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A Semantic Proof that Reducibility Candidates entail Cut Elimination

Authors: Denis Cousineau and Olivier Hermant

Published in: LIPIcs, Volume 15, 23rd International Conference on Rewriting Techniques and Applications (RTA'12) (2012)


Abstract
Two main lines have been adopted to prove the cut elimination theorem: the syntactic one, that studies the process of reducing cuts, and the semantic one, that consists in interpreting a sequent in some algebra and extracting from this interpretation a cut-free proof of this very sequent. A link between those two methods was exhibited by studying in a semantic way, syntactical tools that allow to prove (strong) normalization of proof-terms, namely reducibility candidates. In the case of deduction modulo, a framework combining deduction and rewriting rules in which theories like Zermelo set theory and higher order logic can be expressed, this is obtained by constructing a reducibility candidates valued model. The existence of such a pre-model for a theory entails strong normalization of its proof-terms and, by the usual syntactic argument, the cut elimination property. In this paper, we strengthen this gate between syntactic and semantic methods, by providing a full semantic proof that the existence of a pre-model entails the cut elimination property for the considered theory in deduction modulo. We first define a new simplified variant of reducibility candidates à la Girard, that is sufficient to prove weak normalization of proof-terms (and therefore the cut elimination property). Then we build, from some model valued on the pre-Heyting algebra of those WN reducibility candidates, a regular model valued on a Heyting algebra on which we apply the usual soundness/strong completeness argument. Finally, we discuss further extensions of this new method towards normalization by evaluation techniques that commonly use Kripke semantics.

Cite as

Denis Cousineau and Olivier Hermant. A Semantic Proof that Reducibility Candidates entail Cut Elimination. In 23rd International Conference on Rewriting Techniques and Applications (RTA'12). Leibniz International Proceedings in Informatics (LIPIcs), Volume 15, pp. 133-148, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2012)


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@InProceedings{cousineau_et_al:LIPIcs.RTA.2012.133,
  author =	{Cousineau, Denis and Hermant, Olivier},
  title =	{{A Semantic Proof that Reducibility Candidates entail Cut Elimination}},
  booktitle =	{23rd International Conference on Rewriting Techniques and Applications (RTA'12)},
  pages =	{133--148},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-939897-38-5},
  ISSN =	{1868-8969},
  year =	{2012},
  volume =	{15},
  editor =	{Tiwari, Ashish},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.RTA.2012.133},
  URN =		{urn:nbn:de:0030-drops-34899},
  doi =		{10.4230/LIPIcs.RTA.2012.133},
  annote =	{Keywords: cut elimination, reducibility candidates, (pre-)Heyting algebras}
}
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