O'Hearn, Reynolds and Yang introduced Separation Logic to provide modular reasoning about simple, mutable data structures in memory. They were able to construct small specifications of programs, by reasoning about the local parts of memory accessed by programs. Gardner, Calcagno and Zarfaty generalised this work, introducing Context Logic to reason about more complex data structures. In particular, they developed a formal, compositional specification of the Document Object Model, a W3C XML update library. Whilst keeping to the spirit of local reasoning, they were not able to retain small specifications. We introduce Segment Logic, which provides a more fine-grained analysis of the tree structure and yields small specifications. As well as being aesthetically pleasing, small specifications are important for reasoning about concurrent tree update.
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