Yeah, the log(n) is only the absolute minimum. If you’re specifying yourself mostly by location, then for there to be n different locations you need at least log(n) bits on average (but in practice more), for example.
But I think it’s plausible that the details can be elided when comparing two very similar theories—if the details of the bridging laws are basically the same and we only care about the difference in complexity, that difference might be about log(n).
Another thing, I don’t think Solomonoff Induction would give an advantage of log(n) to theories with n observers. In the post you mention taking the discrete integral of 2−log(n)=1n to get log scaling, but this seems to be based on the plain Kolmogorov complexity C(n), for which log(n) is approximately an upper bound. Solomonoff induction uses prefix complexity K(n), and the discrete integral of 2−K(n) converges to a constant. This means having more copies in the universe can give you at most a constant advantage.
(Based on reading some other comments it sounds like you might already know this. In any case, it means S.I. is even more anti-PP than implied in the post)
Yeah, the log(n) is only the absolute minimum. If you’re specifying yourself mostly by location, then for there to be n different locations you need at least log(n) bits on average (but in practice more), for example.
But I think it’s plausible that the details can be elided when comparing two very similar theories—if the details of the bridging laws are basically the same and we only care about the difference in complexity, that difference might be about log(n).
Another thing, I don’t think Solomonoff Induction would give an advantage of log(n) to theories with n observers. In the post you mention taking the discrete integral of 2−log(n)=1n to get log scaling, but this seems to be based on the plain Kolmogorov complexity C(n), for which log(n) is approximately an upper bound. Solomonoff induction uses prefix complexity K(n), and the discrete integral of 2−K(n) converges to a constant. This means having more copies in the universe can give you at most a constant advantage.
(Based on reading some other comments it sounds like you might already know this. In any case, it means S.I. is even more anti-PP than implied in the post)
Actually I had forgotten about that :)