Completeness of monad-based dynamic logic

Monads are used in functional programming as a means of modeling and encapsulating computational effects at an appropriate level of abstraction. In previous work, we have introduced monad-based dynamic logic and have used it for the specification and verification of various monadic functional-imperative algorithms over different monads. The semantics of this logic was originally defined over partial cartesian closed categories; completeness issues were explicitly left out of consideration. Here, we slightly simplify the logic and reformulate it in the weaker setting of a strong monad over a cartesian category with some additional infrastructure. We provide an extended proof calculus and prove its soundness and completeness. The presence of computational effects, e.g. state, store, exceptions, input, output, non-determinism, or backtracking, substantially complicates reasoning about programs. Dealing with such features has in the past typically required dedicated logics, designed specifically for a particular combination of effects. As a unifying concept for modeling computational effects (in particular, all of those mentioned above) in an elegant way, monads, a well-known tool from category theory, have been established by Moggi in the seminal paper [5]. Based on this work, monads now form the basis of the treatment of encapsulated side effects in the functional programming language Haskell [11], which thanks to this concept admits clean programming in an imperative style. Monads have also appeared in programming semantics, e.g. for Java [3]. In previous work, we have introduced a monad-based Hoare calculus [9] and a dynamic logic [10] that allow reasoning about such generic effects. The semantics and proof calculus of both logics are parametrized over a monad (and hence, over a notion of side effect). The particular (combination of) effects needed for an application can then be axiomatically specified in the general framework. Monad-based dynamic logic (MDL) is similar to Pitts’ evaluation logic [7], but equipped with a rather different semantics which is induced directly by the underlying monad. So far, no completeness result has been proved for either of these logics. Here, we fill in this gap in the theory. We simplify MDL by detaching it from the HasCasl framework as used in [10]. Instead, we introduce a term language and semantics based on cartesian categories. The resulting calculus, slightly extended in relation to the calculus presented in [10], is shown to be