  In this paper, we will consider subfractals of hyperbolic iterated function
systems which satisfy the open set condition. The subfractals will consist of
points associated with infinite strings from a subshift of finite type or sofic
subshift on the symbolic space. We find that the zeros of the lower and upper
topological pressure functions are lower and upper bounds, respectively, for
the Hausdorff, packing, lower and upper box dimensions of the subfractal.
