  We consider the learning--unlearning paradigm defined as follows. First given
a dataset, the goal is to learn a good predictor, such as one minimizing a
certain loss. Subsequently, given any subset of examples that wish to be
unlearnt, the goal is to learn, without the knowledge of the original training
dataset, a good predictor that is identical to the predictor that would have
been produced when learning from scratch on the surviving examples.
  We propose a new ticketed model for learning--unlearning wherein the learning
algorithm can send back additional information in the form of a small-sized
(encrypted) ``ticket'' to each participating training example, in addition to
retaining a small amount of ``central'' information for later. Subsequently,
the examples that wish to be unlearnt present their tickets to the unlearning
algorithm, which additionally uses the central information to return a new
predictor. We provide space-efficient ticketed learning--unlearning schemes for
a broad family of concept classes, including thresholds, parities,
intersection-closed classes, among others.
  En route, we introduce the count-to-zero problem, where during unlearning,
the goal is to simply know if there are any examples that survived. We give a
ticketed learning--unlearning scheme for this problem that relies on the
construction of Sperner families with certain properties, which might be of
independent interest.
