Efficiently Simulating Higher-Order Arithmetic by a First-Order Theory Modulo

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 higherorder and first-order arithmetics that were stated by G

• Type: Journal
• Year: 2008
• Where: CORR
• Authors: Guillaume Burel
• Keywords: CORR 2008, Deduction Modulo, Education, Instance Natural Deduction, Rewrite System