Directed percolation conjecture for cellular automata revisited

The directed percolation (DP) hypothesis for stochastic, range-4 cellular automata with acceptance rule $y \le\sum_{j=-4}^4 s_{i-j} \le 6$, in cases of $y < 6$ was investigated in one and two dimensions. Simulations, mean-field renormalization group and coherent anomaly calculations show that in one dimension the phase transitions for $y<4$ are continuous and belong to the DP class for $y=4,5$ they are discontinuous. The same rules in two dimensions for $y=1$ show $2+1$ dimensional DP universality; but in cases of $y > 1$ the transitions become first order.