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Semantics of the second order lambda calculus†

Published online by Cambridge University Press:  04 March 2009

Bart Jacobs
Bart Jacobs
Affiliation:
Dep. Comp. Sc., Toernooiveld, 6525 ED Nijmegen, The Netherlands
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Abstract

In the literature ther are two main notins of model for the second order λ-calculus: one by Bruce, Meyer and Mitchell (the BMM-model, for short) in set-theoretical formulation and one category-theoretical by Seely. Here we generalise Seely's notion, using semifunctors and semi-adjunctions from Hayashi, and introduce λ2-algebras, λη2-algebras, λ2-models and λη-models, similarly to the untyped λ-calculus. Non-extensional abstraction of both term and type variables is described by semi-adjunctions (essentially as in Martini's thesis). We show that also for second order sume, the β-(and commutation!) conversions correspond to semi-functoriality and that the (additional) η-conversion corresponds to ordinary fuctioriality.

In the above framework various examples – well known ones and variations – are described. Also, we determine the place of the BMM-models; an earlier version of the latter has been reported by Jacobs.

Type
Research Article
Information
Mathematical Structures in Computer Science , Volume 1 , Issue 3 , November 1991 , pp. 327 - 360
Copyright
Copyright © Cambridge University Press 1991

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