Unbounded Proof-Length Speed-up in Deduction Modulo - Archive ouverte HAL Accéder directement au contenu
Autre Publication Scientifique Année : 2007

Unbounded Proof-Length Speed-up in Deduction Modulo

Résumé

In 1973, Parikh proved a speed-up theorem conjectured by Gödel 37 years before: there exists arithmetical formulæ that are provable in first order arithmetic, but whose shorter proof in second order arithmetic is arbitrarily smaller than any proof in first order. On the other hand, resolution for higher order logic can be simulated step by step in a first order narrowing and resolution method based on deduction modulo, whose paradigm is to separate deduction and computation to make proofs clearer and shorter. We first prove that it is possible to find formulæ whose smaller proof in natural deduction modulo a very simple rewrite system is unboundedly smaller than any proof in pure natural deduction. Then, we show that i+1-th order arithmetic can be linearly simulated into i-th order arithmetic modulo some confluent and terminating rewrite system. We also prove that there exists a speed-up between i-th order arithmetic modulo this system and i-th order arithmetic without modulo. These two results allows us to prove that the speed-up conjectured by Gödel does not come from the deductive part of the proofs, but can be expressed as simple computation, therefore justifying the use of deduction modulo as an efficient first order setting simulating higher order.
Fichier principal
Vignette du fichier
speedup_LICS.pdf (295.14 Ko) Télécharger le fichier
Origine : Fichiers produits par l'(les) auteur(s)

Dates et versions

inria-00138195 , version 1 (23-03-2007)
inria-00138195 , version 2 (06-04-2007)
inria-00138195 , version 3 (03-07-2007)

Identifiants

  • HAL Id : inria-00138195 , version 2

Citer

Guillaume Burel. Unbounded Proof-Length Speed-up in Deduction Modulo. 2007. ⟨inria-00138195v2⟩
143 Consultations
165 Téléchargements

Partager

Gmail Facebook X LinkedIn More