Abstracting gradual typing moving forward: precise and space-efficient

Abstract

Abstracting Gradual Typing (AGT) is a systematic approach to designing gradually-typed languages. Languages developed using AGT automatically satisfy the formal semantic criteria for gradual languages identified by Siek et al. Nonetheless, vanilla AGT semantics can still have important shortcomings. First, a gradual language's runtime checks should preserve the space-efficiency guarantees inherent to the underlying static and dynamic languages. To the contrary, the default operational semantics of AGT break proper tail calls. Second, a gradual language's runtime checks should enforce basic modular type-based invariants expected from the static type discipline. To the contrary, the default operational semantics of AGT may fail to enforce some invariants in surprising ways. We demonstrate this in the GTFL ≲ language of Garcia et al. This paper addresses both problems at once by refining the theory underlying AGT's dynamic checks. Garcia et al. observe that AGT involves two abstractions of static types: one for the static semantics and one for the dynamic semantics. We recast the latter as an abstract interpretation of subtyping itself, while gradual types still abstract static types. Then we show how forward-completeness (Giacobazzi and Quintarelli) is key to supporting both space-efficient execution and reliable runtime type enforcement.

Parameterized cast calculi and reusable meta-theory for gradually typed lambda calculi

“…Siek & Wadler (2010) propose another approach that compresses a sequence of casts into two casts where the middle type is the least upper bound with respect to precision. The AGT methodology uses a similar representation (Garcia et al, 2016) and was proved space-efficient (Toro & Tanter, 2020;Bañados Schwerter et al, 2021).…”

“…• subtyping (Siek & Taha, 2007;Garcia et al, 2016;Chung et al, 2018;Bañados Schwerter et al, 2021), • classes and objects (Takikawa et al, 2012;Allende et al, 2013;Vitek, 2016;Takikawa, 2016;Muehlboeck & Tate, 2017;Chung et al, 2018), • parametric polymorphism (Ahmed et al, 2011(Ahmed et al, , 2017Toro et al, 2019;New et al, 2019), • type inference (Siek & Vachharajani, 2008;Garcia & Cimini, 2015;Castagna et al, 2019), • set-theoretic types (Castagna & Lanvin, 2017),…”