Lower Bounds for Constant Depth Circuits for Prefix Problems

A prefix-or circuit has n inputs and n outputs; the ith output is the OR of the first i inputs. A prefix-carry circuit has 2n inputs, interpreted as two n-bit numbers, and n outputs; the ith output is the carry in the ith position of the sum of the two numbers. We show a nonlinear lower bound for constant-depth, unboundedfanin implementations of prefix-or. However, with negation, linear size circuits are possible. For prefix-carry, we show nonlinear lower bounds for arbitrary circuits. In both cases the lower bounds exhibit a size/depth tradeoff: the circuit size must be at least Ω(nf d −1 d(n)) for depth a constant times d. Here the functions f d form an increasing hierarchy coextensive with the primitive recursive functions. The lower bounds match the known upper bounds for these problems, to within a constant factor for depth.