Tomáš Gavenčiak

A researcher in CS theory, AI safety and other stuff.


The Solomonoff Prior is Malign

Complexity indeed matters: the universe seems to be bounded in both time and space, so running anything like Solomonoff prior algorithm (in one of its variants) or AIXI may be outright impossible for any non-trivial model. This for me significantly weakens or changes some of the implications.

A Fermi upper bound of the direct Solomonoff/AIXI algorithm trying TMs in the order of increasing complexity: even if checking one TM took one Planck time on one atom, you could only check cca 10^250=2^800 machines within a lifetime of the universe (~10^110 years until Heat death), so the machines you could even look at have description complexity a meager 800 bits.

  • You could likely speed the greedy search up, but note that most algorithmic speedups do not have a large effect on the exponent (even multiplying the exponent with constants is not very helpful).
  • Significantly narrowing down the space of TMs to a narrow subclass may help, but then we need to take look at the particular (small) class of TMs rather than have intuitions about all TMs. (And the class would need to be really narrow - see below).
  • Due to the Church-Turing thesis, any limiting the scope of the search is likely not very effective, as you can embed arbitrary programs (and thus arbitrary complexity) in anything that is strong enough to be a TM interpreter (which the universe is in multiple ways).
  • It may be hypothetically possible to search for the "right" TMS without examining them individually (witch some future tech, e.g. how sci-fi imagined quantum computing), but if such speedup is possible, any TMs modelling the universe would need to be able to contain this. This would increase any evaluation complexity of the TMs, making them more significantly costly than the Planck time I assumed above (would need a finer Fermi estimate with more complex assumptions?).