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Untyped λ-calculus with relative typing

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Typed Lambda Calculi and Applications (TLCA 1995)

Part of the book series: Lecture Notes in Computer Science ((LNCS,volume 902))

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Abstract

A system of untyped λ-calculus with a restriction on function abstraction using relative typing analogous to the restriction on set comprehension found in Quine's set theory “New Foundations” is discussed. The author has shown elsewhere that this system is equiconsistent with Jensen's NFU (“New Foundations” with urelements)+Infinity, which is in turn equiconsistent with the simple theory of types with infinity. The definition of the system is given and the construction of a model is described. A semantic motivation for the stratification criterion for function abstraction is given, based on an abstract model of computation. The same line of semantic argument is used to motivate an analogy between the notion of “strongly Cantorian set” found in “New Foundations” and the notion of “data type”; an implementation of absolute types as domains of retractions with strongly Cantorian ranges is described. The implementation of these concepts in a theorem prover developed by the author is sketched.

Supported by the U. S. Army Research Office, grant no. DAAH04-94-G-0247

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Authors and Affiliations

  1. Math. Dept., Boise State University, 83725, Boise, Idaho, USA

    M. Randall Holmes

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  1. M. Randall Holmes
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Mariangiola Dezani-Ciancaglini Gordon Plotkin

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© 1995 Springer-Verlag Berlin Heidelberg

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Holmes, M.R. (1995). Untyped λ-calculus with relative typing. In: Dezani-Ciancaglini, M., Plotkin, G. (eds) Typed Lambda Calculi and Applications. TLCA 1995. Lecture Notes in Computer Science, vol 902. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0014056

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  • DOI: https://doi.org/10.1007/BFb0014056

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  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-59048-4

  • Online ISBN: 978-3-540-49178-1

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