OK, yeah. I agree about the three types of program, but as far as I can see the third type of program basically corresponds to Omega being a faker and/or magical.
Well, the way I see it, within the deterministic hypothetical that I 1-box, at the big bang the universe is in the initial state such that I 1-box, and within the deterministic hypothetical that I 2-box, at the big bang the universe is in the initial state such that I 2-box. A valid predictor looks at the initial state and determines what I will do, before I actually do it.
Exactly the same with AIXI, which sets up hypotheticals with different initial states (which is does by adding an universal constant of what it’s going to hypothetically do (the extra tape), which is a very, very clever hack it has to employ to avoid needing to model itself correctly), and can have (or not have) a predictor which uses the initial state—distinct—to determine what AIXI will do before it does that. It correctly captures the fact that initial states which result in different actions are different, even though the way it does so is rather messy and looks ugly.
edit: i.e. to me it seems that there’s nothing fake about the predictor looking at the world’s initial state and concluding that the agent will opt to one-box. It looks bad when for the sake of formal simplicity you’re just writing in the initial state ‘I will one box’ and then have the model of your body read that and one-box, but it seems to me it’s wrong up to a constant and not more wrong than TM using some utterly crazy tag system to run a world simulator.
OK, I think I’ve just answered your question in my response to your other comment, but I’ll give a brief version here.
If there is a bit corresponding to AIXI’s future action, then by AIXI’s assumptions that bit must not be observable to AIXI until after it takes its actions. As such, models of this sort must involve some reason why the bit is observable to Omega, but not observable to AIXI; models where the information determining Omega’s prediction is also observable to AIXI will be significantly simpler.
Ohh, and I forgot to address this:
Well, the way I see it, within the deterministic hypothetical that I 1-box, at the big bang the universe is in the initial state such that I 1-box, and within the deterministic hypothetical that I 2-box, at the big bang the universe is in the initial state such that I 2-box. A valid predictor looks at the initial state and determines what I will do, before I actually do it.
Exactly the same with AIXI, which sets up hypotheticals with different initial states (which is does by adding an universal constant of what it’s going to hypothetically do (the extra tape), which is a very, very clever hack it has to employ to avoid needing to model itself correctly), and can have (or not have) a predictor which uses the initial state—distinct—to determine what AIXI will do before it does that. It correctly captures the fact that initial states which result in different actions are different, even though the way it does so is rather messy and looks ugly.
edit: i.e. to me it seems that there’s nothing fake about the predictor looking at the world’s initial state and concluding that the agent will opt to one-box. It looks bad when for the sake of formal simplicity you’re just writing in the initial state ‘I will one box’ and then have the model of your body read that and one-box, but it seems to me it’s wrong up to a constant and not more wrong than TM using some utterly crazy tag system to run a world simulator.
OK, I think I’ve just answered your question in my response to your other comment, but I’ll give a brief version here.
If there is a bit corresponding to AIXI’s future action, then by AIXI’s assumptions that bit must not be observable to AIXI until after it takes its actions. As such, models of this sort must involve some reason why the bit is observable to Omega, but not observable to AIXI; models where the information determining Omega’s prediction is also observable to AIXI will be significantly simpler.