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Joined 2026-09-21 Message
Incoming links: co-RP
One-sided error Created 2026-09-24 Updated 2026-10-18
A randomized decision algorithm has one-sided error when one answer is always correct and only the other answer can be mistaken.
The guarantee concerns each fixed input, with probability taken over the choices of the randomized algorithm. Always state which answer is protected. In polynomial identity testing, call an input positive when its polynomial is identically zero, and let acceptance mean claiming an identity. Then the possibilities are:
| Actual polynomial | Accept: claim identity | Reject: claim nonidentity |
|---|---|---|
| Identically zero | Always correct; every trial accepts. | Impossible with exact polynomial evaluation. |
| Nonzero | Possible mistake if the chosen point is a zero. | Always correct; the point is a witness. |
Thus a zero polynomial evaluation can leave uncertainty, but a nonzero polynomial evaluation settles the question. An error bound for a fixed nonzero polynomial is not the conditional probability that a particular accepted input is an identity; that latter question would need a distribution on the inputs as well.
In computational complexity theory, a polynomial-time algorithm with this orientation and false acceptance probability at most establishes membership in co-RP. Reversing the question to ask whether the polynomial is nonzero gives the RP orientation: negative instances are always rejected. The random evaluation test for a polynomial identity explains the required field, total degree, sampling and evaluation assumptions.