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DOI: 10.4230/LIPIcs.CSL.2024.8

Infinitary Cut-Elimination via Finite Approximations

Authors: Matteo Acclavio, Gianluca Curzi, and Giulio Guerrieri

Published in: LIPIcs, Volume 288, 32nd EACSL Annual Conference on Computer Science Logic (CSL 2024)

Abstract

We investigate non-wellfounded proof systems based on parsimonious logic, a weaker variant of linear logic where the exponential modality ! is interpreted as a constructor for streams over finite data. Logical consistency is maintained at a global level by adapting a standard progressing criterion. We present an infinitary version of cut-elimination based on finite approximations, and we prove that, in presence of the progressing criterion, it returns well-defined non-wellfounded proofs at its limit. Furthermore, we show that cut-elimination preserves the progressing criterion and various regularity conditions internalizing degrees of proof-theoretical uniformity. Finally, we provide a denotational semantics for our systems based on the relational model.

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Matteo Acclavio, Gianluca Curzi, and Giulio Guerrieri. Infinitary Cut-Elimination via Finite Approximations. In 32nd EACSL Annual Conference on Computer Science Logic (CSL 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 288, pp. 8:1-8:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{acclavio_et_al:LIPIcs.CSL.2024.8,
  author =	{Acclavio, Matteo and Curzi, Gianluca and Guerrieri, Giulio},
  title =	{{Infinitary Cut-Elimination via Finite Approximations}},
  booktitle =	{32nd EACSL Annual Conference on Computer Science Logic (CSL 2024)},
  pages =	{8:1--8:19},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-310-2},
  ISSN =	{1868-8969},
  year =	{2024},
  volume =	{288},
  editor =	{Murano, Aniello and Silva, Alexandra},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CSL.2024.8},
  URN =		{urn:nbn:de:0030-drops-196510},
  doi =		{10.4230/LIPIcs.CSL.2024.8},
  annote =	{Keywords: cut-elimination, non-wellfounded proofs, parsimonious logic, linear logic, proof theory, approximation, sequent calculus, non-uniform proofs}
}

Document

DOI: 10.4230/LIPIcs.CSL.2023.16

Non-Uniform Complexity via Non-Wellfounded Proofs

Authors: Gianluca Curzi and Anupam Das

Published in: LIPIcs, Volume 252, 31st EACSL Annual Conference on Computer Science Logic (CSL 2023)

Abstract

Cyclic and non-wellfounded proofs are now increasingly employed to establish metalogical results in a variety of settings, in particular for type systems with forms of (co)induction. Under the Curry-Howard correspondence, a cyclic proof can be seen as a typing derivation "with loops", closer to low-level machine models, and so comprise a highly expressive computational model that nonetheless enjoys excellent metalogical properties. In recent work, we showed how the cyclic proof setting can be further employed to model computational complexity, yielding characterisations of the polynomial time and elementary computable functions. These characterisations are "implicit", inspired by Bellantoni and Cook’s famous algebra of safe recursion, but exhibit greater expressivity thanks to the looping capacity of cyclic proofs. In this work we investigate the capacity for non-wellfounded proofs, where finite presentability is relaxed, to model non-uniformity in complexity theory. In particular, we present a characterisation of the class FP/poly of functions computed by polynomial-size circuits. While relating non-wellfoundedness to non-uniformity is a natural idea, the precise amount of irregularity, informally speaking, required to capture FP/poly is given by proof-level conditions novel to cyclic proof theory. Along the way, we formalise some (presumably) folklore techniques for characterising non-uniform classes in relativised function algebras with appropriate oracles.

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Gianluca Curzi and Anupam Das. Non-Uniform Complexity via Non-Wellfounded Proofs. In 31st EACSL Annual Conference on Computer Science Logic (CSL 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 252, pp. 16:1-16:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)

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@InProceedings{curzi_et_al:LIPIcs.CSL.2023.16,
  author =	{Curzi, Gianluca and Das, Anupam},
  title =	{{Non-Uniform Complexity via Non-Wellfounded Proofs}},
  booktitle =	{31st EACSL Annual Conference on Computer Science Logic (CSL 2023)},
  pages =	{16:1--16:18},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-264-8},
  ISSN =	{1868-8969},
  year =	{2023},
  volume =	{252},
  editor =	{Klin, Bartek and Pimentel, Elaine},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CSL.2023.16},
  URN =		{urn:nbn:de:0030-drops-174770},
  doi =		{10.4230/LIPIcs.CSL.2023.16},
  annote =	{Keywords: Cyclic proofs, non-wellfounded proof-theory, non-uniform complexity, polynomial time, safe recursion, implicit complexity}
}

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Infinitary Cut-Elimination via Finite Approximations

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Non-Uniform Complexity via Non-Wellfounded Proofs

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