This means that there is a non-deterministic machine which, for a given input of size n, will run in time O(f(n)) (i.e. within a constant multiple of f(n), for n greater than some value), and will always "reject" the input if the answer to the decision problem is "no" for that input, while if the answer is "yes" the machine will "accept" that input for at least one computation path. Equivalently, there is a deterministic Turing machineM that runs in time O(f(n)) and is able to check an O(f(n))-length certificate for an input; if the input is a "yes" instance, then at least one certificate is accepted, if the input is a "no" instance, no certificate can make the machine accept.

The space available to the machine is not limited, although it cannot exceed O(f(n)), because the time available limits how much of the tape is reachable.