The encodability hierarchy for PCF types
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Working with the simple types over a base type of natural numbers (including product types), we consider the question of when a type σ is encodable as a definable retract of τ: that is, when there are λ-terms e:σ→τ and d:τ→σ with d ∘ e = id. In general, the answer to this question may vary according to both the choice of λ-calculus and the notion of equality considered; however, we shall show that the encodability relation ≼ between types actually remains stable across a large class of languages and equality relations, ranging from a very basic language with infinitely many distinguishable constants 0,1,... (but no arithmetic) considered modulo computational equality, up to the whole of Plotkin's PCF considered modulo observational equivalence. We show that σ≼τ≼σ iff στ via trivial isomorphisms, and that for any σ,τ we have either σ≼τ or τ≼σ. Furthermore, we show that the induced linear order on isomorphism classes of types is actually a well-ordering of type ϵ_0, and indeed that there is a close syntactic correspondence between simple types and Cantor normal forms for ordinals below ϵ_0. This means that the relation ≼ is readily decidable, and that terms witnessing a retraction στ are readily constructible when σ≼τ holds.
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