You don't need Kolmogorov complexity for this; simple hypothesis testing will do. The null hypothesis is that the coin is fair and the alternative is that it's

You don't need Kolmogorov complexity for this; simple hypothesis testing will do. The null hypothesis is that the coin is fair and the alternative is that it's biased. If Bob was correct, then there would simply never be any way to refute the null hypothesis of a fair coin, no matter what, since it can simply output anything at all with equal probability as anything else. In reality, that isn't how hypothesis testing works, and pretty much any standard technique (computing p-values, likelihood ratios, etc) will agree that 20 tails in a row is extremely unlikely given the null hypothesis in a way that 10 tails and 10 heads is not.