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We show that the vast majority of extensions of the description logic$\mathcal{EL}$ do not enjoy the Craig interpolation nor the projective Bethdefinability property. This is the case, for example, for $\mathcal{EL}$ withnominals, $\mathcal{EL}$ with the universal role, $\mathcal{EL}$ with a roleinclusion of the form $r\circ s\sqsubseteq s$, and for $\mathcal{ELI}$. Itfollows in particular that the existence of an explicit definition of a conceptor individual name cannot be reduced to subsumption checking via implicitdefinability. We show that nevertheless the existence of interpolants andexplicit definitions can be decided in polynomial time for standard tractableextensions of $\mathcal{EL}$ (such as $\mathcal{EL}^{++}$) and in ExpTime for$\mathcal{ELI}$ and various extensions. It follows that these existenceproblems are not harder than subsumption which is in sharp contrast to thesituation for expressive DLs. We also obtain tight bounds for the size ofinterpolants and explicit definitions and the complexity of computing them:single exponential for tractable standard extensions of $\mathcal{EL}$ anddouble exponential for $\mathcal{ELI}$ and extensions. We close with adiscussion of Horn-DLs such as Horn-$\mathcal{ALCI}$.

Interpolants and Explicit Definitions in Extensions of the Description Logic EL