[antics]

You are 100% mistaken that this implies we had a good definition of random sequences before Knuth. In the article you link, they discuss the uniform distribution, but any distribution (and the modern notion of probability in general) absolutely depends on a mathematically precise notion of random sequences.

If you don't believe me, read the chapter. Early probability theorists (e.g., von Mises, Kolmogorov) literally started thinking about randomness in order to define probability.

EDIT: And, I don't suppose it's worth pointing out that pseudorandomness is not at all the same thing as randomness. The fact that you seem to use them interchangeably is not a good sign IMHO.

EDIT 2: Why the unexplained downvote, HN? :(

[sidww2]

I skimmed through the extract presented (didn't have time to go into detail) but I don't see a formal definition of any kind presented in the extract. Could you point me to where it is? And if it's not in the extract, then could you quote it here?

Pseudorandomness is not the same thing as randomness but most algorithms today work on pseudorandom numbers so the concept is important. My impression was that that's what you were referring to.

PS :- FYI I didn't downvote your comment. Actually upvoted as your post made me discover some new math (various notions of randomness by kolmogorov, von mises, martin-lof) :)

[antics]

So, the excerpt here is chapter 3, section 1. The actual definition happens in chapter 3 section 5 ("What Is a Random Sequence?"). I have the book at home, though, so I can't quote it here. Sorry. But the intuition is, if you have an infinite sequence of random numbers, then the numbers in all infinite subsequences should be equidistributed. So, like, if you a stream of random 0's and 1's, then if you pick only every other number, the 0's and 1's have to be equiprobable, and if you pick every third number, they still have to be equiprobable, etc. This is slightly wrong, but it's on the right track to the actual definition.

re: Pseudorandomness, the point of pseudorandomness is the following.

1. A lot of algorithms use randomness to make pathologically bad cases extremely unlikely. For example, choosing a random pivot in quicksort makes the worst case very unlikely.

2. But in a lot of cases, this leads to huge amounts of space consumption. For example, most frequency moment estimations involve a matrix of random numbers. So if you're getting those numbers from a "truly random" source, then you have to store the entire matrix, which can be huge.

3. So, a better solution is to use a pseudorandom number generator! That way you can store a seed of s bits, and do something clever, like deterministically re-generate the matrix as you need it, rather than storing it outright.

Notice though, that this is not independent of the notion of randomness! In fact they are quite intimately tied together.

[sidww2]

Your definition relies on the notion of probability though. So I'm not sure why you seemingly view Knuth's work as more fundamental than Kolmogorov's, etc.

[antics]

Because the trick Knuth pulls is to express this intuition without appealing to the definition of probability. It's quite clever.

[tel]

What's the sketch of the trick? I can define randomness by appealing to some of the same basic theory used to develop probability, but it's not really independent despite looking that way from the outside. Does Knuth do this uniquely?