A Characterization of F-Complete Type Assignments

The aim of this paper is to investigate the soundness and completeness of the intersection type discipline (for terms of the (untyped ?-calculus) with respect to the F-semantics (F-soundness and F-completeness).As pointed out by Scott, if D is the domain of a ?-model, there is a subset F of D whose elements are the `canonical? representatives of functions. The F-semantics of types takes into account that theintuitive meaning of “???” is `the type of functions with domain ? and range ?? and interprets ??? as a subset of F.The type theories which induce F-complete type assignments are characterized. It follows that a type assignment is F-complete iff equal terms get equal types and, whenever M has a type ???n, where ? is a type variable and ? is the `universal? type, the term ?z1?zn?Mz1?zn has type ?. Here we assume that z1?z.n do not occur free in M.

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