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

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.