[Bur11] Efficiently Simulating Higher-Order Arithmetic by a First-Order Theory Modulo

Revue Internationale avec comité de lecture : Journal Logical Methods in Computer Science, vol. 7(3:1), pp. 1-31, 2011, (doi:10.2168/LMCS-7(1:3)2011)

Mots clés: deduction modulo,proof length,rewriting,arithmetics

Résumé: In deduction modulo, a theory is not represented by a set of axioms but by a congruence on propositions modulo which the inference rules of standard deductive systems---such as for instance natural deduction---are applied. Therefore, the reasoning that is intrinsic of the theory does not appear in the length of proofs. In general, the congruence is defined through a rewrite system over terms and propositions. We define a rigorous framework to study proof lengths in deduction modulo, where the congruence must be computed in polynomial time. We show that even very simple rewrite systems lead to arbitrary proof-length speed-ups in deduction modulo, compared to using axioms. As higher-order logic can be encoded as a first-order theory in deduction modulo, we also study how to reinterpret, thanks to deduction modulo, the speed-ups between higher-order and first-order arithmetics that were stated by Gödel. We define a first-order rewrite system with a congruence decidable in polynomial time such that proofs of higher-order arithmetic can be linearly translated into first-order arithmetic modulo that system. We also present the whole higher-order arithmetic as a first-order system without resorting to any axiom, where proofs have the same length as in the axiomatic presentation.

Equipe: sys

BibTeX

 @article { Bur11, title = "{Efficiently Simulating Higher-Order Arithmetic by a First-Order Theory Modulo}", author = "G. Burel", journal = "Logical Methods in Computer Science", year = 2011, volume = 7, number = 3:1, pages = "1-31", doi = "10.2168/LMCS-7(1:3)2011", }