On the Non-Monotonic Description Logic $\mathcal{ALC}$+T$_{\mathsf{min}}$

In the last 20 years many proposals have been made to incorporate non-monotonic reasoning into description logics, ranging from approaches based on default logic and circumscription to those based on preferential semantics. In particular, the non-monotonic description logic $\mathcal{ALC}$+T$_{\mathsf{min}}$ uses a combination of the preferential semantics with minimization of a certain kind of concepts, which represent atypical instances of a class of elements. One of its drawbacks is that it suffers from the problem known as the \emph{property blocking inheritance}, which can be seen as a weakness from an inferential point of view. In this paper we propose an extension of $\mathcal{ALC}$+T$_{\mathsf{min}}$, namely $\mathcal{ALC}$+T$^+_{\mathsf{min}}$, with the purpose to solve the mentioned problem. In addition, we show the close connection that exists between $\mathcal{ALC}$+T$^+_{\mathsf{min}}$ and concept-circumscribed knowledge bases. Finally, we study the complexity of deciding the classical reasoning tasks in $\mathcal{ALC}$+T$^+_{\mathsf{min}}$.