Logic circuits [from cellular automata]

The rules for the cellular automaton shown here are

{{0, 1, 1 | 3}  1, {0, 3, 3}  3, {1, 0, 0 | 1 | 3}  1, {1, 1, 3}  4, {1, 3, 0}  3, {1, 3, 3}  2, {2, 1, 3}  3, {2, 3, 0}  2, {2, 0, _}  4, {3, 3, 0}  3, {4, 0, 0 | 1 | 2 | 4}  2, {4, 3, 3}  3, {4, 1, 3}  1, {4, 3, 0}  4, {_, _, _}  0}

The initial conditions are given by

Flatten[Block[{And, Or}, Map[{0, 2 (# + 1)} &, expr, {-1}] //. {!x_  {0, x, 0}, And[x__]  {0, 0, 1, 0, x, 1, 3, 0, 0}, Or[x__]  {0, 0, 1, 0, x, 0, 1, 3, 0}}]]

and in terms of these initial conditions the cellular automaton must be run for Length[list //. {0, x__}  {x}] - 1 steps in order to find the result.