Small Specifications for Tree Update

Abstract

Abstract. O'Hearn, Reynolds and Yang introduced local Hoare reasoning about mutable data structures using Separation Logic. They reason about the local parts of the memory accessed by programs, and thus construct their smallest complete specifications. Gardner et al. generalised their work, using Context Logic to reason about structured data at the same level of abstraction as the data itself. In particular, they developed a formal specification of the Document Object Model, a W3C XML update library. Whilst they kept to the spirit of local reasoning, they were not able to retain small specifications: for example, the specification of appendChild was not small. We show how to obtain small specifications by working with a more fine-grained context structure, allowing us to work with arbitrary tree fragments.

Combining Form and Function: Static Types for JQuery Programs

“…Statically, maintaining tree shapes may be analyzable by a tree logic [7]. The primary challenge is a reachability analysis identifying all the possible nodes that might be modified by arbitrary JS; we leave this entirely to future work.…”

Abstract Local Reasoning for Program Modules

“…fragment of the W3C DOM (Gardner, Smith, Wheelhouse, and Zarfaty, 2008), and tree segments (Gardner and Wheelhouse, 2009). …”

“…We thus give the axiomatisation of the tree update commands in terms of such complete trees, although they may actually be subtrees of bigger trees. (Gardner and Wheelhouse (2009) specify tree commands in terms of tree segments, which are fragments of trees that need not be trees in their own right. We work with subtrees here for simplicity of exposition.)…”

“…A separation logic-style parallel rule is only appropriate when the model has a commutative * -like operator; while the heap and list modules have such an operator, the tree model does not. (The segment logic of Gardner and Wheelhouse (2009) is a promising approach to introducing a commutative * to structured models such as trees.) Assume that we have a sound module translation from an abstract module to a concrete one, and that both models include a commutative * connective, such that, for every state d there is some context c such that, for all states d , d * d = c • d .…”