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DOI: 10.4230/LIPIcs.TYPES.2016.10
URN: urn:nbn:de:0030-drops-98523
URL: http://drops.dagstuhl.de/opus/volltexte/2018/9852/
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Espírito Santo, José ; Frade, Maria João ; Pinto, Luís

Permutability in Proof Terms for Intuitionistic Sequent Calculus with Cuts

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LIPIcs-TYPES-2016-10.pdf (0.6 MB)


Abstract

This paper gives a comprehensive and coherent view on permutability in the intuitionistic sequent calculus with cuts. Specifically we show that, once permutability is packaged into appropriate global reduction procedures, it organizes the internal structure of the system and determines fragments with computational interest, both for the computation-as-proof-normalization and the computation-as-proof-search paradigms. The vehicle of the study is a lambda-calculus of multiary proof terms with generalized application, previously developed by the authors (the paper argues this system represents the simplest fragment of ordinary sequent calculus that does not fall into mere natural deduction). We start by adapting to our setting the concept of normal proof, developed by Mints, Dyckhoff, and Pinto, and by defining natural proofs, so that a proof is normal iff it is natural and cut-free. Natural proofs form a subsystem with a transparent Curry-Howard interpretation (a kind of formal vector notation for lambda-terms with vectors consisting of lists of lists of arguments), while searching for normal proofs corresponds to a slight relaxation of focusing (in the sense of LJT). Next, we define a process of permutative conversion to natural form, and show that its combination with cut elimination gives a concept of normalization for the sequent calculus. We derive a systematic picture of the full system comprehending a rich set of reduction procedures (cut elimination, flattening, permutative conversion, normalization, focalization), organizing the relevant subsystems and the important subclasses of cut-free, normal, and focused proofs.

BibTeX - Entry

@InProceedings{espritosanto_et_al:LIPIcs:2018:9852,
  author =	{Jos{\'e} Esp{\'i}rito Santo and Maria Jo{\~a}o Frade and Lu{\'i}s Pinto},
  title =	{{Permutability in Proof Terms for Intuitionistic Sequent Calculus with Cuts}},
  booktitle =	{22nd International Conference on Types for Proofs and  Programs (TYPES 2016)},
  pages =	{10:1--10:27},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-065-1},
  ISSN =	{1868-8969},
  year =	{2018},
  volume =	{97},
  editor =	{Silvia Ghilezan and Herman Geuvers and Jelena Ivetić},
  publisher =	{Schloss Dagstuhl--Leibniz-Zentrum fuer Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{http://drops.dagstuhl.de/opus/volltexte/2018/9852},
  URN =		{urn:nbn:de:0030-drops-98523},
  doi =		{10.4230/LIPIcs.TYPES.2016.10},
  annote =	{Keywords: sequent calculus, permutative conversion, Curry-Howard isomorphism, vector of arguments, generalized application, normal proof, natural proof, cut eli}
}

Keywords: sequent calculus, permutative conversion, Curry-Howard isomorphism, vector of arguments, generalized application, normal proof, natural proof, cut eli
Seminar: 22nd International Conference on Types for Proofs and Programs (TYPES 2016)
Issue Date: 2018
Date of publication: 24.10.2018


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