Amplifying lower bounds by means of self-reducibility

Eric Allender, Michal Koucký
2010 Journal of the ACM  
We observe that many important computational problems in NC 1 share a simple self-reducibility property. We then show that, for any problem A having this self-reducibility property, A has polynomial size TC 0 circuits if and only if it has TC 0 circuits of size n 1+ for every > 0 (counting the number of wires in a circuit as the size of the circuit). As an example of what this observation yields, consider the Boolean Formula Evaluation problem (BFE), which is complete for NC 1 . It follows from
more » ... a lower bound of Impagliazzo, Paturi, and Saks, that BFE requires depth d TC 0 circuits of size n 1+ d . If one were able to improve this lower bound to show that there is some constant > 0 such that every TC 0 circuit family recognizing BFE has size n 1+ , then it would follow that TC 0 = NC 1 . We also show that problems with small uniform constant-depth circuits have algorithms that simultaneously have small space and time bounds. We then make use of known time-space tradeoff lower bounds to show that SAT requires uniform depth d TC 0 and AC 0 [6] circuits of size n 1+c for some constant c depending on d.
doi:10.1145/1706591.1706594 fatcat:on7yqocrijckhbbamxiui3aolu