A theory of non-monotone memory (or: Contexts for free)

We develop a general method of proving contextual properties-including (but not limited to) observational equivalence, space improvement,and memory safety under arbitrary contexts-for programs in untyped call-by-value ?-calculus with first-class, higher-order references(ref, := and !) and deallocation (free). The method significantly generalizes Sumii et al.'s environmental bisimulation technique, and gives a sound and complete characterization of each proved property, in thesense that the "bisimilarity" (the largest set satisfying the bisimulation like conditions) equals the set of terms with the property to be proved. We give examples of contextual properties concerning typical data structures such as linked lists, binary search trees, and directed acyclic graphs with reference counts, all with deletion operations that release memory. This shows the scalability of the environmental approach from contextual equivalence to other binary relations (such as space improvement)and unary predicates (such as memory safety), as well as to languages with non-monotone store, where Kripke-style logical relations have difficulties.