Probabilistic Description Logics Based on the Aggregating Semantics and the Principle of Maximum Entropy

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Description Logics (DLs) are a well-investigated family of logic-based knowledge representation languages which are tailored towards representing terminological knowledge. This terminological knowledge is represented in the TBox using general concept inclusions (GCIs), whereas knowledge about individuals (assertional knowledge) is stated in the ABox. Probabilistic extensions of DLs are motivated by the fact that, in many application domains, knowledge is not always certain. In such extensions, there is a need for treating assertional knowledge differently from terminological knowledge. In principle, probabilistic terminological knowledge has a statistical flavour whereas probabilistic assertional knowledge has a subjective flavour. However, in order to reason with respect to a knowledge base containing both kinds of knowledge, one needs a common semantic framework covering both aspects. Previous work in this area has not addressed this dual need in a completely satisfactory way.The main idea underlying this project is to adapt and extend the recently developed aggregating semantics from a restricted first-order case to DLs by respectively generalizing ABox assertions and GCIs to closed and open probabilistic conditionals. This semantics combines subjective probabilities with population-based statements on the basis of a possible-worlds semantics, thus providing a common semantic framework for both subjective and statistical probabilities. As a second main feature, we apply the principle of maximum entropy on top of aggregating semantics. This overcomes the pitfall of obtaining large and uninformative intervals for inferred probabilities which is a common feature of many of the approaches that reason with respect to sets of probability distributions. Whereas the semantic properties of the approach have been investigated in some detail for a fragment of first-order logic, only preliminary work has been done on algorithmic and computational properties. To be useful in practice, the probabilistic DL obtained by applying this approach need to be equipped with effective reasoning procedures. Thus, the main emphasis of this project will be on investigating computational properties (decidability and complexity) of the probabilistic logics obtained by instantiating the approach in particular with DLs of different expressive power. In addition to showing decidability and complexity results, we will develop practical algorithms for some of the investigated DLs and provide prototypical implementations. Another major challenge will be to extend the approach from universes of fixed finite size to the infinite case by either considering limit probabilities for universes of growing size or considering a countably infinite universe. Furthermore, in addition to the basic approach, we will also investigate extensions, such as using probabilities also within concepts, allowing for additional constraints in the knowledge base and for independence assumptions.