Fix a version of Solomonoff's universal distribution $\mathbf M$ and consider the following procedure for generating an infinite binary sequence $\omega$.

Start with some $\omega_0$. Each subsequent element is given by $\omega_n=\arg \max_a\mathbf M(a|\omega_{1:n-1})$.

What are the properties of $\omega$? For example, is it necessarily computable? Does the answer depend on the choice of $\mathbf M$?


The equivalent formulation is the following but it does not really help me advance with the problem.

Is there a universal lower semicomputable $\lambda$-supermartingale $t$ and a binary sequence $\tilde\omega$ such that $t$ always "goes up" along $\tilde\omega$ but never succeeds?

This type of problem arises when a player plays against a reactive environment and the sequence the player observes is not exogenous. A more general question is what is the short-run behavior of $\mathbf M$ since asymptotic results do not seem to help here.

  • $\begingroup$ Would ties be broken in favor of zero? $\;$ $\endgroup$
    – user6973
    Commented Aug 23, 2013 at 0:47
  • $\begingroup$ Yes, for example, ties are be broken in favor of zero. $\endgroup$
    – Gleb
    Commented Aug 23, 2013 at 1:27

1 Answer 1


It is computable.

Although $M$ is incomputable, since $M(x)\sim 2^{-K(x)}$, the sequence $\omega$ generated by $argmax M$ would have quite low Kolmogorov complexity.

If the sequence is generated by $argmin M$, then it would be Martin-lof random.


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