[Without having looked at the link in your response to my other comment, and I also stopped reading cubefox’s comment once it seemed that it was going in a similar direction. ETA: I realized after posting that I have seen that article before, but not recently.]
I’ll assume that the robot has a special “memory” sensor which stores the exact experience at the time of the previous tick. It will recognize future versions of itself by looking for agents in its (timeless) 0P model which has a memory of its current experience.
For p(“I will see O”), the robot will look in its 0P model for observers which have the t=0 experience in their immediate memory, and selecting from those, how many have judged “I see O” as Here. There will be two such robots, the original and the copy at time 1, and only one of those sees O. So using a uniform prior (not forced by this framework), it would give a 0P probability of 1⁄2. Similarly for p(“I will see C”).
Then it would repeat the same process for t=1 and the copy. Conditioned on “I will see C” at t=1, it will conclude “I will see CO” with probability 1⁄2 by the same reasoning as above. So overall, it will assign:
p(“I will see OO”) = 1⁄2,
p(“I will see CO”) = 1⁄4,
p(“I will see CC”) = 1⁄4
The semantics for these kinds of things is a bit confusing. I think that it starts from an experience (the experience at t=0) which I’ll call E. Then REALIZATION(E) casts E into a 0P sentence which gets taken as an axiom in the robot’s 0P theory.
A different robot could carry out the same reasoning, and reach the same conclusion since this is happening on the 0P side. But the semantics are not quite the same, since the REALIZATION(E) axiom is arbitrary to a different robot, and thus the reasoning doesn’t mean “I will see X” but instead means something more like “They will see X”. This suggests that there’s a more complex semantics that allows worlds and experiences to be combined—I need to think more about this to be sure what’s going on. Thus far, I still feel confident that the 0P/1P distinction is more fundamental than whatever the more complex semantics is.
(I call the 0P → 1P conversion SENSATIONS, and the 1P → 0P conversion REALIZATION, and think of them as being adjoints though I haven’t formalized this part well enough to feel confident that this is a good way to describe it: there’s a toy example here if you are interested in seeing how this might work.)
Then it would repeat the same process for t=1 and the copy. Conditioned on “I will see C” at t=1, it will conclude “I will see CO” with probability 1⁄2 by the same reasoning as above. So overall, it will assign:p(“I will see OO”) = 1⁄2,p(“I will see CO”) = 1⁄4,p(“I will see CC”) = 1⁄4
If we look at the situation in 0P, the three versions of you at time 2 all seem equally real and equally you, yet in 1P you weigh the experiences of the future original twice as much as each of the copies.
Suppose we change the setup slightly so that copying of the copy is done at time 1 instead of time 2. And at time 1 we show O to the original and C to the two copies, then at time 2 we show them OO, CO, CC like before. With this modified setup, your logic would conclude P(“I will see O”)=P(“I will see OO”)=P(“I will see CO”)=P(“I will see CC”)=1/3 and P(“I will see C”)=2/3. Right?
Similarly, if we change the setup from the original so that no observation is made at time 1, the probabilities also become P(“I will see OO”)=P(“I will see CO”)=P(“I will see CC”)=1/3.
Suppose we change the setup from the original so that at time 1, we make 999 copies of you instead of just 1 and show them all C before deleting all but 1 of the copies. Then your logic would imply P(“I will see C”)=.999 and therefore P(“I will see CO”)=P(“I will see CC”)=0.4995, and P(“I will see O”)=P(“I will see OO”)=.001.
This all make me think there’s something wrong with the 1⁄2,1/4,1/4 answer and with the way you define probabilities of future experiences. More specifically, suppose OO wasn’t just two letters but an unpleasant experience, and CO and CC are both pleasant experiences, so you prefer “I will experience CO/CC” to “I will experience OO”. Then at time 0 you would be willing to pay to switch from the original setup to (2) or (3), and pay even more to switch to (4). But that seems pretty counterintuitive, i.e., why are you paying to avoid making observations in (3), or paying to make and delete copies of yourself in (4). Both of these seem at best pointless in 0P.
But every other approach I’ve seen or thought of also has problems, so maybe we shouldn’t dismiss this one too easily based on these issues. I would be interested to see you work out everything more formally and address the above objections (to the extent possible).
[Without having looked at the link in your response to my other comment, and I also stopped reading cubefox’s comment once it seemed that it was going in a similar direction. ETA: I realized after posting that I have seen that article before, but not recently.]
I’ll assume that the robot has a special “memory” sensor which stores the exact experience at the time of the previous tick. It will recognize future versions of itself by looking for agents in its (timeless) 0P model which has a memory of its current experience.
For p(“I will see O”), the robot will look in its 0P model for observers which have the t=0 experience in their immediate memory, and selecting from those, how many have judged “I see O” as Here. There will be two such robots, the original and the copy at time 1, and only one of those sees O. So using a uniform prior (not forced by this framework), it would give a 0P probability of 1⁄2. Similarly for p(“I will see C”).
Then it would repeat the same process for t=1 and the copy. Conditioned on “I will see C” at t=1, it will conclude “I will see CO” with probability 1⁄2 by the same reasoning as above. So overall, it will assign: p(“I will see OO”) = 1⁄2, p(“I will see CO”) = 1⁄4, p(“I will see CC”) = 1⁄4
The semantics for these kinds of things is a bit confusing. I think that it starts from an experience (the experience at t=0) which I’ll call E. Then REALIZATION(E) casts E into a 0P sentence which gets taken as an axiom in the robot’s 0P theory.
A different robot could carry out the same reasoning, and reach the same conclusion since this is happening on the 0P side. But the semantics are not quite the same, since the REALIZATION(E) axiom is arbitrary to a different robot, and thus the reasoning doesn’t mean “I will see X” but instead means something more like “They will see X”. This suggests that there’s a more complex semantics that allows worlds and experiences to be combined—I need to think more about this to be sure what’s going on. Thus far, I still feel confident that the 0P/1P distinction is more fundamental than whatever the more complex semantics is.
(I call the 0P → 1P conversion SENSATIONS, and the 1P → 0P conversion REALIZATION, and think of them as being adjoints though I haven’t formalized this part well enough to feel confident that this is a good way to describe it: there’s a toy example here if you are interested in seeing how this might work.)
If we look at the situation in 0P, the three versions of you at time 2 all seem equally real and equally you, yet in 1P you weigh the experiences of the future original twice as much as each of the copies.
Suppose we change the setup slightly so that copying of the copy is done at time 1 instead of time 2. And at time 1 we show O to the original and C to the two copies, then at time 2 we show them OO, CO, CC like before. With this modified setup, your logic would conclude P(“I will see O”)=P(“I will see OO”)=P(“I will see CO”)=P(“I will see CC”)=1/3 and P(“I will see C”)=2/3. Right?
Similarly, if we change the setup from the original so that no observation is made at time 1, the probabilities also become P(“I will see OO”)=P(“I will see CO”)=P(“I will see CC”)=1/3.
Suppose we change the setup from the original so that at time 1, we make 999 copies of you instead of just 1 and show them all C before deleting all but 1 of the copies. Then your logic would imply P(“I will see C”)=.999 and therefore P(“I will see CO”)=P(“I will see CC”)=0.4995, and P(“I will see O”)=P(“I will see OO”)=.001.
This all make me think there’s something wrong with the 1⁄2,1/4,1/4 answer and with the way you define probabilities of future experiences. More specifically, suppose OO wasn’t just two letters but an unpleasant experience, and CO and CC are both pleasant experiences, so you prefer “I will experience CO/CC” to “I will experience OO”. Then at time 0 you would be willing to pay to switch from the original setup to (2) or (3), and pay even more to switch to (4). But that seems pretty counterintuitive, i.e., why are you paying to avoid making observations in (3), or paying to make and delete copies of yourself in (4). Both of these seem at best pointless in 0P.
But every other approach I’ve seen or thought of also has problems, so maybe we shouldn’t dismiss this one too easily based on these issues. I would be interested to see you work out everything more formally and address the above objections (to the extent possible).