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The Complexity of the Consistency Problem in the Probabilistic Description Logic \(\mathcal {ALC} ^\mathsf {ME}\)

  • Franz BaaderEmail author
  • Andreas Ecke
  • Gabriele Kern-Isberner
  • Marco Wilhelm
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 11715)

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.

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Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  • Franz Baader
    • 1
    Email author
  • Andreas Ecke
    • 1
  • Gabriele Kern-Isberner
    • 2
  • Marco Wilhelm
    • 2
  1. 1.Department of Computer ScienceTU DresdenDresdenGermany
  2. 2.Department of Computer ScienceTU DortmundDortmundGermany

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