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 Charguéraud and Pottier's type-and-capability system for an ML-like higher-order language. Moreover, the method provides a high-level understanding of the essence of recent approaches based on step indexing.

Fine-grained concurrent data structures (or FCDs) reduce the granularity of critical sections in both time and space, thus making it possible for clients to access different parts of a mutable data structure in parallel. However, the tradeoff is that the implementations of FCDs are very subtle and tricky to reason about directly. Consequently, they are carefully designed to be contextual refinements of their coarse-grained counterparts, meaning that their clients can reason about them as if all access to them were sequentialized.In this paper, we propose a new semantic model, based on Kripke logical relations, that supports direct proofs of contextual refinement in the setting of a type-safe high-level language. The key idea behind our model is to provide a simple way of expressing the "local life stories" of individual pieces of an FCD's hidden state by means of protocols that the threads concurrently accessing that state must follow. By endowing these protocols with a simple yet powerful transition structure, as well as the ability to assert invariants on both heap states and specification code, we are able to support clean and intuitive refinement proofs for the most sophisticated types of FCDs, such as conditional compare-and-set (CCAS).

Abstract. We present a realizability model for a call-by-value, higherorder programming language with parametric polymorphism, general first-class references, and recursive types. The main novelty is a relational interpretation of open types (as needed for parametricity reasoning) that include general reference types. The interpretation uses a new approach to modeling references.The universe of semantic types consists of world-indexed families of logical relations over a universal predomain. In order to model general reference types, worlds are finite maps from locations to semantic types: this introduces a circularity between semantic types and worlds that precludes a direct definition of either. Our solution is to solve a recursive equation in an appropriate category of metric spaces. In effect, types are interpreted using a Kripke logical relation over a recursively defined set of worlds.We illustrate how the model can be used to prove simple equivalences between different implementations of imperative abstract data types.

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Fine-grained concurrent data structures (or FCDs) reduce the granularity of critical sections in both time and space, thus making it possible for clients to access different parts of a mutable data structure in parallel. However, the tradeoff is that the implementations of FCDs are very subtle and tricky to reason about directly. Consequently, they are carefully designed to be contextual refinements of their coarse-grained counterparts, meaning that their clients can reason about them as if all access to them were sequentialized. In this paper, we propose a new semantic model, based on Kripke logical relations, that supports direct proofs of contextual refinement in the setting of a type-safe high-level language. The key idea behind our model is to provide a simple way of expressing the "local life stories" of individual pieces of an FCD's hidden state by means of protocols that the threads concurrently accessing that state must follow. By endowing these protocols with a simple yet powerful transition structure, as well as the ability to assert invariants on both heap states and specification code, we are able to support clean and intuitive refinement proofs for the most sophisticated types of FCDs, such as conditional compare-and-set (CCAS).