Safety of abstract interpretations for free, via logical relations and Galois connections

Algebraic properties of logical relations on partially ordered sets are studied. It is shown how to construct a logical relation that extends a collection of base Galois connections to a Galois connection of arbitrary higher-order type. “Theorems-for-free” is used to show that the construction ensures safe abstract interpretation of parametrically polymorphic functions. The properties are used to show how abstract interpretations of program libraries can be constructed.