License: Creative Commons Attribution 3.0 Unported license (CC BY 3.0)
When quoting this document, please refer to the following
DOI: 10.4230/LIPIcs.TLCA.2015.165
URN: urn:nbn:de:0030-drops-51626
URL: https://drops.dagstuhl.de/opus/volltexte/2015/5162/
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Espírito Santo, José

Curry-Howard for Sequent Calculus at Last!

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Abstract

This paper tries to remove what seems to be the remaining stumbling blocks in the way to a full understanding of the Curry-Howard isomorphism for sequent calculus, namely the questions: What do variables in proof terms stand for? What is co-control and a co-continuation? How to define the dual of Parigot's mu-operator so that it is a co-control operator? Answering these questions leads to the interpretation that sequent calculus is a formal vector notation with first-class co-control. But this is just the "internal" interpretation, which has to be developed simultaneously with, and is justified by, an equivalent, "external" interpretation, offered by natural deduction: the sequent calculus corresponds to a bi-directional, agnostic (w.r.t. the call strategy), computational lambda-calculus. Next, the formal duality between control and co-control is studied, in the context of classical logic. The duality cannot be observed in the sequent calculus, but rather in a system unifying sequent calculus and natural deduction.

BibTeX - Entry

@InProceedings{espritosanto:LIPIcs:2015:5162,
  author =	{Jos{\'e} Esp{\'i}rito Santo},
  title =	{{Curry-Howard for Sequent Calculus at Last!}},
  booktitle =	{13th International Conference on Typed Lambda Calculi and Applications (TLCA 2015)},
  pages =	{165--179},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-939897-87-3},
  ISSN =	{1868-8969},
  year =	{2015},
  volume =	{38},
  editor =	{Thorsten Altenkirch},
  publisher =	{Schloss Dagstuhl--Leibniz-Zentrum fuer Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{http://drops.dagstuhl.de/opus/volltexte/2015/5162},
  URN =		{urn:nbn:de:0030-drops-51626},
  doi =		{10.4230/LIPIcs.TLCA.2015.165},
  annote =	{Keywords: co-control, co-continuation, vector notation, let-expression, formal sub- stitution, context substitution, computational lambda-calculus, classical lo}
}

Keywords: co-control, co-continuation, vector notation, let-expression, formal sub- stitution, context substitution, computational lambda-calculus, classical lo
Collection: 13th International Conference on Typed Lambda Calculi and Applications (TLCA 2015)
Issue Date: 2015
Date of publication: 15.06.2015


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