We present a bisimulation method for proving the contextual equivalence of packages in λ-calculus with full existential and recursive types. Unlike traditional logical relations (either semantic or syntactic), our development is “elementary, ” using only sets and relations and avoiding advanced machinery such as domain theory, admissibility, and ⊤⊤-closure. Unlike other bisimulations, ours is complete even for existential types. The key idea is to consider sets of relations—instead of just relations—as bisimulations.

We present a local relational reasoning method for reasoning about contextual equivalence of expressions in a λ-calculus with recursive types and general references. Our development builds on the work of Benton and Leperchey, who devised a nominal semantics and a local relational reasoning method for a language with simple types and simple references. Their method uses a parameterized logical relation. Here we extend their approach to recursive types and general references. For the extension, we build upon Pitts ’ and Shinwell’s work on relational reasoning about recursive types (but no references) in nominal semantics. The extension is non-trivial because of general references (higher-order store) and makes use of some new ideas for proving the existence of the parameterized logical relation and for the choice of parameters.

We present a possible world semantics for a call-by-value higherorder programming language with impredicative polymorphism, general references, and recursive types. The model is one of the first relationally parametric models of a programming language with all these features. To model impredicative polymorphism we define the semantics of types via parameterized (world-indexed) logical relations over a universal domain. It is well-known that it is non-trivial to show the existence of logical relations in the presence of recursive types. Here the problems are exacerbated because of general references. We explain what the problems are and present our solution, which makes use of a novel approach to modeling references. We prove that the resulting semantics is adequate with respect to a standard operational semantics and include simple examples of reasoning about contextual equivalence via parametricity.

We present a sound and complete method for reasoning about contextual equivalence between different implementations of classes in an imperative subset of Java. To the extent of our knowledge this is the first such method for a language with unrestricted inheritance, where the context can arbitrarily extend classes to distinguish otherwise equivalent implementations. Similar reasoning techniques for class-based languages [1, 12] don’t consider inheritance at all, or forbid the context from extending related classes. Other techniques that do consider inheritance [3] study whole-program equivalence. Our technique also handles public, private, and protected interfaces of classes, imperative fields, and invocations of callbacks. Using our technique we were able to prove equivalences in examples with higher-order behavior, where previous methods for functional calculi admit limitations [21, 24]. Adding inheritance to a class-based language increases the distinguishing power of the context. Here we show how this extra distinguishing power is reflected in the conditions for equivalence of our technique. Furthermore we show that adding a cast operator is a conservative extension of the language. 1.

We introduce a Sumii-Pierce-Koutavas-Wand-style bisimulation for Pitts and Stark’s nu-calculus, a simply-typed lambda calculus with fresh name generation. This bisimulation coincides with contextual equivalence and provides a usable and elementary method for establishing all the subtle equivalences given by Stark [11]. We also describe the formalization of soundness and of the examples in the Coq proof assistant.

By combining and simplifying two of the most prominent theories for HOπ of Sangiorgi et al. and Jeffrey and Rathke [15, 4], we present an effective first-order theory for a higher-order picalculus. There are two significant aspects to our theory. The first is that higher-order inputs are treated in a first-order manner, hence eliminating the need to reason about arbitrarily complicated higher-order contexts, or to use up-to context techniques, when establishing equivalences between processes. The second is that we use augmented processes to record directly the knowledge of the observer. This has the benefit of making ordinary firstorder weak bisimulation fully abstract w.r.t. contextual equivalence. It also simplifies the handling of names, giving rise to a truly propositional Hennessy-Milner characterisation of higher-order contextual equivalence. Furthermore, we illustrate the simplicity of our approach in proving several interesting equivalences by exhibiting first-order witness weak bisimulations, and inequivalences by using the propositional Hennessy-Milner Logic. Finally we show that contextual equivalence

We present a series of examples that illuminate an important aspect of the semantics of higher-order functions with local state. Namely that certain behaviour of such functions can only be observed by providing them with arguments that contain the functions themselves. This provides evidence for the necessity of complex conditions for functions in modern semantics for state, such as logical relations and Kripke-like bisimulations, where related functions are applied to related arguments (that may contain the functions). It also suggests that simpler semantics, such as those based on applicative bisimulations where functions are applied to identical arguments, would not scale to higher-order languages with local state.