The W3C Document Object Model (DOM) specifies an XML update library. DOM is written in English, and is therefore not compositional and not complete. We provide a first step towards a compositional specification of DOM. Unlike DOM, we are able to work with a minimal set of commands and obtain a complete reasoning for straight-line code. Our work transfers O'Hearn, Reynolds and Yang's local Hoare reasoning for analysing heaps to XML, viewing XML as an in-place memory store as does DOM. In particular, we apply recent work by Calcagno, Gardner and Zarfaty on local Hoare reasoning about simple tree update to this real-world DOM application. Our reasoning not only formally specifies a significant subset of DOM Core Level 1, but can also be used to verify, for example, invariant properties of simple Javascript programs.

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.

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 integrat this work by analyzing dynamic updates to tree-like structures with pointers (such as XML with identifiers and idrefs). Naíve adaptations of the Ambient Logic are not expressive enough to capture such local updates. Instead we must explicitly reason about arbitrary tree contexts in order to capture updates throughout the tree. We introduce Context Logic, study its proof theory and models, and show how it generalizes Separation Logic and its general theory BI. We use it to reason locally about a small imperative programming language for updating trees, using a Hoare logic in the style of O'Hearn, Reynolds and Yang, and show that weakest preconditions are derivable. We demonstrate the robustness of our approach by using Context Logic to capture the locality of term rewrite systems.