On Circuit Complexity Classes and Iterated Matrix Multiplication

OF THE DISSERTATION On Circuit Complexity Classes and Iterated Matrix Multiplication by Fengming Wang Dissertation Director: Eric Allender In this thesis, we study small, yet important, circuit complexity classes within NC, such as ACC and TC. 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+ for some > 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+ 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).