Two semantic interpretations of probabilities in description logics of typicality

In this work we focus on extensions of Description Logics (DLs) of typicality by means of probabilities. We introduce a novel extension of the logic of typicality $\mathcal{ALC}+\mathbf{T}_{\mathbf{R}}$, able to represent and reason about typical properties and defeasible inheritance in DLs. The novel logic ($\mathcal{ALC}\mathbf{T}<^>{\textsf{P}}$: Typical $\mathcal{ALC}$ with Probabilities as Proportions) allows inclusions of the form $\mathbf{T}(C) \sqsubseteq <^>{p} D$, with probability $p$ representing a proportion, meaning that "all the typical $C$s are $D$s, and the probability that a $C$ element is not a $D$ element is $1-p$". We also compare and confront this novel logic with a similar one already presented in the literature ($\mathbf{T}<^>{\textsf{CL}}$, introduced in Lieto and Pozzato (2020, J. Exp. Theor. Artif. Intell., 32, 769-804)), inspired by the DISPONTE semantics and that allows inclusions of the form $p \:: \ \mathbf{T}(C) \sqsubseteq D$ with probability $p$, where $p$ represents a degree of belief, whose meaning is that "we believe with a degree $p$ that typical $C$s' are also $D$s.". We then show that the proposed $\mathcal{ALC}\mathbf{T}<^>{\textsf{P}}$ extension (like the previous $\mathbf{T}<^>{\textsf{CL}}$) can be applied in order to tackle a specific and challenging problem in the field of common-sense reasoning, namely the combination of prototypical concepts, that have been shown to be problematic to model for other symbolic approaches like fuzzy logic. We show that, for the proposed extension, the complexity of reasoning remains ExpTime-complete as for the underlying standard monotonic DL $\mathcal{ALC}$.