Given such a signature \(\tau\), we define \(\text{Mod}(\tau)\) to be the class of all \(\tau\)-structures with finite domain. In the context of descriptive complexity theory, a logic \(\mathcal{L}\) is taken to be an extension of the language of first-order logic with one or more classes of additional expressions such as higher-order quantifiers or fixed-point operators. These are treated semantically as logical symbols. If \(\mathcal{L}\) is such a logic, and \(\tau\) a signature, we write \(\text{Sent}_{\mathcal{L}(\tau)}\) to denote the class of well-formed sentences constructed using the logical symbols from \(\mathcal{L}\) and the relational symbols from \(\tau\). A sentence \(\phi \in \text{Sent}_{\mathcal{L}(\tau)}\) is said to define a problem \(X\) just in case \(X\) is coextensive with the set of \(\tau\)-structures satisfying \(\phi\) – i.e.