Abstract
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+ϵ for every ϵ> 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 and has the self-reducibility property. It follows from a lower bound of Impagliazzo, Paturi, and Saks, that BFE requires depth d TC 0 circuits of size n 1+ϵ d . If one were able to improve this lower bound to show that there is some constant ϵ> 0 (independent of the depth d ) such that every TC 0 circuit family recognizing BFE has size at least n 1+ϵ , then it would follow that TC 0 ≠ NC 1 . We show that proving lower bounds of the form n 1+ϵ is not ruled out by the Natural Proof framework of Razborov and Rudich and hence there is currently no known barrier for separating classes such as ACC 0 , TC 0 and NC 1 via existing “natural” approaches to proving circuit lower bounds. 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 .