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Recent digital rights frameworks give users the right to delete their data from systems that store and process their personal information (e.g., the "right to be forgotten" in the GDPR). How should deletion be formalized in complex systems that interact with many users and store derivative information? We argue that prior approaches fall short. Definitions of machine unlearning Cao and Yang [2015] are too narrowly scoped and do not apply to general interactive settings. The natural approach of deletion-as-confidentiality Garg et al. [2020] is too restrictive: by requiring secrecy of deleted data, it rules out social functionalities. We propose a new formalism: deletion-as-control. It allows users' data to be freely used before deletion, while also imposing a meaningful requirement after deletion--thereby giving users more control. Deletion-as-control provides new ways of achieving deletion in diverse settings. We apply it to social functionalities, and give a new unified view of various machine unlearning definitions from the literature. This is done by way of a new adaptive generalization of history independence. Deletion-as-control also provides a new approach to the goal of machine unlearning, that is, to maintaining a model while honoring users' deletion requests. We show that publishing a sequence of updated models that are differentially private under continual release satisfies deletion-as-control. The accuracy of such an algorithm does not depend on the number of deleted points, in contrast to the machine unlearning literature.