Description Logics Go Second-Order -- Extending EL with Universally Quantified Concepts

The study of Description Logics have been historically mostly focused on features that can be translated to decidable fragments of first-order logic. In this paper, we leave this restriction behind and look for useful and decidable extensions outside first-order logic. We introduce universally quantified concepts, which take the form of variables that can be replaced with arbitrary concepts, and define two semantics of this extension. A schema semantics allows replacements of concept variables only by concepts from a particular language, giving us axiom schemata similar to modal logics. A second-order semantics allows replacement of concept variables with arbitrary subsets of the domain, which is similar to quantified predicates in second-order logic. To study the proposed semantics, we focus on the extension of the description logic $\mathcal{EL}$. We show that for a useful fragment of the extension, the conclusions entailed by the different semantics coincide, allowing us to use classical $\mathcal{EL}$ reasoning algorithms even for the second-order semantics. For a slightly smaller, but still useful, fragment, we were also able to show polynomial decidability of the extension. This fragment, in particular, can express a generalized form of role chain axioms, positive self restrictions, and some forms of (local) role-value-maps from KL-ONE, without requiring any additional constructors.