Probabilistic Description Logic as a Fragment of Probabilistic First-Order Logic

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Description logics are core formalisms in formal knowledge representation. They are concerned on the one hand with terminological knowledge, i.e. with interrelations among concepts, and on the other hand with assertional knowledge, i.e. knowledge about interrelations between concrete individuals and their description by abstract concepts. Knowledge is often subject to various forms of uncertainty; e.g. knowledge can be statistical ("90% of all birds fly") or subjectively uncertain due to lack of complete knowledge about the actual state of the world ("Patient X exhibits erythema, and therefore suffers from Lyme borreliosis with 80% probability"). Probabilistic description logics aim to formalize such uncertain knowledge using probabilistic methods. Technically, probabilistic description logics are many-dimensional modal logics; models are based on probability distributions on a set of worlds, each of which is associated with a standard interpretation of the description logical dimension. The latter represents the set of individuals, possibly equipped with a statistical distribution. By dint of this notion of model, probabilistic description logic embeds as a fragment into Halpern's probabilistic first-order logic, in analogy to the embedding of classical modal logic into classical first-order logic. This puts the semantics of probabilistic description logics on a firm footing and provides a framework for the estimation and comparison of their expressive power. As in the classical case, central questions concern the decidability and complexity of the core reasoning problems in probabilistic description logics of various levels of expressivity. This includes in particular lightweight dialects such probabilistic EL, which limits expressivity rather substantially but in return allows for tractable reasoning.The aims of the second project phase include a broadening of the metatheory to cover also questions of axiomatization and formal bounds on expressivity, as well as an extension of the expressive means made available; in particular, we will target fixpoint extensions and probabilistic logics featuring fuzzy truth values, such as the fuzzy logic of `probably'. Moreover, the relationship between subjective and statistical probabilities remains in the focus of attention. We will aim to design logics that support intuitively expected reasoning patterns such as direct inference of subjective from statistical probabilities while at the same time having decidable reasoning problems.