Intensional sets, i.e., sets given by a property rather than by enumerating elements, are widely recognized as a key feature to describe complex problems (see, e.g., specification languages such as B and Z). Notwithstanding, very few tools exist supporting high-level automated reasoning on general formulas involving intensional sets. In this paper we present a decision procedure for a first-order logic language offering both extensional and (a restricted form of) intensional sets (RIS). RIS are introduced as first-class citizens of the language, and set-theoretical operators on RIS are dealt with as constraints. Syntactic restrictions on RIS guarantee that the denoted sets are finite. The language of RIS, called \({\mathcal {L}}_\mathcal {RIS}\), is parametric with respect to any first-order theory \({\mathcal {X}}\) providing at least equality and a decision procedure for \({\mathcal {X}}\)-formulas. In particular, we consider the instance of \({\mathcal {L}}_\mathcal {RIS}\) when \({\mathcal {X}}\) is the theory of hereditarily finite sets and binary relations. We also present a working implementation of this instance as part of the \(\{log\}\) tool, and we show through a number of examples and two case studies that, although RIS are a subclass of general intensional sets, they are still sufficiently expressive as to encode and solve many interesting problems. Finally, an extensive empirical evaluation provides evidence that the tool can be used in practice.

内涵集，即由属性而不是通过枚举元素给出的集合，被广泛认为是描述复杂问题的关键特征（例如，参见诸如B和Z之类的规范语言）。尽管如此，很少有工具支持涉及内涵集的通用公式的高级自动推理。在本文中，我们提出了针对一阶逻辑语言的决策过程，该逻辑语言同时提供了扩展集和（约束形式的）内涵集（RIS）。RIS是作为该语言的一等公民引入的，而RIS上的集合理论运算符则作为约束条件来处理。对RIS的语法限制保证了所表示的集合是有限的。RIS的语言称为\（{\ mathcal {L}} _ \ mathcal {RIS} \），对于任何一阶理论都是参数化的\（{\ mathcal {X}} \）至少为\（{\ mathcal {X}} \）公式提供相等性和决策程序。特别地，当\（{\ mathcal {X}} \）是遗传有限集和二元关系的理论时，我们考虑\（{\ mathcal {L}} _ \ mathcal {RIS} \）的实例。我们还作为\（\ {log \} \）工具的一部分介绍了此实例的有效实现，并且通过大量示例和两个案例研究表明，尽管RIS是通用内涵集的子类，但它们是仍然具有足够的表现力，可以编码和解决许多有趣的问题。最后，广泛的经验评估提供了可以在实践中使用该工具的证据。