Exact solution for a 1 + 1 etching model

We present a method to derive analytically the growths exponents of a surface of 1 + 1 dimensions whose dynamics is ruled by cellular automata. Starting from the automata, we write down the time evolution for the height's average and height's variance (roughness). We discuss the existence of a Probability distribution for the congurations. We apply the method to the etching model[1,2] than we obtain the dynamical exponents, which perfectly match the numerical results obtained from simulations. Those exponents are exact and they are the same as those exhibited by the KPZ model[3] for this dimension. Therefore, it shows that the etching model and KPZ belong to the same universality class[4]. Moreover, we proof that in the continuous limit the majors terms leads to KPZ [5].