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Retractions of dI-domains as a model for Type:Type. (English) Zbl 0728.68036
Summary: We introduce an extension, with the Type: Type assumption, for a higher- order functional language: in the framework of \(\lambda\beta\) p (the extension of the type-free lambda calculus introduced in R. Amadio and G. Longo (1986b, in “IFIP Conference ‘Formal description of programming concepts’, Ebberup (DK), August 1986”)). Then, we find (using the stable functions and Berry’s dI-domains) a model for \(\lambda\beta\) p. This settles the consistency problem for \(\lambda\beta\) p, up to now open, and gives very simple mathematical models for higher- order calculi with Type:Type. Such models are based on the relevant property that the range of a stable retraction on a dI-domain is still a dI-domain. Finally, we give the categorical description of the models of \(\lambda\beta\) p, based on Moggi-Asperti’s notion of internalization in A. Asperti [Models of higher order calculi internal categories, Carnegie-Mellon report (1988) (in preparation].

MSC:
68N15 Theory of programming languages
03B15 Higher-order logic; type theory (MSC2010)
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