The enumerability of P collapses P to NC

Abstract

We show that one cannot rule out even a single possibility for the value of an arithmetic circuit on a given input using an NC algorithm, unless P collapses to NC (i.e., unless all problems with polynomial-time sequential solutions can be efficiently parallelized). Thus excluding any possible solution in this case is as hard as finding the solution exactly. The result is robust with respect to NC algorithms that err (i.e., exclude the correct value) with small probability. We also show that P collapses all the way down to NC 1 when the characteristic of the field that the problem is over is sufficiently large (but in this case under a stronger elimination hypothesis that depends on the characteristic). © Springer-Verlag 2004.