Discrete space and discrete time are too restrictive in some cases. The posted link is the HTML format of a paper by Prof.

MacLennan at the University of Tennessee, Knoxville. He describes CSA in the abstract as made of nodes forming a continuum with states from a continuum. However, in order to specify a rule that is applied continuously, you need some differential equations. Thus, the automata space should have a smooth structure.
If I may, I will propose my own definition. For simplicity, define the state of every point in the manifold to be a vector in its tangent space. So the game evolves along a path in the section space. Also, an initial condition should be deterministic, thus, a rule is a section of the vector bundle of the section space of a manifold. That sure does sound weird, but it means that once you fixed a vector field, it will vary smoothly according to the specified rule.

MacLennan chooses this rule to respect local information, so your manifold has to have a metric. A Riemann or Lorenz metric are suitable in this case. A certain radius is fixed and the state of a point is based in some way on the integral over the ball centered at the point.