Amplifying Lower Bounds by Means of Self-Reducibility

We observe that many important computational problems in NC¹ share a simple self-reducibility property. We then show that, for any problem A having this self-reducibility property, A has polynomial size TC⁰ circuits if and only if it has TC⁰ circuits of size n^1+isin for every isin>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 NC¹. It follows from a lower bound of Impagliazzo, Paturi, and Saks, that BFE requires depth d TC⁰ circuits of size n^{1+isin d}. If one were able to improve this lower bound to show that there is some constant isin>0 such that every TC⁰ circuit family recognizing BFE has size n^1+isin, then it would follow that TC⁰neNC¹. 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 TC⁰ and AC⁰ [6] circuits of size n^1+c for some constant c depending on d.