License: Creative Commons Attribution 4.0 International license (CC BY 4.0)
When quoting this document, please refer to the following
DOI: 10.4230/LIPIcs.FSCD.2022.19
URN: urn:nbn:de:0030-drops-163004
URL: https://drops.dagstuhl.de/opus/volltexte/2022/16300/
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Heijltjes, Willem B. ; Hughes, Dominic J. D. ; Straßburger, Lutz

Normalization Without Syntax

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LIPIcs-FSCD-2022-19.pdf (0.8 MB)

Abstract

We present normalization for intuitionistic combinatorial proofs (ICPs) and relate it to the simply-typed lambda-calculus. We prove confluence and strong normalization. Combinatorial proofs, or "proofs without syntax", form a graphical semantics of proof in various logics that is canonical yet complexity-aware: they are a polynomial-sized representation of sequent proofs that factors out exactly the non-duplicating permutations. Our approach to normalization aligns with these characteristics: it is canonical (free of permutations) and generic (readily applied to other logics). Our reduction mechanism is a canonical representation of reduction in sequent calculus with closed cuts (no abstraction is allowed below a cut), and relates to closed reduction in lambda-calculus and supercombinators. While we will use ICPs concretely, the notion of reduction is completely abstract, and can be specialized to give a reduction mechanism for any representation of typed normal forms.

BibTeX - Entry

@InProceedings{heijltjes_et_al:LIPIcs.FSCD.2022.19,
  author =	{Heijltjes, Willem B. and Hughes, Dominic J. D. and Stra{\ss}burger, Lutz},
  title =	{{Normalization Without Syntax}},
  booktitle =	{7th International Conference on Formal Structures for Computation and Deduction (FSCD 2022)},
  pages =	{19:1--19:19},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-233-4},
  ISSN =	{1868-8969},
  year =	{2022},
  volume =	{228},
  editor =	{Felty, Amy P.},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/opus/volltexte/2022/16300},
  URN =		{urn:nbn:de:0030-drops-163004},
  doi =		{10.4230/LIPIcs.FSCD.2022.19},
  annote =	{Keywords: combinatorial proofs, intuitionistic logic, lambda-calculus, Curry-Howard, proof nets}
}

Keywords: combinatorial proofs, intuitionistic logic, lambda-calculus, Curry-Howard, proof nets
Collection: 7th International Conference on Formal Structures for Computation and Deduction (FSCD 2022)
Issue Date: 2022
Date of publication: 28.06.2022

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