Mitchell’s notion of representation independence is a particularly useful application of Reynolds ’ relational parametricity — two different implementations of an abstract data type can be shown contextually equivalent so long as there exists a relation between their type representations that is preserved by their operations. There have been a number of methods proposed for proving representation independence in various pure extensions of System F (where data abstraction is achieved through existential typing), as well as in Algol- or Java-like languages (where data abstraction is achieved through the use of local mutable state). However, none of these approaches addresses the interaction of existential type abstraction and local state. In particular, none allows one to prove representation independence results for generative ADTs — i.e., ADTs that both maintain some local state and define abstract types whose internal

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.

Developing a theory of bisimulation in higher-order languages can be hard. Particularly challenging can be: (1) the proof of congruence, as well as enhancements of the bisimulation proof method with “up-to context ” techniques, and (2) obtaining definitions and results that scale to languages with different features. To meet these challenges, we present environmental bisimulations, a form of bisimulation for higher-order languages, and its basic theory. We consider four representative calculi: pure λ-calculi (call-by-name and call-byvalue), call-by-value λ-calculus with higher-order store, and then Higher-Order π-calculus. In each case: we present the basic properties of environmental bisimilarity, including congruence; we show that it coincides with contextual equivalence; we develop some up-to techniques, including up-to context, as examples of possible enhancements of the associated bisimulation method. Unlike previous approaches (such as applicative bisimulations, logical relations, Sumii-Pierce-Koutavas-Wand), our method does not require induction/indices on evaluation derivation/steps (which may complicate the proofs of congruence, transitivity, and the combination with up-to techniques), or sophisticated methods such as Howe’s for proving congruence. It also scales from the pure λ-calculi to the richer calculi with simple congruence proofs. 1

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 define logical relations between the denotational semantics of a simply typed functional language with recursion and the operational behaviour of low-level programs in a variant SECD machine. The relations, which are defined using biorthogonality and stepindexing, capture what it means for a piece of low-level code to implement a mathematical, domain-theoretic function and are used to prove correctness of a simple compiler. The results have been formalized in the Coq proof assistant.

Self-adjusting computation enables writing programs that can automatically and efficiently respond to changes to their data (e.g., inputs). The idea behind the approach is to store all data that can change over time in modifiable references and to let computations construct traces that can drive change propagation. After changes have occurred, change propagation updates the result of the computation by re-evaluating only those expressions that depend on the changed data. Previous approaches to selfadjusting computation require that modifiable references be written at most once during execution—this makes the model applicable only in a purely functional setting. In this paper, we present techniques for imperative self-adjusting computation where modifiable references can be written multiple times. We define a language SAIL (Self-Adjusting Imperative Language) and prove consistency, i.e., that change propagation and from-scratch execution are observationally equivalent. Since SAIL programs are imperative, they can create cyclic data structures. To prove equivalence in the presence of cycles in the store, we formulate and use an untyped, step-indexed logical relation, where step indices are used to ensure well-foundedness. We show that SAIL accepts an asymptotically efficient implementation by presenting algorithms and data structures for its implementation. When the number of operations (reads and writes) per modifiable is bounded by a constant, we show that change propagation becomes as efficient as in the non-imperative case. The general case incurs a slowdown that is logarithmic in the maximum number of such operations. We describe a prototype implementation of SAIL as a Standard ML library. 1

Abstract. We introduce a Floyd-Hoare-style framework for specification and verification of machine code programs, based on relational parametricity (rather than unary predicates) and using both step-indexing and a novel form of separation structure. This yields compositional, descriptive and extensional reasoning principles for many features of lowlevel sequential computation: independence, ownership transfer, unstructured control flow, first-class code pointers and address arithmetic. We demonstrate how to specify and verify the implementation of a simple memory manager and, independently, its clients in this style. The work has been fully machine-checked within the Coq proof assistant. 1

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. 1.

The method of logical relations is a classic technique for proving the equivalence of higher-order programs that implement the same observable behavior but employ different internal data representations. Although it was originally studied for pure, strongly normalizing languages like System F, it has been extended over the past two decades to reason about increasingly realistic languages. In particular, Appel and McAllester’s idea of step-indexing has been used recently to develop syntactic Kripke logical relations for MLlike languages that mix functional and imperative forms of data abstraction. However, while step-indexed models are powerful tools, reasoning with them directly is quite painful, as one is forced to engage in tedious step-index arithmetic to derive even simple results. In this paper, we propose a logic LADR for equational reasoning about higher-order programs in the presence of existential type abstraction, general recursive types, and higher-order mutable state. LADR exhibits a novel synthesis of features from Plotkin-Abadi logic, Gödel-Löb logic, S4 modal logic, and relational separation logic. Our model of LADR is based on Ahmed, Dreyer, and Rossberg’s state-of-the-art step-indexed Kripke logical relation, which was designed to facilitate proofs of representation independence for “state-dependent ” ADTs. LADR enables one to express such proofs at a much higher level, without counting steps or reasoning about the subtle, step-stratified construction of possible worlds.

We describe a semantic type soundness result, formalized in the Coq proof assistant, for a compiler from a simple imperative language with heap-allocated data into an idealized assembly language. Types in the high-level language are interpreted as binary relations, built using both second-order quantification and a form of separation structure, over stores and code pointers in the low-level machine.