Size-depth trade-offs for monotone arithmetic circuits

Consider the problem of computing the product a1A(1)⋯A(t)b, where A(1),...,A(t) are n × n matrices, a and b are vectors. We show that the size s and depth d of monotone arithmetic circuits for this problem are related as s + n3d = Ω(tn3) Thus, a reduction to depth d = o(t) requires an increase from (optimal) size n2t to size n3t. A similar trade-off is shown for the evaluation of linear recurrences. © 1991.