  The first-order theory of a string automatic structure is known to be
decidable, but there are examples of string automatic structures with
nonelementary first-order theories. We prove that the first-order theory of a
string automatic structure of bounded degree is decidable in doubly exponential
space (for injective automatic presentations, this holds even uniformly). This
result is shown to be optimal since we also present a string automatic
structure of bounded degree whose first-order theory is hard for 2EXPSPACE. We
prove similar results also for tree automatic structures. These findings close
the gaps left open in a previous paper of the second author by improving both,
the lower and the upper bounds.
