Abstract
The probabilistic Description Logic \(\mathcal {ALC} ^\mathsf {ME}\) is an extension of the Description Logic \(\mathcal {ALC} \) that allows for uncertain conditional statements of the form “if C holds, then D holds with probability p,” together with probabilistic assertions about individuals. In \(\mathcal {ALC} ^\mathsf {ME}\), probabilities are understood as an agent’s degree of belief. Probabilistic conditionals are formally interpreted based on the so-called aggregating semantics, which combines a statistical interpretation of probabilities with a subjective one. Knowledge bases of \(\mathcal {ALC} ^\mathsf {ME}\) are interpreted over a fixed finite domain and based on their maximum entropy (\(\mathsf {ME}\)) model. We prove that checking consistency of such knowledge bases can be done in time polynomial in the cardinality of the domain, and in exponential time in the size of a binary encoding of this cardinality. If the size of the knowledge base is also taken into account, the combined complexity of the consistency problem is NP-complete for unary encoding of the domain cardinality and NExpTime-complete for binary encoding.
This work was supported by the German Research Foundation (DFG) within the Research Unit FOR 1513 “Hybrid Reasoning for Intelligent Systems”.
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Notes
- 1.
- 2.
We will see later (proof of Corollary 14) that setting all probabilities to 1 in a pKB basically yields a classical KB, and thus \(\mathcal {ALC} {^\mathsf {ME}}\) indeed is an extension of \(\mathcal {ALC} \).
- 3.
More precisely, this fact is used in the identities marked with \(*\) below.
References
Baader, F., Borgwardt, S., Koopmann, P., Ozaki, A., Thost, V.: Metric temporal description logics with interval-rigid names. In: Dixon, C., Finger, M. (eds.) FroCoS 2017. LNCS (LNAI), vol. 10483, pp. 60–76. Springer, Cham (2017). https://doi.org/10.1007/978-3-319-66167-4_4
Baader, F., Calvanese, D., McGuinness, D.L., Nardi, D., Patel-Schneider, P.F. (eds.): The Description Logic Handbook: Theory, Implementation, and Applications. Cambridge University Press, Cambridge (2003)
Baader, F., Horrocks, I., Lutz, C., Sattler, U.: An Introduction to Description Logic. Cambridge University Press, Cambridge (2017). https://doi.org/10.1017/9781139025355
Baader, F., Koopmann, P., Turhan, A.-Y.: Using ontologies to query probabilistic numerical data. In: Dixon, C., Finger, M. (eds.) FroCoS 2017. LNCS (LNAI), vol. 10483, pp. 77–94. Springer, Cham (2017). https://doi.org/10.1007/978-3-319-66167-4_5
Chvatal, V.: Linear Programming. W.H Freeman (1983)
Gaggl, S.A., Rudolph, S., Schweizer, L.: Fixed-domain reasoning for description logics. In: Proceedings of the ECAI 2016, pp. 819–827. IOS Press (2016)
Grove, A., Halpern, J., Koller, D.: Random worlds and maximum entropy. J. Artif. Intell. Res. 2, 33–88 (1994)
Halpern, J.Y.: An analysis of first-order logics of probability. Artif. Intell. 46(3), 311–350 (1990)
Hoehndorf, R., Schofield, P.N., Gkoutos, G.V.: The role of ontologies in biological and biomedical research: a functional perspective. Brief. Bioinform. 16(6), 1069–1080 (2015)
Kern-Isberner, G. (ed.): Conditionals in Nonmonotonic Reasoning and Belief Revision. LNCS (LNAI), vol. 2087. Springer, Heidelberg (2001). https://doi.org/10.1007/3-540-44600-1
Kern-Isberner, G., Thimm, M.: Novel semantical approaches to relational probabilistic conditionals. In: Proceedings of the KR 2010, pp. 382–392. AAAI Press (2010)
Lutz, C., Schröder, L.: Probabilistic description logics for subjective uncertainty. In: Proceedings of the KR 2010, pp. 393–403. AAAI Press (2010)
Paris, J.B.: Common sense and maximum entropy. Synthese 117(1), 75–93 (1999)
Paris, J.B.: The Uncertain Reasoner’s Companion: A Mathematical Perspective. Cambridge University Press, Cambridge (2006)
Peñaloza, R., Potyka, N.: Towards statistical reasoning in description logics over finite domains. In: Moral, S., Pivert, O., Sánchez, D., Marín, N. (eds.) SUM 2017. LNCS (LNAI), vol. 10564, pp. 280–294. Springer, Cham (2017). https://doi.org/10.1007/978-3-319-67582-4_20
Pratt, V.R.: Models of program logics. In: Proceedings of the FOCS 1979, pp. 115–122. IEEE Computer Society (1979)
Rudolph, S., Krötzsch, M., Hitzler, P.: Type-elimination-based reasoning for the description logic SHIQbs using decision diagrams and disjunctive datalog. Logical Methods in Computer Science 8(1), 38 p. (2012)
Rudolph, S., Schweizer, L.: Not too big, not too small... complexities of fixed-domain reasoning in first-order and description logics. In: Oliveira, E., Gama, J., Vale, Z., Lopes Cardoso, H. (eds.) EPIA 2017. LNCS (LNAI), vol. 10423, pp. 695–708. Springer, Cham (2017). https://doi.org/10.1007/978-3-319-65340-2_57
Stanley, R.: Enumerative Combinatorics: Cambridge Studies in Advanced Mathematics. Cambridge University Press, Cambridge (1997)
Thimm, M., Kern-Isberner, G.: On probabilistic inference in relationalconditional logics. Logic J. IGPL 20(5), 872–908 (2012)
Tobies, S.: The complexity of reasoning with cardinality restrictions and nominals in expressive description logics. J. Artif. Intell. Res. 12, 199–217 (2000)
Wilhelm, M., Kern-Isberner, G., Ecke, A.: Basic independence results for maximum entropy reasoning based on relational conditionals. In: Proceedings of the GCAI 2017, EPiC Series in Computing, vol. 50, pp. 36–50. EasyChair (2017)
Wilhelm, M., Kern-Isberner, G., Ecke, A., Baader, F.: Counting strategies for the probabilistic description logic \(\cal{ALC}^{\sf ME}\) under the principle of maximum entropy. In: Calimeri, F., Leone, N., Manna, M. (eds.) JELIA 2019. LNCS (LNAI), vol. 11468, pp. 434–449. Springer, Cham (2019). https://doi.org/10.1007/978-3-030-19570-0_28
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Baader, F., Ecke, A., Kern-Isberner, G., Wilhelm, M. (2019). The Complexity of the Consistency Problem in the Probabilistic Description Logic \(\mathcal {ALC} ^\mathsf {ME}\). In: Herzig, A., Popescu, A. (eds) Frontiers of Combining Systems. FroCoS 2019. Lecture Notes in Computer Science(), vol 11715. Springer, Cham. https://doi.org/10.1007/978-3-030-29007-8_10
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