Contraction-free proofs and finitary games for Linear Logic

André Hirschowitz (1) , Michel Hirschowitz (2, 3) , Tom Hirschowitz (4)

1 JAD - Laboratoire Jean Alexandre Dieudonné
2 LIX - Laboratoire d'informatique de l'École polytechnique [Palaiseau]
3 LIST (CEA) - Laboratoire d'Intégration des Systèmes et des Technologies
4 LAMA - Laboratoire de Mathématiques

In the standard sequent presentations of Girard's Linear Logic (LL), there are two "non-decreasing" rules, where the premises are not smaller than the conclusion, namely the cut and the contraction rules. It is a universal concern to eliminate the cut rule. We show that, using an admissible modification of the tensor rule, contractions can be eliminated, and that cuts can be simultaneously limited to a single initial occurrence. This view leads to a consistent, but incomplete game model for LL with exponentials, which is finitary, in the sense that each play is finite. The game is based on a set of inference rules which does not enjoy cut elimination. Nevertheless, the cut rule is valid in the model.

Mots clés: Linear logic; Game semantics; Contraction elimination

Domaines: Logique en informatique [cs.LO]; Logique [math.LO]

André Hirschowitz, Michel Hirschowitz, Tom Hirschowitz. Contraction-free proofs and finitary games for Linear Logic. MFPS 2009, 2009, Oxford, United Kingdom. pp.287-305, ⟨10.1016/j.entcs.2009.07.095⟩. ⟨hal-00387452⟩