Detect period of unknown source

Question

How to detect repeating digits in an infinite sequence? I tried Floyd & Brent detection algorithm but come to nothing... I have a generator that yields

Answer 1

Evgeny Kluev's answer provides a way to get an answer that might be right.

Definition

Let's assume you have some sequence D that is a repeating sequence. That is there is some sequence d of length L such that: D_i = d_{i mod L}, where t_i is the ith element of sequence t that is numbered from 0. We will say sequence d generates D.

Theorem

Given a sequence D which you know is generated by some finite sequence t. Given some d it is impossible to decide in finite time whether it generates D.

Proof

Since we are only allowed a finite time we can only access a finite number of elements of D. Let us suppose we access the first F elements of D. We chose the first F because if we are only allowed to access a finite number, the set containing the indices of the elements we've accessed is finite and hence has a maximum. Let that maximum be M. We can then let F = M+1, which is still a finite number.

Let L be the length of d and that D_i = d_{i mod L} for i < F. There are two possibilities for D_F it is either the same as d_{F mod L} or it is not. In the former case d seems to work, but in the latter case it does not. We cannot know which case is true until we access D_F. This will however require accessing F+1 elements, but we are limited to F element accesses.

Hence, for any F we won't have enough information to decide whether d generates D and therefore it is impossible to know in finite time whether d generates D.

Conclusions

It is possible to know in finite time that a sequence d does not generate D, but this doesn't help you. Since you want to find a sequence d that does generate D, but this involves amongst other things being able to prove that some sequence generates D.

Unless you have more information about D this problem is unsolvable. One bit of information that will make this problem decidable is some upper bound on the length of the shortest d that generates D. If you know the function generating D only has a known amount of finite state you can calculate this upper bound.

Hence, my conclusion is that you cannot solve this problem unless you change the specification a bit.