 
Summary: Amplifying Lower Bounds by Means of
SelfReducibility
ERIC ALLENDER
Rutgers University
and
MICHAL KOUCK ´Y
Academy of Sciences of the Czech Republic
We observe that many important computational problems in NC1 share a simple selfreducibility
property. We then show that, for any problem A having this selfreducibility property, A has
polynomialsize TC0 circuits if and only if it has TC0 circuits of size n1+ 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 NC1 and
has the selfreducibility property. It follows from a lower bound of Impagliazzo, Paturi, and Saks,
that BFE requires depth d TC0 circuits of size n1+ d . If one were able to improve this lower bound
to show that there is some constant > 0 (independent of the depth d) such that every TC0 circuit
family recognizing BFE has size at least n1+ , then it would follow that TC0 = NC1. We show
that proving lower bounds of the form n1+ 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 ACC0, TC0 and NC1 via existing "natural" approaches to proving circuit lower bounds.
We also show that problems with small uniform constantdepth circuits have algorithms that
