The class of decision problems solvable in time O(t(n)) by a nondeterministic Turing machine, as follows. The machine is given oracle access to a proof string of unbounded length.
Credited in [For94] to S. Arora, R. Impagliazzo, and U. Vazirani.
An interesting question is whether NP = PFCHK(log n) relative to all possible oracles. Fortnow [For94] observes that the answer depends on what oracle access mechanism is used.
No class.