Weak length induction and slow growing depth boolean circuits
<|reference_start|>Weak length induction and slow growing depth boolean circuits: 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.<|reference_end|>
@article{kuroda1999weak,
    title={Weak length induction and slow growing depth boolean circuits},
    author={Satoru Kuroda},
    journal={arXiv preprint arXiv:cs/9907022},
    year={1999},
    archivePrefix={arXiv},
    eprint={cs/9907022},
    primaryClass={cs.LO}
}