Probabilistic Reasoning in the Description Logic \(\mathcal {ALCP}\) with the Principle of Maximum Entropy

  • Rafael PeñalozaEmail author
  • Nico Potyka
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 9858)


A central question for knowledge representation is how to encode and handle uncertain knowledge adequately. We introduce the probabilistic description logic \(\mathcal {ALCP}\) that is designed for representing context-dependent knowledge, where the actual context taking place is uncertain. \(\mathcal {ALCP}\) allows the expression of logical dependencies on the domain and probabilistic dependencies on the possible contexts. In order to draw probabilistic conclusions, we employ the principle of maximum entropy. We provide reasoning algorithms for this logic, and show that it satisfies several desirable properties of probabilistic logics.


Maximum Entropy Description Logic Probabilistic Logic Probabilistic Constraint Propositional Variable 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


  1. 1.
    Baader, F., Calvanese, D., McGuinness, D.L., Nardi, D., Patel-Schneider, P.F. (eds.): The Description Logic Handbook: Theory, Implementation, and Applications, 2nd edn. Cambridge University Press, Cambridge (2007)zbMATHGoogle Scholar
  2. 2.
    Baader, F., Knechtel, M., Peñaloza, R.: Context-dependent views to axioms and consequences of semantic web ontologies. J. Web Semant. 12–13, 22–40 (2012)CrossRefGoogle Scholar
  3. 3.
    Barnett, O., Paris, J.B.: Maximum entropy inference with quantified knowledge. Log. J. IGPL 16(1), 85–98 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Beierle, C., Kern-Isberner, G., Finthammer, M., Potyka, N.: Extending and completing probabilistic knowledge and beliefs without bias. KI 29(3), 255–262 (2015)Google Scholar
  5. 5.
    Ceylan, I.I., Rafael, P.: The Bayesian description logic \({\cal BEL}\). In: Demri, S., Kapur, D., Weidenbach, C. (eds.) IJCAR 2014. LNCS, vol. 8562, pp. 480–494. Springer, Switzerland (2014)Google Scholar
  6. 6.
    d’Amato, C., Fanizzi, N., Lukasiewicz, T.: Tractable reasoning with Bayesian description logics. In: Greco, S., Lukasiewicz, T. (eds.) SUM 2008. LNCS (LNAI), vol. 5291, pp. 146–159. Springer, Heidelberg (2008)CrossRefGoogle Scholar
  7. 7.
    De Bona, G., Cozman, F.G., Finger, M.: Towards classifying propositional probabilistic logics. J. Appl. Log. 12(3), 349–368 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Domingos, P.M., Lowd, D.: Markov Logic: An Interface Layer for Artificial Intelligence. Synthesis Lectures on Artificial Intelligence and Machine Learning. Morgan & Claypool Publishers, Los Altos (2009)zbMATHGoogle Scholar
  9. 9.
    Donini, F.M., Massacci, F.: ExpTime tableaux for \({\cal ALC}\). Artif. Intell. 124(1), 87–138 (2000)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Halpern, J.Y.: An analysis of first-order logics of probability. Artif. Intell. 46, 311–350 (1990)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Halpern, J.Y., Koller, D.: Representation dependence in probabilistic inference. JAIR 21, 319–356 (2004)MathSciNetzbMATHGoogle Scholar
  12. 12.
    Hansen, P., Perron, S.: Merging the local and global approaches to probabilistic satisfiability. Int. J. Approx. Reason. 47(2), 125–140 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Kern-Isberner, G.: Conditionals in Nonmonotonic Reasoning and Belief Revision: Considering Conditionals as Agents. LNCS (LNAI), vol. 2087. Springer, Heidelberg (2001)zbMATHGoogle Scholar
  14. 14.
    Kern-Isberner, G., Thimm, M.: Novel semantical approaches to relational probabilistic conditionals. In: Proceedings of KR 2010, pp. 382–391. AAAI Press (2010)Google Scholar
  15. 15.
    Kern-Isberner, G., Lukasiewicz, T.: Combining probabilistic logic programming with the power of maximum entropy. Artif. Intell. 157(1–2), 139–202 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Klinov, P., Parsia, B.: A hybrid method for probabilistic satisfiability. In: Bjørner, N., Sofronie-Stokkermans, V. (eds.) CADE 2011. LNCS, vol. 6803, pp. 354–368. Springer, Heidelberg (2011)CrossRefGoogle Scholar
  17. 17.
    Lukasiewicz, T.: Probabilistic deduction with conditional constraints over basic events. JAIR 10, 380–391 (1999)MathSciNetzbMATHGoogle Scholar
  18. 18.
    Lukasiewicz, T., Straccia, U.: Managing uncertainty and vagueness in description logics for the semantic web. J. Web Semant. 6(4), 291–308 (2008)CrossRefGoogle Scholar
  19. 19.
    Lutz, C., Schröder, L.: Probabilistic description logics for subjective uncertainty. In: Proceedings of KR 2010. AAAI Press (2010)Google Scholar
  20. 20.
    Nilsson, N.J.: Probabilistic logic. Artif. Intell. 28, 71–88 (1986)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Nocedal, J., Wright, S.J.: Numerical Optimization, 2nd edn. Springer, Berlin (2006)zbMATHGoogle Scholar
  22. 22.
    Paris, J.: The Uncertain Reasoner’s Companion - A Mathematical Perspective. Cambridge University Press, Cambridge (1994)zbMATHGoogle Scholar
  23. 23.
    Peñaloza, R., Potyka, N.: Probabilistic reasoning in the description logic ALCP with the principle of maximum entropy (full version). CoRR abs/1606.09521 (2016).
  24. 24.
    Potyka, N.: Reasoning over linear probabilistic knowledge bases with priorities. In: Beierle, C., Dekhtyar, A. (eds.) SUM 2015. LNCS, vol. 9310, pp. 121–136. Springer, Heidelberg (2015)CrossRefGoogle Scholar
  25. 25.
    Potyka, N.: Relationships between semantics for relational probabilistic conditional logics. In: Computational Models of Rationality, Essays dedicated to Gabriele Kern-Isberner, pp. 332–347. College Publications (2016)Google Scholar
  26. 26.
    Riguzzi, F., Bellodi, E., Lamma, E., Zese, R.: Epistemic and statistical probabilistic ontologies. In: Proceedings of URSW 2012, vol. 900, pp. 3–14. CEUR-WS (2012)Google Scholar
  27. 27.
    Schmidt-Schauß, M., Smolka, G.: Attributive concept descriptions with complements. Artif. Intell. 48(1), 1–26 (1991)MathSciNetCrossRefzbMATHGoogle Scholar
  28. 28.
    Yeung, R.W.: Information Theory and Network Coding. Springer Science & Business Media, Berlin (2008)zbMATHGoogle Scholar

Copyright information

© Springer International Publishing Switzerland 2016

Authors and Affiliations

  1. 1.Free University of Bozen-BolzanoBolzanoItaly
  2. 2.University of OsnabrückOsnabrückGermany

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