An entity using Solomonoff induction cannot, no matter how many digits it sees, assign the obvious hypothesis any likelyhood.
Uncomputable sequences have computable approxiomations. Solomonoff induction could do very well at predicting such sequences. However, it wouldn’t assume that they are uncomputable. Why should it? That is a bizarre hypothesis—not an “obvious” one. It is surely more likely that the observed finite prefix was generated by some computable method.
Solomonoff induction only deals with finite sequences. It doesn’t assign p=0 to any sequence consistent with its observations so far. Uncomputable sequences are necessarily inifinite—and though Solomonoff induction can’t handle them, neither can the observable universe. I think that the case that they matter remains to be made.
Uncomputable sequences have computable approxiomations. Solomonoff induction could do very well at predicting such sequences. However, it wouldn’t assume that they are uncomputable. Why should it? That is a bizarre hypothesis—not an “obvious” one. It is surely more likely that the observed finite prefix was generated by some computable method.
Being a less likely hypothesis is not the same thing as having probability zero.
Solomonoff induction only deals with finite sequences. It doesn’t assign p=0 to any sequence consistent with its observations so far. Uncomputable sequences are necessarily inifinite—and though Solomonoff induction can’t handle them, neither can the observable universe. I think that the case that they matter remains to be made.