  We define a hierarchy of circuit complexity classes LD^i, whose depth are the
inverse of a function in Ackermann hierarchy. Then we introduce extremely weak
versions of length induction and construct a bounded arithmetic theory L^i_2
whose provably total functions exactly correspond to functions computable by
LD^i circuits. Finally, we prove a non-conservation result between L^i_2 and a
weaker theory AC^0CA which corresponds to the class AC^0. Our proof utilizes
KPT witnessing theorem.
