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The encodability hierarchy for PCF types
Working with the simple types over a base type of natural numbers (including
product types), we consider the question of when a type is encodable
as a definable retract of : that is, when there are -terms
and with .
In general, the answer to this question may vary according to both the choice
of -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
(but no arithmetic) considered modulo computational equality, up to the whole
of Plotkin's PCF considered modulo observational equivalence. We show that
iff via trivial
isomorphisms, and that for any we have either or . Furthermore, we show that the induced linear
order on isomorphism classes of types is actually a well-ordering of type
, and indeed that there is a close syntactic correspondence between
simple types and Cantor normal forms for ordinals below . This
means that the relation is readily decidable, and that terms
witnessing a retraction are readily constructible when
holds.Comment: 19 page