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DOI: 10.4230/LIPIcs.CSL.2013.45

Realizability and Strong Normalization for a Curry-Howard Interpretation of HA + EM1

Authors: Federico Aschieri, Stefano Berardi, and Giovanni Birolo

Published in: LIPIcs, Volume 23, Computer Science Logic 2013 (CSL 2013)

Abstract

We present a new Curry-Howard correspondence for HA + EM_1, constructive Heyting Arithmetic with the excluded middle on \Sigma^0_1-formulas. We add to the lambda calculus an operator ||_a which represents, from the viewpoint of programming, an exception operator with a delimited scope, and from the viewpoint of logic, a restricted version of the excluded middle. We motivate the restriction of the excluded middle by its use in proof mining; we introduce new techniques to prove strong normalization for HA + EM_1 and the witness property for simply existential statements. One may consider our results as an application of the ideas of Interactive realizability, which we have adapted to the new setting and used to prove our main theorems.

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Federico Aschieri, Stefano Berardi, and Giovanni Birolo. Realizability and Strong Normalization for a Curry-Howard Interpretation of HA + EM1. In Computer Science Logic 2013 (CSL 2013). Leibniz International Proceedings in Informatics (LIPIcs), Volume 23, pp. 45-60, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2013)

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@InProceedings{aschieri_et_al:LIPIcs.CSL.2013.45,
  author =	{Aschieri, Federico and Berardi, Stefano and Birolo, Giovanni},
  title =	{{Realizability and Strong Normalization for a Curry-Howard Interpretation of HA + EM1}},
  booktitle =	{Computer Science Logic 2013 (CSL 2013)},
  pages =	{45--60},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-939897-60-6},
  ISSN =	{1868-8969},
  year =	{2013},
  volume =	{23},
  editor =	{Ronchi Della Rocca, Simona},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CSL.2013.45},
  URN =		{urn:nbn:de:0030-drops-41894},
  doi =		{10.4230/LIPIcs.CSL.2013.45},
  annote =	{Keywords: Interactive realizability, classical Arithmetic, witness extraction, delimited exceptions}
}

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Realizability and Strong Normalization for a Curry-Howard Interpretation of HA + EM1

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