Skip to main content Accessibility help
×
Home
Hostname: page-component-5c569c448b-hlvcg Total loading time: 0.221 Render date: 2022-07-04T00:25:07.553Z Has data issue: true Feature Flags: { "shouldUseShareProductTool": true, "shouldUseHypothesis": true, "isUnsiloEnabled": true, "useRatesEcommerce": false, "useNewApi": true } hasContentIssue true

A MALL geometry of interaction based on indexed linear logic

Published online by Cambridge University Press:  14 June 2021

Masahiro Hamano*
Affiliation:
Miin Wu School of Computing, National Cheng Kung University, No1, University Road, Tainan City 70101, Taiwan
Rights & Permissions[Opens in a new window]

Abstract

HTML view is not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

We construct a geometry of interaction (GoI: dynamic modelling of Gentzen-style cut elimination) for multiplicative-additive linear logic (MALL) by employing Bucciarelli–Ehrhard indexed linear logic MALL(I) to handle the additives. Our construction is an extension to the additives of the Haghverdi–Scott categorical formulation (a multiplicative GoI situation in a traced monoidal category) for Girard’s original GoI 1. The indices are shown to serve not only in their original denotational level, but also at a finer grained dynamic level so that the peculiarities of additive cut elimination such as superposition, erasure of subproofs, and additive (co-) contraction can be handled with the explicit use of indices. Proofs are interpreted as indexed subsets in the category Rel, but without the explicit relational composition; instead, execution formulas are run pointwise on the interpretation at each index, with respect to symmetries of cuts, in a traced monoidal category with a reflexive object and a zero morphism. The sets of indices diminish overall when an execution formula is run, corresponding to the additive cut-elimination procedure (erasure), and allowing recovery of the relational composition. The main theorem is the invariance of the execution formulas along cut elimination so that the formulas converge to the denotations of (cut-free) proofs.

Type
Paper
Creative Commons
Creative Common License - CCCreative Common License - BY
This is an Open Access article, distributed under the terms of the Creative Commons Attribution licence (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted re-use, distribution, and reproduction in any medium, provided the original work is properly cited.
Copyright
© The Author(s), 2021. Published by Cambridge University Press

References

Bucciarelli, A. and Ehrhard, T. (2000). On phase semantics and denotational semantics in multiplicative-additive linear logic. Annals of Pure and Applied Logic 102 (3) 247282.CrossRefGoogle Scholar
Bucciarelli, A. and Ehrhard, T. (2001). On phase semantics and denotational semantics: the exponentials. Annals of Pure and Applied Logic 109 (3) 205241.CrossRefGoogle Scholar
Danos, V. and Regnier, L. (1995). Proof-nets and the Hilbert space. In: Girard, J.-Y., Lafont, Y. and Regnier, L. (eds.) Advances in Linear Logic, Cambridge University Press, UK, 307328.CrossRefGoogle Scholar
Duchesne, E. (2009). La Localisation en Logique: Géométrie de l’Interaction et Sémantique Dénotationnelle. Thèse de doctorat, Université Aix-Marseille II.Google Scholar
Girard, J.-Y. (1989). Geometry of interaction I: interpretation of system F. In: Logic Colloquium’88, North-Holland, Amsterdam, 221–260.CrossRefGoogle Scholar
Girard, J.-Y. (1995). Geometry of interaction III: accommodating the additives. In: Advances in Linear Logic, LNS, vol. 222, Camridge University Press, UK, 329–389.CrossRefGoogle Scholar
Girard, J.-Y. (2011). Geometry of interaction V: logic in the hyperfinite factor. Theoretical Computer Science 412 (20) 18601883 CrossRefGoogle Scholar
Haghverdi, E. and Scott, P. (2006). A categorical model for the geometry of interaction. Theoretical Computer Science 350 (2–3) 252274.CrossRefGoogle Scholar
Haghverdi, E. and Scott, P. (2011). Geometry of interaction and the dynamics of proof reduction: a tutorial. In: New Structures in Physics, Lectures Notes in Physics, 339–397.Google Scholar
Hamano, M. (2018). Geometry of interaction for MALL via Hughes-vanGlabbeek proof-nets. ACM Transactions on Computational Logic 19 (4) 125:38.CrossRefGoogle Scholar
Hamano, M. and Takemura, R. (2008). An indexed system for multiplicative additive polarized linear logic. In: Proceedings of CSL’08, LNCS, vol. 5213, 262–277.CrossRefGoogle Scholar
Joyal, A., Street, R. and Verity, D. (1996). Traced monoidal categories. Mathematical Proceedings of the Cambridge Philosophical Society 119, 447468.CrossRefGoogle Scholar
Seiller, T. (2016). Interaction graphs: additives. Annals of Pure and Applied Logic 167, 95154.CrossRefGoogle Scholar
You have Access Open access

Save article to Kindle

To save this article to your Kindle, first ensure coreplatform@cambridge.org is added to your Approved Personal Document E-mail List under your Personal Document Settings on the Manage Your Content and Devices page of your Amazon account. Then enter the ‘name’ part of your Kindle email address below. Find out more about saving to your Kindle.

Note you can select to save to either the @free.kindle.com or @kindle.com variations. ‘@free.kindle.com’ emails are free but can only be saved to your device when it is connected to wi-fi. ‘@kindle.com’ emails can be delivered even when you are not connected to wi-fi, but note that service fees apply.

Find out more about the Kindle Personal Document Service.

A MALL geometry of interaction based on indexed linear logic
Available formats
×

Save article to Dropbox

To save this article to your Dropbox account, please select one or more formats and confirm that you agree to abide by our usage policies. If this is the first time you used this feature, you will be asked to authorise Cambridge Core to connect with your Dropbox account. Find out more about saving content to Dropbox.

A MALL geometry of interaction based on indexed linear logic
Available formats
×

Save article to Google Drive

To save this article to your Google Drive account, please select one or more formats and confirm that you agree to abide by our usage policies. If this is the first time you used this feature, you will be asked to authorise Cambridge Core to connect with your Google Drive account. Find out more about saving content to Google Drive.

A MALL geometry of interaction based on indexed linear logic
Available formats
×
×

Reply to: Submit a response

Please enter your response.

Your details

Please enter a valid email address.

Conflicting interests

Do you have any conflicting interests? *