A path to DOT: formalizing fully path-dependent types

Abstract

The Dependent Object Types (DOT) calculus aims to formalize the Scala programming language with a focus on path-dependent types -types such as x .a 1 . . . a n .T that depend on the runtime value of a path x .a 1 . . . a n to an object. Unfortunately, existing formulations of DOT can model only types of the form x .A which depend on variables rather than general paths. This restriction makes it impossible to model nested module dependencies. Nesting small components inside larger ones is a necessary ingredient of a modular, scalable language. DOT's variable restriction thus undermines its ability to fully formalize a variety of programming-language features including Scala's module system, family polymorphism, and covariant specialization.This paper presents the pDOT calculus, which generalizes DOT to support types that depend on paths of arbitrary length, as well as singleton types to track path equality. We show that naive approaches to add paths to DOT make it inherently unsound, and present necessary conditions for such a calculus to be sound. We discuss the key changes necessary to adapt the techniques of the DOT soundness proofs so that they can be applied to pDOT. Our paper comes with a Coq-mechanized type-safety proof of pDOT. With support for paths of arbitrary length, pDOT can realize DOT's full potential for formalizing Scala-like calculi.

Scala step-by-step: soundness for DOT with step-indexed logical relations in Iris

“…To demonstrate usefulness of mutual information hiding, consider the example in Fig. 1, adapted from Rapoport and Lhoták [2019], and inspired by the actual implementation of the Scala 3 compiler (Dotty). The example models a system with mutually recursive modules types and symbols, encoded as members of the object pcore and representing separate compilation units.…”

“…Alas, pDOT cannot type this code, as pDOT requires that all (recursive) objects ( : ). { } must have a precise self type [Rapoport and Lhoták 2019]. Informally, is precise if the bounds >: <: of all type members that appear hereditarily in satisfy = Ð i.e., if the recursively defined object does not contain any abstract type members (we define this notion formally in Fig.…”

“…4 and 5). We focus on the typing rules that are essential to the rest of the paper, and defer to Rapoport and Lhoták [2019] for the remaining ones. A term in pDOT can be typed either with a dependent function type ∀ : .…”

“…To address these issues more rigorously, the compiler of the new Scala 3 language (called Dotty) has been designed hand in hand with a new foundational type system Ð the Dependent Object Types (DOT) calculus. This development led to a number of increasingly expressive versions of DOT and type soundness proofs thereof Kabir and Lhoták 2018;Rapoport et al 2017;Rapoport and Lhoták 2016;, culminating in the pDOT calculus [Rapoport and Lhoták 2019], and has helped to fix various soundness bugs in Scala 3 .…”

“…Question (2) is challenging because current syntactic type soundness proofs for DOT are intricate, and thus hard to scale to new DOT variants. While Rapoport et al [2017] describe a recipe for syntactic proofs for DOT, applying this recipe to pDOT Ð one of the most expressive versions of DOT to date Ð involves 7 carefully designed variants of pDOT's typing judgment [Rapoport and Lhoták 2019]. Moreover, subtyping for recursive types is not supported by pDOT but, to date, only by the soundness proofs by and Amin and Rompf [2017], which have not been extended in the published literature, and lack other crucial pDOT features.…”