A characterization of F-complete type assignments

AbstractThe aim of this paper is to investigate the soundness and completeness of the intersection type discipline (for terms of the (untyped λ-calculus) with respect to the F-semantics (F-soundness and F-completeness).As pointed out by Scott, if D is the domain of a γ-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 “σ→τ” is ‘the type of functions with domain σ and range τ’ and interprets σ→τ 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 ϕ∧ωn, where ϕ is a type variable and ϕ is the ‘universal’ type, the term λz1…zn…Mz1…zn has type ϕ. Here we assume that z1…z.n do not occur free in M