A Monadic Formalization of ML5

Abstract

ML5 is a programming language for spatially distributed computing, based on a Curry-Howard correspondence with the modal logic S5. Despite being designed by a correspondence with S5 modal logic, the ML5 programming language differs from the logic in several ways. In this paper, we explain these discrepancies between ML5 and S5 by translating ML5 into a slightly different logic: intuitionistic S5 extended with a lax modality that encapsulates effectful computations in a monad. This translation both explains the existing ML5 design and suggests some simplifications and generalizations. We have formalized our translation within the Agda proof assistant. Rather than formalizing lax S5 as a proof theory, we \emph{embed} it as a universe within the the dependently typed host language, with the universe elimination given by implementing the modal logic's Kripke semantics. This representation technique saves us the work of defining a proof theory for the logic and proving it correct, and additionally allows us to inherit the equational theory of the meta-language, which can be exploited in proving that the semantics validates the operational semantics of ML5.