Knuth's $R4$ definition of a random sequence is called "too weak" and he presents an example of why --- second paragraph after the definition on the previous link.
Definition $R4$. A $[0..1)$ sequence $⟨U_n⟩$ is said to be “random” if, for every effective algorithm that specifies an infinite sequence of distinct nonnegative integers $s_n$ for $n ≥ 0$, the subsequence $U_{s_0},U_{s_1},U_{s_2}$, ... corresponding to this algorithm is $∞$-distributed.
Can you explain why he concludes that the definition is too weak? You should read definition $R4$ and the second paragraph after $R4$, where he describes a sequence based on a famous gambling system and seems to conclude that definition $R4$ allows for this sequence, which a good definition of random sequence should not allow.
That's my understanding. My trouble is. I don't see how the sequence built out of the gambling system is allowed by definition $R4$. It must be because any subsequence extracted from the gambling system sequence is indeed $k$-distributed for any natural $k \ge 1$. How does this follow?
To answer this, you'll need to have the gambling sequence clearly in mind and it's better if you read his words than my interpretation of it, but I must tell you what I think the sequence is like so you can help me see my misunderstandings. I think an example should make everything clearer. Bear with me as I try to come up with a correct example.
Let $U_n$ be a sequence defined over the reals $[0,1)$. Using a real random sequence, let's produce a second one (the non-random one) by modifying the random one in the following way: for each $n = 1, 2, 3, ...$, after a run of $n$ numbers less than $1/2$, we modify the next number in the random sequence to be always greater or equal to $1/2$, thus obtaining a non-random sequence. (Think of "less than $1/2$" as HEADS and "greater or equal to $1/2$" as TAILS.) Here's an example of such gambling sequence:
$(U_0 = 0.5067308904603183)\\ (U_1 = 0.8476066606357194)\\ (U_2 = 0.10461131407859581)\\ (U_3 = 0.9921332104978403)\\ (U_4 = 0.9416750166258784)\\ (U_5 = 0.8838481765334544)\\ (U_6 = 0.4309303573410759)\\ (U_7 = 0.4120206364198338)\\ (U_8 = 0.6329820276378333)\\ (U_9 = 0.5022764886900573)\\ (U_{10} = 0.1591727701267079)\\ (U_{11} = 0.9636171831359096)\\ (U_{12} = 0.6541579154005383)\\ (U_{13} = 0.679958229286436)\\ (U_{14} = 0.22302974490220356)\\ (U_{15} = 0.06599312478829408)\\ (U_{16} = 0.47280939443603953)\\ (U_{17} = 0.6504496373919595)\\ (U_{18} = 0.01796499377505824)$
In this sequence, after a run of $1$ head, you get a tail. After a run of $2$ heads, you get a tail, after a run of $3$ heads, you get a tail.
Suppose an infinite sequence has this property. We don't want to call this sequence random, but Knuth says $R4$ allows such sequence. I can't see that at all. Let's take the subsequence of only "tail" elements (those that come up after a run of $n$ heads in this crooked sequence). In the example above, that would be $U_0, U_3, U_8, U_{17}, ...$. This sequence will never be $1$-distributed because all elements are always "tails". So this is not his objection to definition $R4$ and, therefore, I don't know how else to object and don't know what's wrong with definition $R4$.