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The Finitely Generated Types of the λ-Calculus

  • Thierry Joly
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 2044)

Abstract

We answer a question raised by Richard Statman (cf. [8]) concerning the simply typed λ-calculus (having o as only ground type): Is it possible to generate from a finite set of combinators all the closed terms of a given type ? (By combinators we mean closed λ-terms of any types).

Let us call complexity of a λ-term t the least number of distinct variables required for its writing up to α-equivalence. We prove here that a type T can be generated from a finite set of combinators iff there is a constant bounding the complexity of every closed normal λ-term of type T. The types of rank ⩽ 2 and the types A 1→(A2→…(A n→o)) such that for all i = 1, … n: A i = o, Ai = o→o or A i = (o→(o→…(o→o)))→o, are thus the only inhabited finitely generated types.

Keywords

Free Variable Distinct Variable Small Term Ground Type Sketch Proof 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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References

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Copyright information

© Springer-Verlag Berlin Heidelberg 2001

Authors and Affiliations

  • Thierry Joly
    • 1
  1. 1.équipe Preuves, Programmes et SystèmesUniversité Paris VIIParis cedex 05France

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