I’ve always thought of it like, it doesn’t rely on the universe being computable, just on the universe having a computable approximation. So if the universe is computable, SI does perfectly, if it’s not, SI does as well as any algorithm could hope to.
Not really. It’s superior to all algorithms running on a Turing machine, and actually is superior to an algorithm running on a accelerating turing machine, because it actually has the complement of the recursively enumerable sets, since it’s a 1st level halting oracle, which is very nice, but compared to the most powerful computers/reasoning engines known to mathematics, it’s way less powerful. It’s optimal in the domain of computable universes, and with the resources available to a Solomonoff Inductor, it can create a halting oracle, which lets it predict the first level uncomputable sequences like Chaitin’s constant, but not anything more, which is a major limitation compared to what the champions/currently optimal machines are for reasoning.
So depending on how much compute we give the human in the form of intuition, it could very easily beat Solonomoff Induction.
which lets it predict the first level uncomputable sequences like Chaitin’s constant
Do you have a proof/source for this? I haven’t heard it before.
I know in particular that is assigns a probability of 0 to Chaitin’s constant (because all the hypotheses are computable). Are you saying it can predict the prefixes of Chaitin’s constant better than random? I haven’t heard this claim either.
Hm, I might have confused what you are allowed to do if you had enough compute to run Solonomoff Induction with Solomonoff induction itself, so that’s maybe the issue I had here.
If I wanted to create my own argument for why Solomonoff induction could do it, it’s because that it’s essentially a halting oracle, which allows it to compute/create all the digits of Chaitin’s constants, given that it can compute in general the recursively enumerable sets, and since it can compute all the digits of Chaitin’s constants, it can basically read them like a book, and thus it has predicted the sequence of digits.
Solomonoff induction is a specific probability distribution. It isn’t making “decisions” per se. It can’t notice that it’s existence implies that there is a halting oracle, and that it therefore can predict one. This is because, in general, Solomonoff induction is not embedded.
If there was a physical process for a halting oracle, that would be pretty sick because then we could just run Solomonoff induction. As shown in my post, we don’t need to worry that there might be an even better strategy in such a universe; the hypotheses of Solomonoff induction can take advantage of the halting oracle just as well as we can!
As shown in my post, we don’t need to worry that there might be an even better strategy in such a universe; the hypotheses of Solomonoff induction can take advantage of the halting oracle just as well as we can!
You do mention that the methods of reasoning have to be computable for this to work, and there I’m quite a bit more skeptical of that condition holding.
Not really. It’s superior to all algorithms running on a Turing machine, and actually is superior to an algorithm running on a accelerating turing machine, because it actually has the complement of the recursively enumerable sets, since it’s a 1st level halting oracle, which is very nice, but compared to the most powerful computers/reasoning engines known to mathematics, it’s way less powerful. It’s optimal in the domain of computable universes, and with the resources available to a Solomonoff Inductor, it can create a halting oracle, which lets it predict the first level uncomputable sequences like Chaitin’s constant, but not anything more, which is a major limitation compared to what the champions/currently optimal machines are for reasoning.
So depending on how much compute we give the human in the form of intuition, it could very easily beat Solonomoff Induction.
Do you have a proof/source for this? I haven’t heard it before.
I know in particular that is assigns a probability of 0 to Chaitin’s constant (because all the hypotheses are computable). Are you saying it can predict the prefixes of Chaitin’s constant better than random? I haven’t heard this claim either.
Hm, I might have confused what you are allowed to do if you had enough compute to run Solonomoff Induction with Solomonoff induction itself, so that’s maybe the issue I had here.
If I wanted to create my own argument for why Solomonoff induction could do it, it’s because that it’s essentially a halting oracle, which allows it to compute/create all the digits of Chaitin’s constants, given that it can compute in general the recursively enumerable sets, and since it can compute all the digits of Chaitin’s constants, it can basically read them like a book, and thus it has predicted the sequence of digits.
Solomonoff induction is a specific probability distribution. It isn’t making “decisions” per se. It can’t notice that it’s existence implies that there is a halting oracle, and that it therefore can predict one. This is because, in general, Solomonoff induction is not embedded.
If there was a physical process for a halting oracle, that would be pretty sick because then we could just run Solomonoff induction. As shown in my post, we don’t need to worry that there might be an even better strategy in such a universe; the hypotheses of Solomonoff induction can take advantage of the halting oracle just as well as we can!
You do mention that the methods of reasoning have to be computable for this to work, and there I’m quite a bit more skeptical of that condition holding.