Unbounded Proof-Length Speed-up in Deduction Modulo

Guillaume Burel (1)

1 PROTHEO - Constraints, automatic deduction and software properties proofs

In 1973, Parikh proved a theorem conjectured by Gödel 37 years before, which says that it is possible to find 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 is unboundedly smaller than any proof in pure natural deduction. Then, we show that a proof in the i+1-th order arithmetic can be transformed into a proof of linear length in the i-th order arithmetic modulo some finite terminating and confluent rewrite system. This allows us to prove that the speed-up conjectured by Gödel does not come from the deduction part of the proofs, but can be expressed as computation, therefore justifying the use of deduction modulo as an efficient first order setting simulating higher order.

Mots clés: arithmetic; higher order logic; natural deduction vs. Hilbert-type systems; proof theory; rewriting

Domaines: Logique en informatique [cs.LO]

Guillaume Burel. Unbounded Proof-Length Speed-up in Deduction Modulo. 2007. ⟨inria-00138195v1⟩