### 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).

Original language | English (US) |
---|---|

Pages (from-to) | 346-355 |

Number of pages | 10 |

Journal | Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics) |

Volume | 3153 |

State | Published - 2004 |

Externally published | Yes |

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### ASJC Scopus subject areas

- Computer Science(all)
- Biochemistry, Genetics and Molecular Biology(all)
- Theoretical Computer Science

### Cite this

**The enumerability of P collapses P to NC.** / Beygelzimer, Alina; Ogihara, Mitsunori.

Research output: Contribution to journal › Article

*Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)*, vol. 3153, pp. 346-355.

}

TY - JOUR

T1 - The enumerability of P collapses P to NC

AU - Beygelzimer, Alina

AU - Ogihara, Mitsunori

PY - 2004

Y1 - 2004

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). 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).

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). 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).

UR - http://www.scopus.com/inward/record.url?scp=35048817665&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=35048817665&partnerID=8YFLogxK

M3 - Article

VL - 3153

SP - 346

EP - 355

JO - Lecture Notes in Computer Science

JF - Lecture Notes in Computer Science

SN - 0302-9743

ER -