Step-Indexed Kripke Models over Recursive Worlds

Over the last decade, there has been extensive research on modelling challenging features in programming languages and program logics, such as higher-order store and storable resource invariants. A recent line of work has identified a common solution to some of these challenges: Kripke models over worlds that are recursively defined in a category of metric spaces. In this paper, we broaden the scope of this technique from the original domain-theoretic setting to an elementary, operational one based on step indexing. The resulting method is widely applicable and leads to simple, succinct models of complicated language features, as we demonstrate in our semantics of Chargueraud and Pottier's type-and-capability system for an ML-like higher-order language. Moreover, the method provides a hie-level understanding of the essence of recent approaches based on step indexing.