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Let $\mathcal{L}_{\mathcal{X}}$ be the language of first-order, decidabletheory $\mathcal{X}$. Consider the language,$\mathcal{L}_{\mathcal{RQ}}(\mathcal{X})$, that extends$\mathcal{L}_{\mathcal{X}}$ with formulas of the form $\forall x \in A: \phi$(restricted universal quantifier, RUQ) and $\exists x \in A: \phi$ (restrictedexistential quantifier, REQ), where $A$ is a finite set and $\phi$ is a formulamade of $\mathcal{X}$-formulas, RUQ and REQ. That is,$\mathcal{L}_{\mathcal{RQ}}(\mathcal{X})$ admits nested restricted quantifiers.In this paper we present a decision procedure for$\mathcal{L}_{\mathcal{RQ}}(\mathcal{X})$ based on the decision procedurealready defined for the Boolean algebra of finite sets extended with restrictedintensional sets ($\mathcal{L}_\mathcal{RIS}$). The implementation of thedecision procedure as part of the $\{log\}$ (`setlog') tool is also introduced.The usefulness of the approach is shown through a number of examples drawn fromseveral real-world case studies.

A Set-Theoretic Decision Procedure for Quantifier-Free, Decidable Languages Extended with Restricted Quantifiers