.: An Introduction to Description Logic. Cambridge University Press, Cambridge (2017)CrossRefGoogle Scholar
  4. Bertsimas, D., Tsitsiklis, J.N.: Introduction to Linear Optimization. Athena Scientific, Belmont (1997)Google Scholar
  5. Bona, G.D., Cozman, F.G., Finger, M.: Towards classifying propositional probabilistic logics. J. Appl. Logic 12(3), 349–368 (2014)Google Scholar
  6. Eckhoff, J.: Helly, Radon, and Carathéodory type theorems. In: Handbook of Convex Geometry, pp. 389–448. Elsevier (1993)Google Scholar
  7. Finger, M., Bona, G.D.: Probabilistic satisfiability: logic-based algorithms and phase transition. In: IJCAI 2011, pp. 528–533 (2011)Google Scholar
  8. Finger, M., De Bona, G.: Probabilistic satisfiability: algorithms with the presence and absence of a phase transition. Ann. Math. Artif. Intell. 75(3), 351–379 (2015)MathSciNetCrossRefGoogle Scholar
  9. Finger, M., Wassermann, R., Cozman, F.G.: Satisfiability in EL with sets of probabilistic ABoxes. In: Rosati et al. (2011)Google Scholar
  10. Gutiérrez-Basulto, V., Jung, J.C., Lutz, C., Schröder, L.: A closer look at the probabilistic description logic Prob-EL. In: AAAI 2011 (2011)Google Scholar
  11. Gutiérrez-Basulto, V., Jung, J.C., Lutz, C., Schröder, L.: Probabilistic description logics for subjective uncertainty. JAIR 58, 1–66 (2017)MathSciNetCrossRefGoogle Scholar
  12. Heinsohn, J.: Probabilistic description logics. In: Proceedings of UAI 1994, pp. 311–318 (1994)Google Scholar
  13. Jung, J.C., Gutiérrez-Basulto, V., Lutz, C., Schröder, L.: The complexity of probabilistic EL. In: Rosati et al. (2011)Google Scholar
  14. Lukasiewicz, T.: Expressive probabilistic description logics. Artif. Intell. 172(6), 852–883 (2008)MathSciNetCrossRefGoogle Scholar
  15. Lutz, C., Schröder, L.: Probabilistic description logics for subjective uncertainty. In: KR 2010. AAAI Press (2010)Google Scholar
  16. Rosati, R., Rudolph, S., Zakharyaschev, M. (eds.): Proceedings of DL 2011. CEUR Workshop Proceedings, vol. 745. (2011)Google Scholar
  17. Warners, J.P.: A linear-time transformation of linear inequalities into conjunctive normal form. Inf. Process. Lett. 68(2), 63–69 (1998)MathSciNetCrossRefGoogle Scholar

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Authors and Affiliations

  1. 1.Department of Computer ScienceUniversity of São PauloSão PauloBrazil

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