research-article

Least and Greatest Fixed Points in Linear Logic

Author:

David BaeldeAuthors Info & Claims

ACM Transactions on Computational Logic (TOCL), Volume 13, Issue 1

Article No.: 2, Pages 1 - 44

https://doi.org/10.1145/2071368.2071370

Published: 01 January 2012 Publication History

Abstract

The first-order theory of MALL (multiplicative, additive linear logic) over only equalities is a well-structured but weak logic since it cannot capture unbounded (infinite) behavior. Instead of accounting for unbounded behavior via the addition of the exponentials (! and ?), we add least and greatest fixed point operators. The resulting logic, which we call μMALL, satisfies two fundamental proof theoretic properties: we establish weak normalization for it, and we design a focused proof system that we prove complete with respect to the initial system. That second result provides a strong normal form for cut-free proof structures that can be used, for example, to help automate proof search. We show how these foundations can be applied to intuitionistic logic.

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Cited By

Dagnino FPadovani L(2024)sMALL CaPS: An Infinitary Linear Logic for a Calculus of Pure SessionsProceedings of the 26th International Symposium on Principles and Practice of Declarative Programming10.1145/3678232.3678234(1-13)Online publication date: 9-Sep-2024
https://dl.acm.org/doi/10.1145/3678232.3678234
Curzi GDas A(2023)Computational expressivity of (circular) proofs with fixed points2023 38th Annual ACM/IEEE Symposium on Logic in Computer Science (LICS)10.1109/LICS56636.2023.10175772(1-13)Online publication date: 26-Jun-2023
https://doi.org/10.1109/LICS56636.2023.10175772
Saurin A(2023)A Linear Perspective on Cut-Elimination for Non-wellfounded Sequent Calculi with Least and Greatest Fixed-PointsAutomated Reasoning with Analytic Tableaux and Related Methods10.1007/978-3-031-43513-3_12(203-222)Online publication date: 14-Sep-2023
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Least and Greatest Fixed Points in Linear Logic

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cover image ACM Transactions on Computational Logic

ACM Transactions on Computational Logic Volume 13, Issue 1

January 2012

267 pages

ISSN:1529-3785

EISSN:1557-945X

DOI:10.1145/2071368

Issue’s Table of Contents

Copyright © 2012 ACM.

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Publication History

Published: 01 January 2012

Accepted: 01 September 2010

Revised: 01 July 2010

Received: 01 October 2009

Published in TOCL Volume 13, Issue 1

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Cited By

Dagnino FPadovani L(2024)sMALL CaPS: An Infinitary Linear Logic for a Calculus of Pure SessionsProceedings of the 26th International Symposium on Principles and Practice of Declarative Programming10.1145/3678232.3678234(1-13)Online publication date: 9-Sep-2024
https://dl.acm.org/doi/10.1145/3678232.3678234
Curzi GDas A(2023)Computational expressivity of (circular) proofs with fixed points2023 38th Annual ACM/IEEE Symposium on Logic in Computer Science (LICS)10.1109/LICS56636.2023.10175772(1-13)Online publication date: 26-Jun-2023
https://doi.org/10.1109/LICS56636.2023.10175772
Saurin A(2023)A Linear Perspective on Cut-Elimination for Non-wellfounded Sequent Calculi with Least and Greatest Fixed-PointsAutomated Reasoning with Analytic Tableaux and Related Methods10.1007/978-3-031-43513-3_12(203-222)Online publication date: 14-Sep-2023
https://doi.org/10.1007/978-3-031-43513-3_12
Rocha PCaires L(2023)Safe Session-Based Concurrency with Shared Linear StateProgramming Languages and Systems10.1007/978-3-031-30044-8_16(421-450)Online publication date: 22-Apr-2023
https://dl.acm.org/doi/10.1007/978-3-031-30044-8_16
Baelde DDoumane AKuperberg DSaurin A(2022)Bouncing Threads for Circular and Non-Wellfounded ProofsProceedings of the 37th Annual ACM/IEEE Symposium on Logic in Computer Science10.1145/3531130.3533375(1-13)Online publication date: 2-Aug-2022
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Ciccone LPadovani L(2022)Fair termination of binary sessionsProceedings of the ACM on Programming Languages10.1145/34986666:POPL(1-30)Online publication date: 12-Jan-2022
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Miller DViel A(2022)The undecidability of proof search when equality is a logical connectiveAnnals of Mathematics and Artificial Intelligence10.1007/s10472-021-09764-090:5(523-535)Online publication date: 1-May-2022
https://dl.acm.org/doi/10.1007/s10472-021-09764-0
Qian ZKavvos GBirkedal L(2021)Client-server sessions in linear logicProceedings of the ACM on Programming Languages10.1145/34735675:ICFP(1-31)Online publication date: 19-Aug-2021
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Ehrhard TJafarrahmani FGorla D(2021)Categorical models of linear logic with fixed points of formulasProceedings of the 36th Annual ACM/IEEE Symposium on Logic in Computer Science10.1109/LICS52264.2021.9470664(1-13)Online publication date: 29-Jun-2021
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Kuznetsov S(2021)Complexity of a Fragment of Infinitary Action Logic with Exponential via Non-well-founded ProofsAutomated Reasoning with Analytic Tableaux and Related Methods10.1007/978-3-030-86059-2_19(317-334)Online publication date: 30-Aug-2021
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