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Most specifications of garbage collectors concentrate on the low-level algorithmic details of {\em how} to find and preserve accessible objects. Often, they focus on bit-level manipulations such as ``scanning stack frames,'' ``marking objects,'' ``tagging data,'' {\em etc.} While these details are important in some contexts, they often obscure the more fundamental aspects of memory management: {\em what} objects are garbage and {\em why}? We develop a series of calculi that are just low-level enough that we can express allocation and garbage collection, yet are sufficiently abstract that we may formally prove the correctness of various memory management strategies. By making the heap of a program syntactically apparent, we can specify memory actions as rewriting rules that allocate values on the heap and automatically dereference pointers to such objects when needed. This formulation permits the specification of garbage collection as a relation that removes portions of the heap without affecting the outcome of the evaluation. Our high-level approach allows us to specify in a compact manner a wide variety of memory management techniques, including standard trace-based garbage collection ({\em i.e.}, the family of copying and mark/sweep collection algorithms), generational collection, and type-based, tag-free collection. Furthermore, since the definition of garbage is based on the {\em semantics} of the underlying language instead of the conservative approximation of inaccessibility, we are able to specify and prove the idea that type inference can be used to collect some objects that are accessible but never used.