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Infinitary Cut-Elimination via Finite Approximations

Authors Matteo Acclavio , Gianluca Curzi , Giulio Guerrieri



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Author Details

Matteo Acclavio
  • University of Southern Denmark, Odense, Denmark
  • University of Sussex, Department of Informatics, Brighton, UK
Gianluca Curzi
  • University of Birmingham, UK
  • University of Gothenburg, Sweden
Giulio Guerrieri
  • University of Sussex, Department of Informatics, Brighton, UK

Acknowledgements

We would like to thank Anupam Das, Abhishek De, Farzad Jafar-Rahmani, Alexis Saurin, Tito (Lê Thành Dung Nguyên), Damiano Mazza and the anonymous reviewers for their useful comments and suggestions.

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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)
https://doi.org/10.4230/LIPIcs.CSL.2024.8

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.

Subject Classification

ACM Subject Classification
  • Theory of computation → Linear logic
  • Theory of computation → Proof theory
Keywords
  • cut-elimination
  • non-wellfounded proofs
  • parsimonious logic
  • linear logic
  • proof theory
  • approximation
  • sequent calculus
  • non-uniform proofs

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