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 preand 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.
Much recent research has been devoted to modeling effects within type theory. Building on this work, we observe that effectful type theories can provide a foundation on which to build semantics for more complex programming constructs and program logics, extending the reasoning principles that apply within the host effectful type theory itself. Concretely, our main contribution is a semantics for concurrent separation logic (CSL) within the F ★ proof assistant in a manner that enables dependently typed, effectful F ★ programs to make use of concurrency and to be specified and verified using a full-featured, extensible CSL. In contrast to prior approaches, we directly derive the partial-correctness Hoare rules for CSL from the denotation of computations in the effectful semantics of non-deterministically interleaved atomic actions. Demonstrating the flexibility of our semantics, we build generic, verified libraries that support various concurrency constructs, ranging from dynamically allocated, storable spin locks, to protocol-indexed channels. We conclude that our effectful semantics provides a simple yet expressive basis on which to layer domainspecific languages and logics for verified, concurrent programming.
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