Staying Productive Under the Palm Trees: On Graded Coeffect Typing in the Tropical Semiring

[2608.02596] Staying Productive Under the Palm Trees: On Graded Coeffect Typing in the Tropical Semiring Skip to main content Search Submit Donate Log in Computer Science > Logic in Computer Science arXiv:2608.02596 (cs) [Submitted on 3 Aug 2026 ( v1 ), last revised 24 Aug 2026 (this version, v2)] Title: Staying Productive Under the Palm Trees: On Graded Coeffect Typing in the Tropical Semiring Authors: Rémy Cerda , Ugo Dal Lago View PDF Abstract: We show that the tropical semiring over the natural numbers, when used as the grading space in graded coeffect typing, faithfully models the passage of time while simultaneously guaranteeing productivity of well-typed programs. A grade a, when assigned to a function parameter, indicates that the parameter is not necessarily available immediately, but will become available after a time steps. We investigate this idea through two formal systems. We first introduce a graded type system featuring recursive and polymorphic types, and show that, in this setting, a natural restriction on recursive types is sufficient to guarantee productivity, while still allowing the definition of streams and recursive programs on them. In particular, we prove that Nakano's later modality can be embedded directly into our system. We then show that tropical grading naturally suggests a novel form of intersection typing, in which the role traditionally played by sets or multisets of types is instead taken by "timed" sets, i.e., functions assigning to each type A the earliest time, represented as a grade, from which the underlying term is available with type A. For the resulting system, we prove not only that productivity is guaranteed, but that it is also characterized: the typable terms are exactly those with hereditarily head normal forms. Remarkably, the system is recursion-theoretically optimal, i.e., typability can be directly proved to be a $\Pi_0^2$ property in the arithmetical hierarchy. Subjects: Logic in Computer Science (cs.LO) ; Programming Languages (cs.PL) Cite as: arXiv:2608.02596 [cs.LO] (or arXiv:2608.02596v2 [cs.LO] for this version) https://doi.org/10.48550/arXiv.2608.02596 Focus to learn more arXiv-issued DOI via DataCite Submission history From: Rémy Cerda [ view email ] [v1] Mon, 3 Aug 2026 17:58:34 UTC (107 KB) [v2] Mon, 24 Aug 2026 18:52:05 UTC (91 KB) Full-text links: Access Paper: View PDF TeX Source view license Additional Features Audio Summary Current browse context: cs.LO < prev | next > new | recent | 2026-08 Change to browse by: cs cs.PL References & Citations SciX Google Scholar Semantic Scholar export BibTeX citation Bookmark Bibliographic Tools Bibliographic and Citation Tools Bibliographic Explorer Toggle Bibliographic Explorer ( What is the Explorer? ) Connected Papers Toggle Connected Papers ( What is Connected Papers? ) Litmaps Toggle Litmaps ( What is Litmaps? ) scite.ai Toggle scite Smart Citations ( What are Smart Citations? ) Code, Data, Media Code, Data and Media Associated with this Article alphaXiv Toggle alphaXiv ( What is alphaXiv? ) Links to Code Toggle CatalyzeX Code Finder for Papers ( What is CatalyzeX? ) DagsHub Toggle DagsHub ( What is DagsHub? ) GotitPub Toggle Gotit.pub ( What is GotitPub? ) Huggingface Toggle Hugging Face ( What is Huggingface? ) ScienceCast Toggle ScienceCast ( What is ScienceCast? ) Demos Demos Replicate Toggle Replicate ( What is Replicate? ) Spaces Toggle Hugging Face Spaces ( What is Spaces? ) Spaces Toggle TXYZ.AI ( What is TXYZ.AI? ) Related Papers Recommenders and Search Tools Link to Influence Flower Influence Flower ( What are Influence Flowers? ) Core recommender toggle CORE Recommender ( What is CORE? ) Author Venue Institution Topic About arXivLabs arXivLabs: experimental projects with community collaborators arXivLabs is a framework that allows collaborators to develop and share new arXiv features directly on our website. Both individuals and organizations that work with arXivLabs have embraced and accepted our values of openness, community, excellence, and user data privacy. arXiv is committed to these values and only works with partners that adhere to them. Have an idea for a project that will add value for arXiv's community? Learn more about arXivLabs . Which authors of this paper are endorsers? | Disable MathJax ( What is MathJax? ) We gratefully acknowledge support from our major funders , member institutions , and all contributors. About · Help · Contact · Subscribe · Copyright · Privacy · Accessibility · Operational Status (opens in new tab) Major funding support from (opens in new tab) (opens in new tab) (opens in new tab)

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