Two coins flipped at equal velocity and trajectory in vacuum will flip an equal number of times. Thus, the process of a coin flip is completely deterministic since it depends on newtonian physics. So, the probability is defined by the randomness of the conditions, not the process itself.

When the conditions are controlled then so is the probability. The results of this research should not be surprising despite the tone of the article.

You also could have air motions, electric fields and so on passing through the trajectories of the coins. Unless you account for everything on atomic level, you have all that uncertainty as random noise in you measured outcome. So indeed it is a random variable. Biased, yes, but random.

Actually, even if you can have data at atomic level you still have chaos theory and butterfly effect preventing you from making prediction. Then even some small fluctuation born by the Plank's "uncertainty principle" can grow up in time to big enough error.

I consider the act of flipping the coin part of the process.

Not correct.

If it was random, the output does not depend on the input (or there is no meaningful input).

But they showed that the output depends on the input. So it's NOT random.

Biased random means the result is not uniform, but it does not depends on the input.

No so here. Here it's not biased - it simply depends on the input.

(Obviously pseudo random number generators depend on the input - but those explicitly are not random, they are simply useful.)

It's not a completely deterministic process, and I have no other way to characterize non-determinism. While the initial conditions bias the result, the result is not determined by the initial conditions. The result is still probabilistic.

Let's assume we have a function true_random() that returns a true random value between 0 and 1. Now let's define f(x) as:

  def f(x):
    r = true_random()
    if r < 0.5: return r
    else: return x

Half the time, it returns a random value between 0 and 1. The other half, it returns its input. This is still a random process.

Half the time it returns a value between 0 and .5 ...

I upvoted, but between .5 is still less than 1. He just over estimated :)

There is a nice way to define randomness by determining if given a string generated by the process (i.e. 010101011110101)can you write a program smaller than the string that will output that string for any string generated by the process.

This is called Kolmogorov randomness and has nothing to do with how input effects the output. In this instance, any string generated by a million coin tosses couldn't be reproduced by an algorithm that is smaller than the string, in general. By this metric I'd argue that the process is still random.

Just because the input has an effect on the output, doesn't make the output less random. "F(x) = x + rand()" is still a random function.