On Go¨del's theorems on lengths of proofs I: number of lines and speedup for arithmetics
Journal of Symbolic Logic
Term rewriting and all that
Journal of Automated Reasoning
HOL-λσ: an intentional first-order expression of higher-order logic
Mathematical Structures in Computer Science
Abstract canonical presentations
Theoretical Computer Science - Clifford lectures and the mathematical foundations of programming semantics
ACM Transactions on Computational Logic (TOCL)
LICS '07 Proceedings of the 22nd Annual IEEE Symposium on Logic in Computer Science
Embedding pure type systems in the lambda-pi-calculus modulo
TLCA'07 Proceedings of the 8th international conference on Typed lambda calculi and applications
Isabelle/HOL: a proof assistant for higher-order logic
Isabelle/HOL: a proof assistant for higher-order logic
RTA'05 Proceedings of the 16th international conference on Term Rewriting and Applications
Types for Proofs and Programs
Combining Deduction Modulo and Logics of Fixed-Point Definitions
LICS '12 Proceedings of the 2012 27th Annual IEEE/ACM Symposium on Logic in Computer Science
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In 1973, Parikh proved a speed-up theorem conjectured by Gödel 37 years before: there exist arithmetical formulæ that are provable in first order arithmetic, but whose shorter proof in second order arithmetic is arbitrarily smaller than any proof in first order. On the other hand, resolution for higher order logic can be simulated step by step in a first order narrowing and resolution method based on deduction modulo, whose paradigm is to separate deduction and computation to make proofs clearer and shorter. We prove that i+1-th order arithmetic can be linearly simulated into i-th order arithmetic modulo some confluent and terminating rewrite system. We also show that there exists a speed-up between i-th order arithmetic modulo this system and i-th order arithmetic without modulo. All this allows us to prove that the speed-up conjectured by Gödel does not come from the deductive part of the proofs, but can be expressed as simple computation, therefore justifying the use of deduction modulo as an efficient first order setting simulating higher order.