Problems in NP have certificates which can be verified in polynomial time.
It seems conceivable that there could be problems in P which have certificates which can be verified in logarithmic time. (Such certificates would also have to be logarithmic in size, naturally.)
However, it is also conceivable that there are no such problems. I've tried to come up with examples, but none of them have worked out so far.
Have problems like this been studied and shown to exist (or not exist)?
I can't seem to find a complexity class that would describe them in the Complexity Zoo, although it's certainly possible that there's one there and I'm just having a hard time identifying it. (L and LOG usually refer to space, and neither LH nor NLOG looks like it necessarily captures the idea.)