# Lower bounds for constant depth circuits for prefix problems

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## Abstract

A prefix-or circuit has *n* inputs and *n* outputs; the *i*th output is the OR of the first *i* inputs. A prefix-carry circuit has 2*n* inputs, interpreted as two *n*-bit numbers, and *n* outputs; the *i*th output is the carry in the *i*th 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.

## Keywords

Boolean Function Boolean Circuit Parallel Random Access Machine Output Vertex Primitive Recursive Function## Preview

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