Dijkstra Monads for All

TL;DR

This paper introduces Dijkstra monads as a flexible framework for verifying programs with various monadic effects, unifying different verification styles through a general semantic approach.

Contribution

It defines a universal construction for Dijkstra monads from monad morphisms and develops a language for correct monad transformers within a dependent type theory.

Findings

Framework supports verification of effects like exceptions, nondeterminism, state, input-output, and recursion.

Implemented in Coq and F* demonstrating practical applicability.

Allows tailoring Dijkstra monads to specific verification tasks.

Abstract

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.