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DOI: 10.4230/LIPIcs.CSL.2017.9

Removing Cycles from Proofs

Authors: Andrea Aler Tubella, Alessio Guglielmi, and Benjamin Ralph

Published in: LIPIcs, Volume 82, 26th EACSL Annual Conference on Computer Science Logic (CSL 2017)

Abstract

If we track atom occurrences in classical propositional proofs in deep inference, we see that they can form cyclic structures between cuts and identity steps. These cycles are an obstacle to a very natural form of normalisation, that simply unfolds all the contractions in a proof. This mechanism, which we call ‘decomposition’, has many points of contact with explicit substitutions in lambda calculi. In the presence of cycles, decomposition does not terminate, and this is an obvious drawback if we want to interpret proofs computationally. One way of eliminating cycles is eliminating cuts. However, we could ask ourselves whether it is possible to eliminate cycles independently of (general) cut elimination. This paper shows an efficient way to do so, therefore establishing the independence of decomposition from cut elimination. In other words, cut elimination in propositional logic can be separated into three separate procedures: 1) cycle elimination, 2) unfolding of contractions, 3) elimination of cuts in the linear fragment.

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Andrea Aler Tubella, Alessio Guglielmi, and Benjamin Ralph. Removing Cycles from Proofs. In 26th EACSL Annual Conference on Computer Science Logic (CSL 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 82, pp. 9:1-9:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2017)

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@InProceedings{alertubella_et_al:LIPIcs.CSL.2017.9,
  author =	{Aler Tubella, Andrea and Guglielmi, Alessio and Ralph, Benjamin},
  title =	{{Removing Cycles from Proofs}},
  booktitle =	{26th EACSL Annual Conference on Computer Science Logic (CSL 2017)},
  pages =	{9:1--9:17},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-045-3},
  ISSN =	{1868-8969},
  year =	{2017},
  volume =	{82},
  editor =	{Goranko, Valentin and Dam, Mads},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CSL.2017.9},
  URN =		{urn:nbn:de:0030-drops-77008},
  doi =		{10.4230/LIPIcs.CSL.2017.9},
  annote =	{Keywords: proof theory, deep inference, proof complexity}
}

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Removing Cycles from Proofs

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