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.CSL.2013.248
URN: urn:nbn:de:0030-drops-42019
URL: http://dagstuhl.sunsite.rwth-aachen.de/volltexte/2013/4201/
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Fortier, Jérôme ; Santocanale, Luigi

Cuts for circular proofs: semantics and cut-elimination

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Abstract

One of the authors introduced in [Santocanale, FoSSaCS, 2002] a calculus of circular proofs for studying the computability arising from the following categorical operations: finite products, finite coproducts, initial algebras, final coalgebras. The calculus presented [Santocanale, FoSSaCS, 2002] is cut-free; even if sound and complete for provability, it lacked an important property for the semantics of proofs, namely fullness w.r.t. the class of intended categorical models (called mu-bicomplete categories in [Santocanale, ITA, 2002]).
In this paper we fix this problem by adding the cut rule to the calculus and by modifying accordingly the syntactical constraint ensuring soundness of proofs. The enhanced proof system fully represents arrows of the canonical model (a free mu-bicomplete category). We also describe a cut-elimination procedure as a a model of computation arising from the above mentioned categorical operations. The procedure constructs a cut-free proof-tree with possibly infinite branches out of a finite circular proof with cuts.

BibTeX - Entry

@InProceedings{fortier_et_al:LIPIcs:2013:4201,
  author =	{J{\'e}r{\^o}me Fortier and Luigi Santocanale},
  title =	{{Cuts for circular proofs: semantics and cut-elimination}},
  booktitle =	{Computer Science Logic 2013 (CSL 2013)},
  pages =	{248--262},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-939897-60-6},
  ISSN =	{1868-8969},
  year =	{2013},
  volume =	{23},
  editor =	{Simona Ronchi Della Rocca},
  publisher =	{Schloss Dagstuhl--Leibniz-Zentrum fuer Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{http://drops.dagstuhl.de/opus/volltexte/2013/4201},
  URN =		{urn:nbn:de:0030-drops-42019},
  doi =		{10.4230/LIPIcs.CSL.2013.248},
  annote =	{Keywords: categorical proof-theory, fixpoints, initial and final (co)algebras, inductive and coinductive types}
}

Keywords: categorical proof-theory, fixpoints, initial and final (co)algebras, inductive and coinductive types
Collection: Computer Science Logic 2013 (CSL 2013)
Issue Date: 2013
Date of publication: 02.09.2013

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