  We describe a class of cellular automata (CAs) that are end-to-end
differentiable. DCAs interpolate the behavior of ordinary CAs through rules
that act on distributions of states. The gradient of a DCA with respect to its
parameters can be computed with an iterative propagation scheme that uses
previously-computed gradients and values. Gradient-based optimization over DCAs
could be used to find ordinary CAs with desired properties.
