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

by   Maximiliano Cristiรก, et al.

Let โ„’_๐’ณ be the language of first-order, decidable theory ๐’ณ. Consider the language, โ„’_โ„›๐’ฌ(๐’ณ), that extends โ„’_๐’ณ with formulas of the form โˆ€ x โˆˆ A: ฯ• (restricted universal quantifier, RUQ) and โˆƒ x โˆˆ A: ฯ• (restricted existential quantifier, REQ), where A is a finite set and ฯ• is a formula made of ๐’ณ-formulas, RUQ and REQ. That is, โ„’_โ„›๐’ฌ(๐’ณ) admits nested restricted quantifiers. In this paper we present a decision procedure for โ„’_โ„›๐’ฌ(๐’ณ) based on the decision procedure already defined for the Boolean algebra of finite sets extended with restricted intensional sets (โ„’_โ„›โ„๐’ฎ). 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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