Plausible sealing for gradual parametricity

Abstract

Graduality and parametricity have proven to be extremely challenging notions to bring together. Intuitively, enforcing parametricity gradually requires possibly sealing values in order to detect violations of uniform behavior. Toro et al. (2019) argue that the two notions are incompatible in the context of System F, where sealing is transparently driven by potentially imprecise type information, while New et al. (2020) reconcile both properties at the cost of abandoning the syntax of System F and requiring user-provided sealing annotations that are not subject to graduality guarantees. Furthermore, all current proposals rely on a global form of dynamic sealing in order to enforce parametric behavior at runtime, which weakens parametric reasoning and breaks equivalences in the static language. Based on the observation that the tension between graduality and parametricity comes from the early commitment to seal values based on type information, we propose plausible sealing as a new intermediate language mechanism that allows postponing such decisions to runtime. We propose an intermediate language for gradual parametricity, Funky, which supports plausible sealing in a simplified setting where polymorphism is restricted to instantiations with base and variable types. We prove that Funky satisfies both parametricity and graduality, mechanizing key lemmas in Agda. Additionally, we avoid global dynamic sealing and instead propose a novel lexically-scoped form of sealing realized using a representation of evidence inspired by the category of spans. As a consequence, Funky satisfies a standard formulation of parametricity that does not break System F equivalences. In order to show the practicality of plausible sealing, we describe a translation from Funk, a source language without explicit sealing, to Funky, that takes care of inserting plausible sealing forms. We establish graduality of Funk, subject to a restriction on type applications, and explain the source-level parametric reasoning it supports. Finally, we provide an interactive prototype along with illustrative examples both novel and from the literature.

Pragmatic Gradual Polymorphism with References

“…In particular, in our work we illustrate how to obtain a polymorphic gradually typed calculus, with gradual references and with the gradual guarantee. In contrast, none of the existing gradually polymorphic calculi supports references and the gradual guarantee is only supported with restrictions [20]; or major modifications in the syntax and semantics of the language [24]; or not supported/proved at all [37,3,17].…”

“…With the user-defined syntax, the gradual guarantee and parametricity are proved. More recently, Labrada et al [20] improve on GSF. They do not change the syntax of System F but insert plausible sealing forms during the elaboration from a gradual source language which is named Funk to a target cast calculus.…”

“…Toro et al [37] even proved that gradual guarantee and parametricity are incompatible. By giving up the traditional System F syntax, New et al [24] proved the gradual guarantee and parametricity by using user-provided sealing annotations, but this requires resorting to syntax that is not based on System F. Finally, Labrada et al [20] proved the gradual guarantee and parametricity by inserting sealing with some restrictions. For instance, only base and variable types can be used to instantiate type applications.…”

“…With dynamic sealing we do not do the substitution directly but record a fresh variable binding. However, even calculi that satisfy parametricity have to compromise on the important gradual guarantee property, or System F syntax, or be equiped with heavy forms of runtime evidence [37,20]. A thorough discussion of various approaches is given in Section 6.…”