A Context Logic for Tree Update

Spatial logics have been used to describe properties of tree-like structures (Ambient Logic) and in a Hoare style to reason about dynamic updates of heap-like structures (Separation Logic). We integrate this work by analyzing dynamic updates to tree structures with pointers (such as XML with identifiers and idrefs). Naı̈ve adaptations of the previous logics are not expressive enough to capture such local updates. Instead we must explicitly reason about arbitrary tree contexts — not just horizontal composition and vertical branching — in order to capture updates throughout the tree. To illustrate the point, we introduce a small imperative programming language for updating our trees, small Hoare-style axioms for the commands in the style of O’Hearn, Reynolds and Yang, and show how weakest preconditions are derivable from the small axioms with a generalized frame rule. We demonstrate the generality of our approach by showing that it collapses to Separation Logic for a heap model.