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- W4230791186 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 Godel. 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." @default.
- W4230791186 created "2022-05-11" @default.
- W4230791186 creator A5019658459 @default.
- W4230791186 date "2011-03-17" @default.
- W4230791186 modified "2023-09-26" @default.
- W4230791186 title "Efficiently Simulating Higher-Order Arithmetic by a First-Order Theory Modulo" @default.
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- W4230791186 doi "https://doi.org/10.2168/lmcs-7(1:3)2011" @default.
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