We show that searching a width k maze is complete for \Pi_k, i.e., for
the k'th level of the AC^0 hierarchy. Equivalently, st-connectivity
for width k grid graphs is complete for \Pi_k. As an application,
we show that there is a data structure solving dynamic st-connectivity
for constant width grid graphs with time bound O(\log \log n) per operation
on a random access machine. The dynamic algorithm is derived from the
parallel one in an indirect way using algebraic tools.