  A one-dimensional cellular automaton with a probabilistic evolution rule can
generate stochastic surface growth in $(1 + 1)$ dimensions. Two such discrete
models of surface growth are constructed from a probabilistic cellular
automaton which is known to show a transition from a active phase to a
absorbing phase at a critical probability associated with two particular
components of the evolution rule. In one of these models, called model $A$ in
this paper, the surface growth is defined in terms of the evolving front of the
cellular automaton on the space-time plane. In the other model, called model
$B$, surface growth takes place by a solid-on-solid deposition process
controlled by the cellular automaton configurations that appear in successive
time-steps. Both the models show a depinning transition at the critical point
of the generating cellular automaton. In addition, model $B$ shows a kinetic
roughening transition at this point. The characteristics of the surface width
in these models are derived by scaling arguments from the critical properties
of the generating cellular automaton and by Monte Carlo simulations.
