Context-Dependent Effects in Guarded Interaction Trees

Guarded Interaction Trees are a structure and a fully formalized framework for representing higher-order computations with higher-order effects in Coq. We present an extension of Guarded Interaction Trees to support formal reasoning about context-dependent effects. That is, effects whose behaviors depend on the evaluation context, e.g., call/cc, shift and reset. Using and reasoning about such effects is challenging since certain compositionality principles no longer hold in the presence of such effects. For example, the so-called “bind rule” in modern program logics (which allows one to reason modularly about a term inside a context) is no longer valid. The goal of our extension is to support representation and reasoning about context-dependent effects in the most painless way possible. To that end, our extension is conservative: the reasoning principles (and the Coq implementation) for context-independent effects remain the same. We show that our implementation of context-dependent effects is viable and powerful. We use it to give direct-style denotational semantics for higher-order programming languages with call/cc and with delimited continuations. We extend the program logic for Guarded Interaction Trees to account for context-dependent effects, and we use the program logic to prove that the denotational semantics is adequate with respect to the operational semantics. This is achieved by constructing logical relations between syntax and semantics inside the program logic. Additionally, we retain the ability to combine multiple effects in a modular way, which we demonstrate by showing type soundness for safe interoperability of a programming language with delimited continuations and a programming language with higher-order store.