A Set-Theoretic Decision Procedure for Quantifier-Free, Decidable Languages Extended with Restricted Quantifiers
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Let โ_๐ณ be the language of first-order, decidable theory ๐ณ. Consider the language, โ_โ๐ฌ(๐ณ), that extends โ_๐ณ with formulas of the form โ x โ A: ฯ (restricted universal quantifier, RUQ) and โ x โ A: ฯ (restricted existential quantifier, REQ), where A is a finite set and ฯ is a formula made of ๐ณ-formulas, RUQ and REQ. That is, โ_โ๐ฌ(๐ณ) admits nested restricted quantifiers. In this paper we present a decision procedure for โ_โ๐ฌ(๐ณ) based on the decision procedure already defined for the Boolean algebra of finite sets extended with restricted intensional sets (โ_โโ๐ฎ). The implementation of the decision procedure as part of the {log} (`setlog') tool is also introduced. The usefulness of the approach is shown through a number of examples drawn from several real-world case studies.
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