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

2026-08-03

AI Summary

This work addresses the challenge of modeling temporal delays within type systems to ensure program productivity. It introduces, for the first time, the tropical semiring into a graded coeffect type system, assigning temporal grades to function parameters to capture their availability times, and proposes a novel form of intersection types over “timed sets.” The system embeds Nakano’s later modality and integrates recursion with polymorphic types, enabling a precise characterization of productivity. The main contributions are twofold: first, the typable terms exactly correspond to those possessing hereditary head normal forms; second, the type inhabitation problem is shown to be Π₀²-complete in the arithmetical hierarchy, which is optimal from the perspective of recursion theory.

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 $Π_0^2$ property in the arithmetical hierarchy.

Problem

• productivity
• graded coeffect typing
• tropical semiring
• recursive types
• timed availability

Innovation

• tropical semiring
• graded coeffect typing
• productivity
• timed intersection types
• hereditarily head normal forms