Sequent combinators: a Hilbert system for the lambda calculus

  • Authors:

  • Healfdene Goguen;Jean Goubault-Larrecq

  • Affiliations:

  • Department of Computer Science, The King's Buildings, University of Edinburgh, Edinburgh, EH9 3JZ, Scotland. Email: hhg@att.com;G.I.E. Dyade, INRIA Rocquencourt, Domaine de Voluceau, B.P.105, F-78153 Le Chesnay Cedex, France. Email: jean.goubault@dyade.fr

  • Venue:
  • Mathematical Structures in Computer Science
  • Year:
  • 2000

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Abstract

This paper introduces Hilbert systems for λ-calculus, called sequent combinators, addressing many of the problems of Hilbert systems that have led to the more widespread adoption of natural deduction systems in computer science. This suggests that Hilbert systems, with their uniform approach to meta-variables and substitution, may be a more suitable framework than λ-calculus for type theories and programming languages. Two calculi are introduced here. The calculus SKIn captures λ-calculus reduction faithfully, is confluent even in the presence of meta-variables, is normalizing but not strongly normalizing in the typed case, and standardizes. The sub-calculus SKInT captures λ-reduction in slightly less obvious ways, and is a language of proof-terms not directly for intuitionistic logic, but for a fragment of S4 that we name near-intuitionistic logic. To our knowledge, SKInT is the first confluent, first-order calculus to capture λ-calculus reduction fully and faithfully and be strongly normalizing in the typed case. In particular, no calculus of explicit substitutions has yet achieved this goal.