Complexity class for subsumption for $\mathcal{AL}(\circ, ^{-})$ According to Baader et al's _Description Logic Handbook_ , subsumption for $\mathcal{AL}(\circ)$ and $\mathcal{AL}(^{-})$ is in $\mathrm{P}$. However, I am not sure what complexity class subsumption for $\mathcal{AL}(\circ, ^{-})$ would fall into. Would it still be $\mathrm{P}$, or does this make subsumption much harder?

As it turns out, $\mathcal{AL}(\circ, ^{-})$ is a super-language of $\mathcal{FL}(\circ,^{-})$, whose subsumption problem is $NP$-hard by Donini's chapter in the Description Logic Handbook.