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Monadic Translation of Intuitionistic Sequent Calculus

  • José Espírito Santo
  • Ralph Matthes
  • Luís Pinto
Part of the Lecture Notes in Computer Science book series (LNCS, volume 5497)

Abstract

This paper proposes and analyses a monadic translation of an intuitionistic sequent calculus. The source of the translation is a typed λ-calculus previously introduced by the authors, corresponding to the intuitionistic fragment of the call-by-name variant of \(\overline{\lambda}\mu\tilde{\mu}\) of Curien and Herbelin, and the target is a variant of Moggi’s monadic meta-language, where the rewrite relation includes extra permutation rules that may be seen as variations of the “associativity” of bind (the Kleisli extension operation of the monad).

The main result is that the monadic translation simulates reduction strictly, so that strong normalisation (which is enjoyed at the target, as we show) can be lifted from the target to the source. A variant translation, obtained by adding an extra monad application in the translation of types, still enjoys strict simulation, while requiring one fewer extra permutation rule from the target.

Finally we instantiate, for the cases of the identity monad and the continuations monad, the meta-language into the simply-typed λ-calculus. By this means, we give a generic account of translations of sequent calculus into natural deduction, which encompasses the traditional mapping studied by Zucker and Pottinger, and CPS translations of intuitionistic sequent calculus.

Keywords

Reduction Step Reduction Rule Natural Deduction Sequent Calculus Typing Rule 
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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Copyright information

© Springer-Verlag Berlin Heidelberg 2009

Authors and Affiliations

  • José Espírito Santo
    • 1
  • Ralph Matthes
    • 2
  • Luís Pinto
    • 1
  1. 1.Departamento de MatemáticaUniversidade do MinhoPortugal
  2. 2.I.R.I.T. (C.N.R.S. and University of Toulouse III)France

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