On functions preserving levels of approximation: A refined model construction for various lambda calculi

Abstract

In this paper dI-domains are enriched by a family of projections which assign to each point a sequence of canonical approximations. The morphisms are stable maps that preserve the levels of approximation generated by the projections. For the computation of an approximation of a given level of the output they require, in addition, only information about the input of at most the same level of approximation. It is shown that the category of these domains and maps is Cartesian closed. The set of morphisms between two such domains is a domain of the same kind, but turns out not to be an exponent in the category. Domains D are constructed which are isomorphic to the exponent D D . Moreover, it is proved that the space of retractions in an exponent D D is a retract of this. Both results together provide new models of Amadio-Longo's extension λβp of the λ-calculus. As has been shown by Amadio and Longo, strong type theories which incorporate the Type: Type assumption can be syntactically interpreted in this calculus.