@article{Burel2010a,
TITLE = {Efficiently Simulating Higher-Order Arithmetic by a First-Order Theory Modulo},
AUTHOR = {Burel, Guillaume},
LANGUAGE = {eng},
ISSN = {1860-5974},
URL = {http://arxiv.org/pdf/0805.1464v4},
DOI = {10.2168/LMCS-7 (1:3) 2011},
LOCALID = {Local-ID: C125716C0050FB51-3D13ACDE62D02282C125783F0031B40F-Burel2010a},
PUBLISHER = {Department of Theoretical Computer Science, Technical University of Braunschweig},
ADDRESS = {Braunschweig},
YEAR = {2011},
DATE = {2011},
ABSTRACT = {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 G\"odel. 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.},
JOURNAL = {Logical Methods in Computer Science},
VOLUME = {7},
NUMBER = {1},
PAGES = {3:1--3:31},
EID = {3},
}