 Where \(\sigma_{i}(t) \in \Sigma = \{0, 1, \ldots, k - 1\}\) is the state of cell number i at time step t; r specifies the range, that is, how many cells on any side count as neighbors for a given cell; and \(\phi\) is defined explicitly by assigning values in \(\Sigma\) to each of the \(k^{2r+1} (2r + 1)\)-tuples representing all the possible neighborhood configurations. For example, with \(r = 1, \Sigma = \{1, 0\}\), a possible transition rule \(\phi\) can be expressed as in  Fig. 4  (with 1 being represented as black, 0 as white): 