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Efficiently Simulating HigherOrder Arithmetic by a FirstOrder Theory Modulo
"... Deduction modulo is a paradigm which consists in applying the inference rules of a deductive system—such as for instance natural deduction—modulo a rewrite system over terms and propositions. It has been shown that higherorder logic can be simulated into the firstorder natural deduction modulo. Ho ..."
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Deduction modulo is a paradigm which consists in applying the inference rules of a deductive system—such as for instance natural deduction—modulo a rewrite system over terms and propositions. It has been shown that higherorder logic can be simulated into the firstorder natural deduction modulo. However, a theorem stated by Gödel and proved by Parikh expresses that proofs in secondorder arithmetic may be unboundedly shorter than proofs in firstorder arithmetic, even when considering only formulæ provable in firstorder arithmetic. We investigate how deduction modulo can be used to translate proofs of higherorder arithmetic into firstorder proofs without inflating their length. First we show how higher orders can be encoded through a quite simple (finite, terminating, confluent, leftlinear) rewrite system. Then, a proof in higherorder arithmetic can be linearly translated into a proof in firstorder arithmetic modulo this system. Second, in the continuation of a work of Dowek and Werner, we show how to express the whole higherorder arithmetic as a rewrite system. Then, proofs of higherorder arithmetic can be linearly translated into proofs in the empty theory modulo this rewrite system. These results show that the speedup between firstand secondorder arithmetic, and more generally between i th and i +1 storder arithmetic, can in fact be expressed as computation, and does not lie in the really deductive part of the proofs.
A Semantic Normalization Proof for System T
"... Abstract. Semantics methods have been used to prove cut elimination theorems for a long time. It is only recently that they have been extended to prove strong normalization results. For instance using the notion of superconsistency that is a semantic criterion for theories expressed in deduction mo ..."
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Abstract. Semantics methods have been used to prove cut elimination theorems for a long time. It is only recently that they have been extended to prove strong normalization results. For instance using the notion of superconsistency that is a semantic criterion for theories expressed in deduction modulo implying strong normalization. However, the strong normalization of System T has always been reluctant to such semantic methods. In this paper we give a semantic normalization proof of system T using the super consistency of some theory. 1
Consistency Implies Cut Admissibility
 PSATTT'11: INTERNATIONAL WORKSHOP ON PROOFSEARCH IN AXIOMATIC THEORIES AND TYPE THEORIES (2011)
, 2011
"... For any finite and consistent firstorder theory, we can find a presentation as a rewriting system that enjoys cut admissibility. ..."
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For any finite and consistent firstorder theory, we can find a presentation as a rewriting system that enjoys cut admissibility.
A Semantic Normalization Proof for Inductive Types
, 2008
"... Abstract. Semantics methods have been used to prove cut elimination theorems for a long time. It is only recently that they have been extended to prove strong normalization results. For instance using the notion of superconsistency that is a semantic criterion for theories expressed in deduction mo ..."
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Abstract. Semantics methods have been used to prove cut elimination theorems for a long time. It is only recently that they have been extended to prove strong normalization results. For instance using the notion of superconsistency that is a semantic criterion for theories expressed in deduction modulo implying strong normalization. However, the strong normalization of System T has always been reluctant to such semantic methods. In this paper we give a semantic normalization proof of system T using the super consistency of some theory. We then extend the result to every strictly positive inductive type and discuss the extension to predicate logic. 1