The lower bound amplification using downward self-reducibility shows that
proving n^{1+epsilon} lower bounds for TC^0 circuits computing the usual NC^1 complete
problems implies separation of TC^0 from NC^1. Similar results can be obtained for other
circuit classes such as ACC^0, CC^0, etc. In this talk I will discuse the implications of these
results for natural proofs. Further using downward selfreducibility I will show a lower bound
on the size of a reduction between NP-complete problems. Time permiting I may demonstrate
that ACC^0 is almost identical with CC^0.