Counting Strategies for the Probabilistic Description Logic $\mathcal {ALC}^\mathsf {ME}$ Under the Principle of Maximum Entropy

Abstract

We present $\mathcal {ALC}^\mathsf {ME}$, a probabilistic variant of the Description Logic $\mathcal {ALC}$ that allows for representing and processing conditional statements of the form “if E holds, then F follows with probability p” under the principle of maximum entropy. Probabilities are understood as degrees of belief and formally interpreted by the aggregating semantics. We prove that both checking consistency and drawing inferences based on approximations of the maximum entropy distribution is possible in $\mathcal {ALC}^\mathsf {ME}$ in time polynomial in the domain size. A major problem for probabilistic reasoning from such conditional knowledge bases is to count models and individuals. To achieve our complexity results, we develop sophisticated counting strategies on interpretations aggregated with respect to the so-called conditional impacts of types, which refine their conditional structure.

This work was supported by the German Research Foundation (DFG) within the Research Unit FOR 1513 “Hybrid Reasoning for Intelligent Systems”.