A filter lambda model and the completeness of type assignment

Abstract

In [6, p. 317] Curry described a formal system assigning types to terms of the type-free λ-calculus. In [11] Scott gave a natural semantics for this type assignment and asked whether a completeness result holds.Inspired by [4] and [5] we extend the syntax and semantics of the Curry types in such a way that filters in the resulting type structure form a domain in the sense of Scott [12]. We will show that it is possible to turn the domain of types into a λ-model, among other reasons because all λ-terms possess a type. This model gives the completeness result for the extended system. By a conservativity result the completeness for Curry's system follows.Independently Hindley [8], [9] has proved both completeness results using term models. His method of proof is in some sense dual to ours.For λ-calculus notation see [1].

Characterizing Convergent Terms in Object Calculi via Intersection Types

“…On the other hand the set of types that can be given to a term describes its functional behaviour, that is its meaning (see e.g. 12,6,22,14]). That convergency is characterized by t ypability within the system by t ypes of some speci c shape is basic with respect to the construction of denotational models using types (see 6,3,4]).…”

Essential intersection type assignment

“…In Curry's system it is, for example, not possible to assign a type to the term (λx.xx); moreover, although the lambda terms (λcd.d) and ((λxyz.xz(yz))(λab.a)) are β-equal, the principal type schemes for these terms are different. The Intersection Type Discipline as presented in [5] (a more enhanced system was presented in [4]) is an extension of Curry's system that does not have these drawbacks. The extension being made consists mainly of allowing for term variables (and terms) to have more than one type.…”

“…The type assignment system presented in [4] (the BCD-system) is based on the system as presented in [5]; it defines the set of intersection types T in a more general way, and is strengthened further by introducing a partial order relation '≤' on types as well as adding the type assignment rule (≤) and a more general form of the rules concerning intersection. The rule (≤), as well as the more general treatment of intersection types were introduced mainly to prove completeness of type assignment.…”

“…These rules not only allow of superfluous steps in derivations, but they also make it possible to give essentially different derivations for the same result. Moreover, in [4] the relation ≤ induced an equivalence relation ∼ on types. Equivalence classes are big (for example: ω ∼ σ→ω, for all types σ) and type assignment is closed for ∼ .…”

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AbstractThis paper introduces a notion of intersection type assignment on the Lambda Calculus that is a restriction of the BCD-system as presented in [4]. This restricted system is essential in the following sense: it is an almost syntax directed system that satisfies all major properties of the BCD-system.

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