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The Bayesian Ontology Language \(\mathcal {BEL}\)

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Abstract

We introduce the new probabilistic description logic (DL) \(\mathcal {BEL} \), which extends the light-weight DL \(\mathcal {EL}\) with the possibility of expressing uncertainty about the validity of some knowledge. Contrary to other probabilistic DLs, \(\mathcal {BEL}\) is designed to represent classical knowledge that depends on an uncertain context; that is, some of the knowledge may hold or not depending on the current situation. The probability distribution of these contexts is expressed by a Bayesian network (BN). We study different reasoning problems in \(\mathcal {BEL}\), providing tight complexity bounds for all of them. One particularly interesting property of our framework is that reasoning can be decoupled between the logical (i.e., \(\mathcal {EL}\)), and the probabilistic (i.e., the BN) components. We later generalize all the notions presented to introduce Bayesian extensions of arbitrary ontology languages. Using the decoupling property, we are able to provide tight complexity bounds for reasoning in the Bayesian extensions of many other DLs. We provide a detailed analysis of our formalism w.r.t. the assumptions made and compare it with the existing approaches.

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Notes

  1. In their general form, BNs allow for arbitrary discrete random variables. We restrict w.l.o.g. to Boolean variables for ease of presentation.

  2. We will often see valuations as contexts containing one literal for each Boolean variable.

  3. Recall that a problem is fixed-parameter tractable if it can be solved in polynomial time, assuming that the parameter has a fixed (i.e., constant) size [25].

  4. We use a different formulation than in our previous work [14] to better align the decision problem with existing literature.

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Acknowledgments

İsmail İlkan Ceylan is supported by DFG within the Research Training Group “RoSI” (GRK 1907) and Rafael Peñaloza is partially supported by DFG within the Cluster of Excellence “cfAED”; part of this work was developed while Rafael Peñaloza was affiliated with TU Dresden, and the Center for Advancing Electronics Dresden, Germany. We would like to thank Gabriele Kern-Isberner for fruitful discussions regarding the material and conditional implications, and Franz Baader for his suggestions for improving the quality and presentation of the paper.

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Authors and Affiliations

  1. Institute for Theoretical Computer Science, Technische Universität Dresden, Dresden, Germany

    İsmail İlkan Ceylan

  2. KRDB Research Centre, Free University of Bozen-Bolzano, Bolzano, Italy

    Rafael Peñaloza

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  1. İsmail İlkan Ceylan
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  2. Rafael Peñaloza
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Correspondence to Rafael Peñaloza.

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Ceylan, İ.İ., Peñaloza, R. The Bayesian Ontology Language \(\mathcal {BEL}\) . J Autom Reasoning 58, 67–95 (2017). https://doi.org/10.1007/s10817-016-9386-0

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