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Infinets: The Parallel Syntax for Non-wellfounded Proof-Theory

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Automated Reasoning with Analytic Tableaux and Related Methods (TABLEAUX 2019)

Part of the book series: Lecture Notes in Computer Science ((LNAI,volume 11714))

Abstract

Logics based on the \(\mu \)-calculus are used to model inductive and coinductive reasoning and to verify reactive systems. A well-structured proof-theory is needed in order to apply such logics to the study of programming languages with (co)inductive data types and automated (co)inductive theorem proving. The traditional proof system suffers some defects, non-wellfounded (or infinitary) and circular proofs have been recognized as a valuable alternative, and significant progress have been made in this direction in recent years. Such proofs are non-wellfounded sequent derivations together with a global validity condition expressed in terms of progressing threads.

The present paper investigates a discrepancy found in such proof systems, between the sequential nature of sequent proofs and the parallel structure of threads: various proof attempts may have the exact threading structure while differing in the order of inference rules applications. The paper introduces infinets, that are proof-nets for non-wellfounded proofs in the setting of multiplicative linear logic with least and greatest fixed-points (\(\mu \mathsf {MLL}^{\infty }\)) and study their correctness and sequentialization.

This project has received funding from the European Union’s Horizon 2020 research and innovation programme under the Marie Skłodowska-Curie grant agreement No 754362. Partially funded by ANR Project RAPIDO, ANR-14-CE25-0007.

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Acknowledgement

We are indebted to anonymous reviewers for providing insightful comments which has immensely enhanced the presentation of the paper.

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Authors and Affiliations

  1. IRIF, Université de Paris, Paris, France

    Abhishek De

  2. IRIF, CNRS, Université de Paris, Paris, France

    Alexis Saurin

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  1. Abhishek De
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  2. Alexis Saurin
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Correspondence to Alexis Saurin .

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Editors and Affiliations

  1. IBISC, Univ. Evry, Université Paris-Saclay, Evry, France

    Serenella Cerrito

  2. Middlesex University London, London, UK

    Andrei Popescu

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De, A., Saurin, A. (2019). Infinets: The Parallel Syntax for Non-wellfounded Proof-Theory. In: Cerrito, S., Popescu, A. (eds) Automated Reasoning with Analytic Tableaux and Related Methods. TABLEAUX 2019. Lecture Notes in Computer Science(), vol 11714. Springer, Cham. https://doi.org/10.1007/978-3-030-29026-9_17

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  • DOI: https://doi.org/10.1007/978-3-030-29026-9_17

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  • Online ISBN: 978-3-030-29026-9

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