We define an observability property for a calculus with pattern matching which is inspired by the notion of solvability for the lambda-calculus. We prove that observability can be characterized by means of typability and inhabitation in an intersection type system P based on non-idempotent types. We show first that the system P characterizes the set of terms having canonical form, i.e. that a term is typable if and only if it reduces to a canonical form. But the set of observable terms is properly contained in the set of canonical. Thus, typability alone is not sufficient to characterize observability, in contrast to the solvability case for lambda-calculus. We then prove that typability, together with inhabitation, provides a full characterization of observability, in the sense that a term is observable if and only if it is typable and the types of all its arguments are inhabited. We complete the picture by providing an algorithm for the inhabitation problem of P.

中文翻译：

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可观察性 = 可打性 + 可居住性

我们为具有模式匹配的微积分定义了可观察性属性，其灵感来自 lambda 演算的可解性概念。我们证明了在基于非幂等类型的交集类型系统 P 中，可观察性可以通过类型化和居住性来表征。我们首先表明系统 P 表征具有规范形式的术语集，即，当且仅当它归约为规范形式时，术语是可类型化的。但是可观察项的集合正确地包含在规范集合中。因此，与 lambda 演算的可解性情况相比，仅可打字性不足以表征可观察性。然后我们证明可打字性与居住性一起提供了可观察性的完整表征，从这个意义上说，一个术语是可观察的，当且仅当它是可类型化的，并且它的所有参数的类型都存在。我们通过提供 P 的居住问题的算法来完成图片。