  We introduce an analog of the theory of Borel equivalence relations in which
we study equivalence relations that are decidable by an infinite time Turing
machine. The Borel reductions are replaced by the more general class of
infinite time computable functions. Many basic aspects of the classical theory
remain intact, with the added bonus that it becomes sensible to study some
special equivalence relations whose complexity is beyond Borel or even
analytic. We also introduce an infinite time generalization of the countable
Borel equivalence relations, a key subclass of the Borel equivalence relations,
and again show that several key properties carry over to the larger class.
Lastly, we collect together several results from the literature regarding Borel
reducibility which apply also to absolutely Delta_1^2 reductions, and hence to
the infinite time computable reductions.
