Dijkstra monads for all

• Published in: Proceedings of ACM on programming languages 2019-07, Vol.3 (ICFP), p.1-29
• Main Authors: Maillard, Kenji, Ahman, Danel, Atkey, Robert, Martínez, Guido, Hriţcu, Cătălin, Rivas, Exequiel, Tanter, Éric
• Format: Article
• Language: English
• Subjects: Computer Science; Cryptography and Security; Logic in Computer Science; Programming Languages; Software Engineering
• ISSN: 2475-1421; 2475-1421
• DOI: 10.1145/3341708

This paper proposes a general semantic framework for verifying programs with arbitrary monadic side-effects using Dijkstra monads, which we define as monad-like structures indexed by a specification monad. We prove that any monad morphism between a computational monad and a specification monad gives rise to a Dijkstra monad, which provides great flexibility for obtaining Dijkstra monads tailored to the verification task at hand. We moreover show that a large variety of specification monads can be obtained by applying monad transformers to various base specification monads, including predicate transformers and Hoare-style pre- and postconditions. For defining correct monad transformers, we propose a language inspired by Moggi's monadic metalanguage that is parameterized by a dependent type theory. We also develop a notion of algebraic operations for Dijkstra monads, and start to investigate two ways of also accommodating effect handlers. We implement our framework in both Coq and F*, and illustrate that it supports a wide variety of verification styles for effects such as exceptions, nondeterminism, state, input-output, and general recursion.