Constructive Galois Connections: Taming the Galois Connection Framework for Mechanized Metatheory

TL;DR

This paper introduces constructive Galois connections, a variant that enhances mechanization and reasoning in abstract interpretation and metatheory, bridging classical theory with proof assistants and enabling verified static analysis and gradual typing.

Contribution

It develops a constructive variant of Galois connections with monadic structure, facilitating mechanized reasoning and extraction of verified algorithms in dependently-typed languages.

Findings

Successfully mechanized abstract interpretation proofs

Enabled verified static analyzer design

Provided semantic basis for gradual typing

Abstract

Galois connections are a foundational tool for structuring abstraction in semantics and their use lies at the heart of the theory of abstract interpretation. Yet, mechanization of Galois connections remains limited to restricted modes of use, preventing their general application in mechanized metatheory and certified programming. This paper presents constructive Galois connections, a variant of Galois connections that is effective both on paper and in proof assistants; is complete with respect to a large subset of classical Galois connections; and enables more general reasoning principles, including the "calculational" style advocated by Cousot. To design constructive Galois connection we identify a restricted mode of use of classical ones which is both general and amenable to mechanization in dependently-typed functional programming languages. Crucial to our metatheory is the addition of monadic structure to Galois connections to control a "specification effect". Effectful calculations may reason classically, while pure calculations have extractable computational content. Explicitly moving between the worlds of specification and implementation is enabled by our metatheory. To validate our approach, we provide two case studies in mechanizing existing proofs from the literature: one uses calculational abstract interpretation to design a static analyzer, the other forms a semantic basis for gradual typing. Both mechanized proofs closely follow their original paper-and-pencil counterparts, employ reasoning principles not captured by previous mechanization approaches, support the extraction of verified algorithms, and are novel.