The Complexity of the Consistency Problem in the Probabilistic Description Logic ^\mathsf ME.

The probabilistic Description Logic ALC(ME) is an extension of the Description Logic 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 ALC(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 ALC(ME) are interpreted over a fixed finite domain and based on their maximum entropy (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.