A generalization of Dijkstra’s calculus to typed program specifications

Abstract

Dijkstra’s predicate transformer calculus in its extended form gives an axiomatic semantics to program specifications including partiality and recursion. However, even the classical theory is based on infinitary first order logic which is needed to guarantee the existence of predicate transformers for weakest (liberal) preconditions. This theory can be generalized to higher-order intuitionistic logic.

Such logics can be interpreted in topoi. Then each topos E canonically corresponds to a definitionally complete theory T such that E is equivalent to the topos $\mathbb{E}(T)$ (T) of definable types over T. Furthermore, each model of T in an arbitrary topos F canonically corresponds to a logical morphism $\mathbb{E}(T) \to F$ (T) → F.

This correspondence enables the definition of a type specification discipline with a semantics based on topoi such that the predicate transformers in the associated logic give an axiomatic semantics for typed program specifications.