Amplifying Lower Bounds by Means of Self-Reducibility

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+\epsilon} for every \epsilon ≫ 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 a lower bound of Impagliazzo, Paturi, and Saks, that BFE requires depth d TC^0 circuits of size n^{1+\epsilon_d}. If one were able to improve this lower bound to show that there is some constant \epsilon≫0 such that every TC^0 circuit family recognizing BFE has size n^{1+\epsilon}, then it would follow that TC^0 \not= \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.