[vkwhlwwc] cellular automata with two dimensions of time

consider a grid universe with one dimension of space and two dimensions of time.  array of cell states S[x,t1,t2].

initial state: (forall x, forall t1, S[x,t1,0]), (forall x, forall t2, S[x,0,t2]).

state transition function form: S[x,t1,t2] = f( S[x-1, t1-1, t2] , S[x-1, t1, t2-1] , S[x , t1-1, t2] , S[x , t1, t2-1] , S[x+1, t1-1, t2] , S[x+1, t1, t2-1] ).  six inputs (cf 8 for Conway's Game of Life).

we've chosen a time neighborhood analogous to the von Neumann neighborhood of space except forbidding looking into the future.  also possible is the Moore time neighborhood which would look at three additional inputs S[x+{-1,0,+1}, t1-1, t2-1] diagonally backward in time.

what are some interesting initial populations and state transition functions?

inspired by anecdotes of general relativity not going mathematically wrong with more than one dimension of time: it only becomes physically difficult or impossible to interpret.  we've simplified to flat space and simplified to discrete space and time.