Foundations of dependent interoperability

Abstract

Full-spectrum dependent types promise to enable the development of correct-by-construction software. However, even certified software needs to interact with simply-typed or untyped programs, be it to perform system calls, or to use legacy libraries. Trading static guarantees for runtime checks, the dependent interoperability framework provides a mechanism by which simply-typed values can safely be coerced to dependent types and, conversely, dependently-typed programs can defensively be exported to a simply-typed application. In this article, we give a semantic account of dependent interoperability. Our presentation relies on and is guided by a pervading notion of type equivalence, whose importance has been emphasized in recent work on homotopy type theory. Specifically, we develop the notions of type-theoretic partial Galois connections as a key foundation for dependent interoperability, which accounts for the partiality of the coercions between types. We explore the applicability of both type-theoretic Galois connections and anticonnections in the setting of dependent interoperability. A partial Galois connection enforces a translation of dependent types to runtime checks that are both sound and complete with respect to the invariants encoded by dependent types. Conversely, picking an anticonnection instead lets us induce weaker, sound conditions that can amount to more efficient runtime checks.Our framework is developed in Coq; it is thus constructive and verified in the strictest sense of the terms. Using our library, users can specify domain-specific partial connections between data structures. Our library then takes care of the (sometimes, heavy) lifting that leads to interoperable programs. It thus becomes possible, as we shall illustrate, to internalize and hand-tune the extraction of dependently-typed programs to interoperable OCaml programs within Coq itself.

Gradual type theory

“…Dependent Contract Checking. We also plan to explore using GTT's specification of casts in a dependently typed setting, building on work using Galois connections for casts between dependent types (Dagand et al, 2018;Eremondi et al, 2019), and work on effectful dependent types based a CBPV-like judgement structure (Ahman et al, 2016;Pédrot & Tabareau, 2020).…”

Graduality from embedding-projection pairs

“…Recently, work on dependent interoperability [Dagand et al 2016[Dagand et al , 2018 has identified Galois connections as a semantic formulation for casting between more and less precise types in a nongradual dependently typed language, and conjectures that this should relate to type dynamism. We confirm their conjecture in showing that the casts in gradual typing satisfy the slightly stronger property of being embedding-projection pairs and have used it to explain the cast semantics of gradual typing and graduality.…”

Approximate normalization for gradual dependent types

“…The programmer knows that as long as vectors are used in vector-expecting code, no crashes can happen, and safety ensures that using a list in a vector operation will always fail gracefully or run successfully. This is significantly different from work on casts to subset types [Tanter and Tabareau 2015] and dependent interoperability [Dagand et al 2018], where the user must explicitly provide decidable properties or (partial) equivalences.…”

“…Osera et al [2012] present dependent interoperability as a multi-language approach to combine both typing disciplines, mediated by runtime checks. Building on the subset cast framework (reformulated in a monadic setting instead of using axiomatic errors), Dagand et al [2016Dagand et al [ , 2018 revisit dependent interoperability in Coq via typetheoretic Galois connections that allow automatic lifting of higher-order programs. Dependent interoperability allows exploiting the connection between pre-existing types, such as Vec and List, imposing the overhead of data structure conversions.…”