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


Burel,  Guillaume
Automation of Logic, MPI for Informatics, Max Planck Society;

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Burel, G. (2011). Efficiently Simulating Higher-Order Arithmetic by a First-Order Theory Modulo. Logical Methods in Computer Science, 7(1): 3, pp. 3:1-3:31. doi:10.2168/LMCS-7 (1:3) 2011.

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.