Weak length induction and slow growing depth boolean circuits

We de ne a hierarchy of circuit complexity classes LD , whose depth are the inverse of a function in Ackermann hierarchy. Then we introduce extremely weak versions of length induction called LIND and construct a bounded arithmetic theory L2 whose provably total functions exactly corresponds to functions computable by LD circuits. Finally, we prove a non-conservation result between L2 and a weaker theory ACCA which corresponds to AC. Our proof utilizes KPT witnessing theorem.