An environment program can just assume a value for the physical variable and then abort by failing to halt if the next action doesn’t match it.
Why would AIXI come up with something like that? Any such program is clearly more complex than one that does the same thing but doesn’t fail to halt.
Or it can assume that the physical simulation branches at time t0, when Omega prepares the box, simulate each branch it until t < t1, when the next AIXI action occurs, and then kill off the branch corresponding to the wrong action.
Once again, possible but unnecessarily complex to explain AIXI’s observations.
Or, as it has already been proposed by somebody else, it could internally represent the physical world as a set of constraints and then run a constraint solver on it, without the need of performing a step-by-step chronological simulation.
Sure, but the point is that those constraints would still be physics-like in nature. Omega’s prediction accuracy is much better explained by constraints that are physics-like rather than an extra constraint that says “Omega is always right”. if you assume a constraint of the latter kind, you are still forced to explain all the other aspects of Omega—things like Omega walking, Omega speaking, and Omega thinking, or more generally Omega doing all those things that ze does. Also, if Omega isn’t always right, but is instead right only 99% of the time, then the constraint-based approach is penalized further.
So it seems that there are plenty of environment programs that can represent the action of Omega without assuming that it violates the known laws of physics. But even if it had to, what is the problem? AIXI doesn’t assume that the laws of physics forbid retro-causality.
It doesn’t assume that, no, but because it assumes that its observations cannot be affected by its future actions AIXI is still very much restricted in that regard.
My point is a simple one:
If AIXI is going to end-up one-boxing, the simplest model of Omega will be one that used its prediction method and already predicted that AIXI would one-box.
If AIXI is going to end up two-boxing, the simplest model of Omega will be one that used its prediction method and already predicted that AIXI would two-box.
However, if Omega predicted one-boxing and AIXI realized that this was the case, AIXI would still evaluate that the two-boxing action results in AIXI getting more money than the one-boxing action, which means that AIXI would two-box.
As long as Omega is capable of reaching this relatively simple logical conclusion, Omega thereby knows that a prediction of one-boxing would turn out to be wrong, and hence Omega should predict two-boxing; this will, of course, turn out to be correct.
The kinds of models you’re suggesting, with branching etc. are significantly more complex and don’t really serve to explain anything.
It doesn’t assume that, no, but because it assumes that its observations cannot be affected by its future actions AIXI is still very much restricted in that regards.
But this doesn’t matter for Newcomb’s problem, since AIXI observes the content of the opaque box only after it has made its decision.
However, if Omega predicted one-boxing and AIXI realized that this was the case, AIXI would still evaluate that the two-boxing action results in AIXI getting more money than the one-boxing action, which means that AIXI would two-box.
Which means that the epistemic model was flawed with high probability. You are insisting that the flawed model is simpler that the correct one. This may be true for certain states of evidence where AIXI is not convinced that Omega works as advertised, but you haven’t shown that this is true for all possible states of evidence.
The kinds of models you’re suggesting, with branching etc. are significantly more complex and don’t really serve to explain anything.
They may be more complex only up to a small constant overhead (how many bits does it take to include a condition “if OmegaPrediction != NextAction then loop forever”?), therefore, a constant amount of evidence should be sufficient to select them.
Which means that the epistemic model was flawed with high probability.
You are insisting that the flawed model is simpler that the correct one. This may be true for certain states of evidence where AIXI is not convinced that Omega works as advertised, but you haven’t shown that this is true for all possible states of evidence.
Yes, AIXI’s epistemic model will be flawed. This is necessarily true because AIXI is not capable of coming up with the true model of Newcomb’s problem, which is one in which its action and Omega’s prediction of its action share a common cause. Since AIXI isn’t capable of having a self-model, the only way it could possibly replicate the behaviour of the true model is by inserting retrocausality and/or magic into its environment.
They may be more complex only up to a small constant overhead (how many bits does it take to include a condition “if OmegaPrediction != NextAction then loop forever”?), therefore, a constant amount of evidence should be sufficient to select them.
I’m not even sure AIXI is capable of considering programs of this kind, but even if it is, what kind of evidence can AIXI have received that would justify the condition “if OmegaPrediction != NextAction then loop forever”? What evidence would justify such a model over a strictly simpler version without that condition?
Essentially, you’re arguing that rather than coming up with a correct model of its environment (e.g. one in which Omega makes a prediction on the basis of the AIXI equation), AIXI will somehow make up for its inability to self-model by coming up with an inaccurate and obviously false retrocausal/magical model of its environment instead.
However, I don’t see why this would be the case. It’s quite clear that either Omega has already predicted one-boxing, or Omega has already predicted two-boxing. At the very least, the evidence should narrow things down to models of either kind, although I think that AIXI should easily have sufficient evidence to work out which of them is actually true (that being the two-boxing one).
Why would AIXI come up with something like that? Any such program is clearly more complex than one that does the same thing but doesn’t fail to halt.
Once again, possible but unnecessarily complex to explain AIXI’s observations.
Sure, but the point is that those constraints would still be physics-like in nature. Omega’s prediction accuracy is much better explained by constraints that are physics-like rather than an extra constraint that says “Omega is always right”. if you assume a constraint of the latter kind, you are still forced to explain all the other aspects of Omega—things like Omega walking, Omega speaking, and Omega thinking, or more generally Omega doing all those things that ze does. Also, if Omega isn’t always right, but is instead right only 99% of the time, then the constraint-based approach is penalized further.
It doesn’t assume that, no, but because it assumes that its observations cannot be affected by its future actions AIXI is still very much restricted in that regard.
My point is a simple one: If AIXI is going to end-up one-boxing, the simplest model of Omega will be one that used its prediction method and already predicted that AIXI would one-box. If AIXI is going to end up two-boxing, the simplest model of Omega will be one that used its prediction method and already predicted that AIXI would two-box.
However, if Omega predicted one-boxing and AIXI realized that this was the case, AIXI would still evaluate that the two-boxing action results in AIXI getting more money than the one-boxing action, which means that AIXI would two-box. As long as Omega is capable of reaching this relatively simple logical conclusion, Omega thereby knows that a prediction of one-boxing would turn out to be wrong, and hence Omega should predict two-boxing; this will, of course, turn out to be correct.
The kinds of models you’re suggesting, with branching etc. are significantly more complex and don’t really serve to explain anything.
But this doesn’t matter for Newcomb’s problem, since AIXI observes the content of the opaque box only after it has made its decision.
Which means that the epistemic model was flawed with high probability.
You are insisting that the flawed model is simpler that the correct one. This may be true for certain states of evidence where AIXI is not convinced that Omega works as advertised, but you haven’t shown that this is true for all possible states of evidence.
They may be more complex only up to a small constant overhead (how many bits does it take to include a condition “if OmegaPrediction != NextAction then loop forever”?), therefore, a constant amount of evidence should be sufficient to select them.
Yes, AIXI’s epistemic model will be flawed. This is necessarily true because AIXI is not capable of coming up with the true model of Newcomb’s problem, which is one in which its action and Omega’s prediction of its action share a common cause. Since AIXI isn’t capable of having a self-model, the only way it could possibly replicate the behaviour of the true model is by inserting retrocausality and/or magic into its environment.
I’m not even sure AIXI is capable of considering programs of this kind, but even if it is, what kind of evidence can AIXI have received that would justify the condition “if OmegaPrediction != NextAction then loop forever”? What evidence would justify such a model over a strictly simpler version without that condition?
Essentially, you’re arguing that rather than coming up with a correct model of its environment (e.g. one in which Omega makes a prediction on the basis of the AIXI equation), AIXI will somehow make up for its inability to self-model by coming up with an inaccurate and obviously false retrocausal/magical model of its environment instead.
However, I don’t see why this would be the case. It’s quite clear that either Omega has already predicted one-boxing, or Omega has already predicted two-boxing. At the very least, the evidence should narrow things down to models of either kind, although I think that AIXI should easily have sufficient evidence to work out which of them is actually true (that being the two-boxing one).