On the Computational Interpretation of Negation

  • Authors:

  • Michel Parigot

  • Affiliations:

  • -

  • Venue:
  • Proceedings of the 14th Annual Conference of the EACSL on Computer Science Logic
  • Year:
  • 2000

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Abstract

We investigate the possibility of giving a computational interpretation of an involutive negation in classical natural deduction. We first show why this cannot be simply achieved by adding ∀∀A = A to typed λ-calculus: the main obstacle is that an involutive negation cannot be a particular case of implication at the computational level. It means that one has to go out typed λ-calculus in order to have a safe computational interpretation of an involutive negation. We then show how to equip λµ-calculus in a natural way with an involutive negation: the abstraction and application associated to negation are simply the operators µ and [] from λµ-calculus. The resulting system is called symmetric λµ-calculus. Finally we give a translation of symmetric λ-calculus in symmetric λµ-calculus, which doesn't make use of the rule of µ-reduction of λµ-calculus (which is precisely the rule which makes the difference between classical and intuitionistic proofs in the context of λµ-calculus). This seems to indicate that an involutive negation generates an original way of computing. Because symmetric λµ-calculus contains both ways, it should be a good framework for further investigations.