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Comparing Infinitary Systems for Linear Logic with Fixed Points

Authors Anupam Das, Abhishek De, Alexis Saurin

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Subject Classification

ACM Subject Classification
  • Theory of computation → Linear logic
  • Theory of computation → Proof theory
Keywords
  • linear logic
  • fixed points
  • non-wellfounded proofs
  • omega-branching proofs
  • analytical hierarchy

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Abstract

Extensions of Girard’s linear logic by least and greatest fixed point operators (μMALL) have been an active field of research for almost two decades. Various proof systems are known viz. finitary and non-wellfounded, based on explicit and implicit (co)induction respectively. In this paper, we compare the relative expressivity, at the level of provability, of two complementary infinitary proof systems: finitely branching non-wellfounded proofs (μMALL^∞) vs. infinitely branching well-founded proofs (μMALL_{ω,∞}). Our main result is that μMALL^∞ is strictly contained in μMALL_{ω,∞}. 
For inclusion, we devise a novel technique involving infinitary rewriting of non-wellfounded proofs that yields a wellfounded proof in the limit. For strictness of the inclusion, we improve previously known lower bounds on μMALL^∞ provability from Π⁰₁-hard to Σ¹₁-hard, by encoding a sort of Büchi condition for Minsky machines.

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Anupam Das, Abhishek De, and Alexis Saurin. Comparing Infinitary Systems for Linear Logic with Fixed Points. In 43rd IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 284, pp. 40:1-40:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023) https://doi.org/10.4230/LIPIcs.FSTTCS.2023.40

Author Details

Anupam Das
  • School of Computer Science, University of Birmingham, UK
Abhishek De
  • School of Computer Science, University of Birmingham, UK
Alexis Saurin
  • IRIF, CNRS, Université Paris Cité, France
  • INRIA $π^3$, Paris, France

Funding

The first and second authors are supported by a UKRI Future Leaders Fellowship, Structure vs. Invariants in Proofs, project reference MR/S035540/1. The third author is partially funded by the ANR project RECIPROG, project reference ANR-21-CE48-019-01.

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