Now that I think about it, it should be obvious the size of dependency cannot be fixed, otherwise pi would be periodic! (there are finitely many fixed size 'parents', so it must recur)
I should also note that the straightforward interpretation in this case is that of a temporal neighborhood for a Cellular Automaton! That is, dependence of several states back in time, and 0 space dimensions. You can also think of a 1D CA if you introduce a special state that signals the "expansion" of the digits of pi (which digit we're currently expanding)
This also enlightens me in the bizarre concept of multiple time dimensions. If you start with a 2D field, and use the same technique of keeping track of the current active expanding cells (i.e. "current time"), starting from a single active cell (time 0) in a top-left corner, then you can expand cells across a diagonal, and they depend on previous states in two different directions.
I should also note that the straightforward interpretation in this case is that of a temporal neighborhood for a Cellular Automaton! That is, dependence of several states back in time, and 0 space dimensions. You can also think of a 1D CA if you introduce a special state that signals the "expansion" of the digits of pi (which digit we're currently expanding)
This also enlightens me in the bizarre concept of multiple time dimensions. If you start with a 2D field, and use the same technique of keeping track of the current active expanding cells (i.e. "current time"), starting from a single active cell (time 0) in a top-left corner, then you can expand cells across a diagonal, and they depend on previous states in two different directions.