  Given a countable transitive model of set theory and a partial order
contained in it, there is a natural countable Borel equivalence relation on
generic filters over the model; two are equivalent if they yield the same
generic extension. We examine the complexity of this equivalence relation for
various partial orders, with particular focus on Cohen and random forcing. We
prove, amongst other results, that the former is an increasing union of
countably many hyperfinite Borel equivalence relations, while the latter is
neither amenable nor treeable.
