Computability in higher types, Pω and the completeness of type assignment

Abstract

Pω, the powerset of the natural numbers, may be turned into an applicative structure by Myhill and Shepherdson, “·”. Then, for A, B ⊆ Pω, set A → B = { d ϵ Pω | ∀ a ϵ A, da ϵ B}. Any effectively given domain (in the sense of Scott (1982)) can be embedded into Pω by a continuous and computable retraction ( notation: X 〈 c A X , for some A X ⊆ Pω, which is also an effectively given domain). We first prove that if X 〈 c A X and Y 〈 c A Y , then also A X → A Y is an effectively given domain and (☆): Cont( X, Y) 〈 c A X → A Y , i.e., the continuous functions can be embedded into A X → A Y . Let now P ⊆ Pω be the collection of single-valued sets, i.e., P is isomorphic to the effectively given domain of the partial functions on ω, and let T be the function-type symbols, with (1) ϵ T. Then, for P 1 = P, P σ→ τ = P σ → P τ extends the classical recursive operators at higher types. By (☆), Ershov's model of the Kleene-Kreisel countable functionals can effectively be embedded, by some G σ 's, into the type structure { P σ } σϵT in Pω. Thus, the recursive functionals correspond to the r.e. sets in the due types, for example, f has type σ → τ iff G σ→ τ ( f) is an r.e. set in P σ → P τ . { P σ } σϵT clearly yields a model for formal type assingnment to terms of λ-calculus, i.e., for any assignment B of types to variables and any σ ϵ T one has B ⊢ σM⇒[ M] ξB ϵ P σ , where ξ B : Var → Pω, according to B. We prove that also the reverse implication holds for typable terms. Thus, a completeness theorem for type checking is established over a model defined by an independent recursion-theoretical motivation.