A kripke logical relation between ML and assembly

Abstract

There has recently been great progress in proving the correctness of compilers for increasingly realistic languages with increasingly realistic runtime systems. Most work on this problem has focused on proving the correctness of a particular compiler, leaving open the question of how to verify the correctness of assembly code that is hand-optimized or linked together from the output of multiple compilers. This has led Benton and other researchers to propose more abstract, compositional notions of when a low-level program correctly realizes a high-level one. However, the state of the art in so-called "compositional compiler correctness" has only considered relatively simple high-level and low-level languages.In this paper, we propose a novel, extensional, compilerindependent notion of equivalence between high-level programs in an expressive, impure ML-like λ-calculus and low-level programs in an (only slightly) idealized assembly language. We define this equivalence by means of a biorthogonal, step-indexed, Kripke logical relation, which enables us to reason quite flexibly about assembly code that uses local state in a different manner than the high-level code it implements (e.g., self-modifying code). In contrast to prior work, we factor our relation in a symmetric, languagegeneric fashion, which helps to simplify and clarify the formal presentation, and we also show how to account for the presence of a garbage collector. Our approach relies on recent developments in Kripke logical relations for ML-like languages, in particular the idea of possible worlds as state transition systems.

Parametric Polymorphism — Universally

“…Reynolds [16] made this mathematically precise by formulating the notion of relational parametricity, and gave a set-theoretic model, where polymorphic programs are required to preserve all relations between instantiated types. Relational parametricity has proven to be one of the key techniques for formally establishing properties of software systems, such as representation independence [2,6], equivalences between programs [11], or deriving useful theorems about programs from their type alone [20].…”

“…They are certainly fundamental, as can be seen by their numerous uses within the programming literature (see e.g. [2,6,11,19,1]). While valuable, this only provides a partial answer, which ought to be complemented by a deeper and more fundamental understanding.…”

Verifying an Open Compiler Using Multi-language Semantics

“…We have already discussed the existing work [10,12] on compositional compiler correctness in §2. Here we focus on other closely related work.…”

“…Benton and Hur verified a compiler from the simply-typed λ-calculus with recursion [10]-and later, from System F with recursion [11]-to an SECD machine, proving that if source component e S compiles to target code e T , then e S and e T are logically related. Later, Hur and Dreyer [12] used essentially the same approach to prove correctness of a compiler from an idealized ML to assembly.…”

Logical Step-Indexed Logical Relations

“…We believe our approach is simpler and more direct than ⊤⊤-closure, but neither approach subsumes the either. On the one hand, ⊤⊤-closure is applicable in more general settings, such as lower-level languages [8,17] or languages with control operators [15], where the behavior of a term depends on its evaluation context. On the other hand, this added generality means that a ⊤⊤-closed relation is incapable of validating some of the inference rules that hold in our more restricted setting.…”