> What do you mean by "exponentially, not asymptotically, but for finite n"? Exponential is by definition asymptotic and continues infinitely, no?
In statistics, we generally separate asymptotic bounds—which usually make guarantees only asymptotically—from finite-decay bounds like Chernoff, which decay exponentially not only in the limit, but also at each particular finite n. The second is much rarer and much more powerful.
This is an important distinction here: we are not talking about an asymptotic limit of coin flips. Each and every round of coin flips reduces the probability of not having an answer exponentially.
> I'm just saying it's not an incredible example of simulating a fair coin, because it just... isn't doing that. As an analogy: communicating with the Voyager spacecraft might be incredible, but it's not an incredible example of infinitely fast communication... because it just isn't.
Ok, no problem, it's up to you to decide if you want to use your own definition here.
But, FYI, in computability theory, it is definitely fair game and very common to "simulate" some computation with something that is either much more memory-intensive or compute-intensive, in either the deterministic or probabilistic case. For example, it's pretty common to add or remove memory and see whether the new "simulated" routine takes more or less compute power than the "standard" algorithm, and that is kind of what people are up to with the PSPACE stuff that is going on right now.
Using that lens—and this is entirely off the cuff, but in the 10 seconds of thinking, I believe probably mostly correct—this algorithm "simulates" a fair coin toss by using 1 extra bit of memory and O(p) compute, where p is the reciprocal of |0.5-<coin's bias>|. You can choose p to be infinite but you can do that for a sorting algorithm too (and that is why both sorting and this algorithm have DoS implications). Is this the best we can do? Well, that's the problem we're studying with randomness exractors: given this imperfect source of randomness, can we extract perfect randomness out of it at all?