Dijkstra Monads in Monadic Computation

Abstract : The Dijkstra monad has been introduced recently for capturing weakest precondition computations within the context of program verification, supported by a theorem prover. Here we give a more general description of such Dijkstra monads in a categorical setting. We first elaborate the recently developed view on program semantics in terms of a triangle of computations, state transformers, and predicate transformers. Instantiations of this triangle for different monads $T$ show how to define the Dijkstra monad associated with $T$, via the logic involved. Technically, we provide a morphism of monads from the state monad transformation applied to $T$, to the Dijkstra monad associated with $T$. This monad map is precisely the weakest precondition map in the triangle, given in categorical terms by substitution.