Abstract. Self-reducibility continues to give us new angles on attacking some of the fundamental questions about computation and complexity. 1

In this thesis, we study small, yet important, circuit complexity classes within NC 1, such as ACC 0 and TC 0. We also investigate the power of a closely related problem called Iterated Matrix Multiplication and its implications in low levels of algebraic complexity theory. More concretely, • We show that extremely modest-sounding lower bounds for certain problems can lead to non-trivial derandomization results. – If the word problem over S5 requires constant-depth threshold circuits of size n1+ɛ for some ɛ> 0, then any language accepted by uniform polynomial-size probabilistic threshold circuits can be solved in subexponential time (and more strongly, can be accepted by a uniform family of deterministic constant-depth threshold circuits of subexponential size.) – If there are no constant-depth arithmetic circuits of size n1+ɛ for the problem of multiplying a sequence of n 3-by-3 matrices, then for every constant d, black-box identity testing for depth-d arithmetic circuits with bounded individual degree can be performed in subexponential time (and even by a uniform family of deterministic constant-depth AC circuits of subexponential size).