TY - JOUR

T1 - The enumerability of P collapses P to NC

AU - Beygelzimer, Alina

AU - Ogihara, Mitsunori

N1 - Funding Information:
We would like to thank Joachim von zur Gathen for referring us to the results on the parallel complexity of polynomial factorization. We would also like to thank Eric Allender for pointing out an alternative interpretation of the results in terms of resource-bounded Kolmogorov complexity. This work was supported in part by NSF Grants EIA-0080124 and EIA-0205061.

PY - 2005/11/22

Y1 - 2005/11/22

N2 - 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). In other words, excluding any possible solution in this case is as hard as actually finding the solution. 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 NC1 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).

AB - 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). In other words, excluding any possible solution in this case is as hard as actually finding the solution. 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 NC1 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).

KW - Circuit value problem

KW - Complexity theory

KW - Enumerative approximations

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U2 - 10.1016/j.tcs.2005.07.010

DO - 10.1016/j.tcs.2005.07.010

M3 - Article

AN - SCOPUS:27644438779

VL - 345

SP - 248

EP - 259

JO - Theoretical Computer Science

JF - Theoretical Computer Science

SN - 0304-3975

IS - 2-3

ER -