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How close would this rank a program p with a universal Turing machine simulating p? My sense is not very close because the "same" computation steps on each program don't align.
My "most naïve formula for logical correlation" would be something like put a probability distribution on binary string inputs, treat and as random variables , and compute their mutual information.
Interesting idea.
I don't think using a classical Turing machine in this way would be the right prior for the multiverse. Classical Turing machines are a way for ape brains to think about computation using the circuitry we have available ("imagine other apes following these social contentions about marking long tapes of paper"). They aren't the cosmically simplest form of computation. For example, the (microscopic non-course-grained) laws of physics are deeply time reversible, where Turing machines are not.
I suspect this computation speed prior would lead to Boltzmann-brain problems. Your brain at this moment might be computed at high fidelity, but everything else in the universe would be approximated for the computational speed-up.
Counterpoint worth considering:
It's hard to get enough of something that almost works.
(Vincent Felitti, as quoted from In the Realm of Hungry Ghosts: Close Encounters with Addiction)
"When you talk about the New York Times, rational thought does not appear to be holding the mic"
--Me, mentally, to many people in the rationalist/tech sphere since Scott took SlateStarCodex down.
Right, I read all that. I still don't understand what it means to append two things to the list.
Here's how I understand modelLewis, modelElga, etc.
"This model represent the world as a probability distribution. To get a more concrete sense of the world model, here's a function which generates a sample from that probability distribution"
Here's how I understand your model.
"This model represents the world as a ????, which like a probability distribution but different. To get a concrete sense of the world model, here's a function which generates a sample from that probability distribution JUST KIDDING here's TWO samples".
Why can you generate two samples at once? What does that even mean?? The world model isn't quite just a stationary probability distribution, fine, what is it then? Your model isn't structured like other models, fine, but how is it structured? I'm drowning in type errors.
EDIT and I'm suggesting be really concrete, if you can, if that will help. Like come up with some concrete situation where Beauty makes a bet, or says a thing, ("Beauty woke up on Monday and said 'I think there's a 50% chance the coin came up on heads, and refuse to say there's a state of affairs about what day it presently is'") and explain what in her model made her make that bet or say that thing. Or maybe draw a picture which what her brain looks like under that circumstance compared to other circumstances.
You don't have to reply, but FYI I don't understand what ListC represents (a total ordering of events defined by a logical clock? A logical clock ordering Beauty's thoughts, or a logical clock ordering what causally can affect what, or logically affect what allowing for Newcomb-like situations? Why is there a clock at all?), how ListC is used, what concatenating multiple entries to ListC means in terms of beliefs, etc. If it's important for readers to understand this you might have to step us through (or point us to an earlier article where you stepped us through).
I don't understand what return ['Tails&Monday','Tails&Tuesday'] and ListC += outcome mean. Can you explain it more? Perhaps operationalize it into some specific way Sleeping Beauty should act in some situation?
For example, if Sleeping Beauty is allowed to make bets every time she is woken up, I claim she should bet as though she believes the coin came up with probability 1/3 for Heads and 2/3 for Tails (≈because over many iterations she'll find herself betting in situations where Tails is true twice as often as in situations where Heads is true).
I don't understand what your solution means for Sleeping Beauty. The best operationalization I can think of for "Ω={Heads&Monday, Tails&Monday&Tuesday}" is something like:
"Thanks for participating in the experiment Ms. Beauty, and just FYI it's Tuesday"
"𝕬𝖓𝖉 𝕸𝖔𝖓𝖉𝖆𝖞"
"No, it's just Tuesday"
"𝕬𝖓𝖉 𝕸𝖔𝖓𝖉𝖆𝖞 𝖆𝖘 𝖜𝖊𝖑𝖑. 𝕿𝖍𝖊𝖗𝖊 𝖈𝖆𝖓 𝖇𝖊 𝖓𝖔 𝕿𝖚𝖊𝖘𝖉𝖆𝖞 𝖜𝖎𝖙𝖍𝖔𝖚𝖙 𝖆 𝕸𝖔𝖓𝖉𝖆𝖞"
"Yep, it was Monday yesterday. Now it's Tuesday"
"𝕭𝖚𝖙 𝖎𝖙 𝖎𝖘 𝖆𝖑𝖘𝖔 𝕸𝖔𝖓𝖉𝖆𝖞, 𝖋𝖔𝖗 𝕴 𝖊𝖝𝖎𝖘𝖙 𝖆𝖘 𝖔𝖓𝖊 𝖎𝖓 𝖆𝖑𝖑 𝖕𝖔𝖎𝖓𝖙𝖘 𝖔𝖋 𝖙𝖎𝖒𝖊 𝖆𝖓𝖉 𝖘𝖕𝖆𝖈𝖊"
"Wow! That explains the unearthly blue glow!"
"𝕴𝖓𝖉𝖊𝖊𝖉"
Daniel Kahneman notes that if sperm are randomized, the chance of Hitler, Stalin, and Mao all being born boys is 1/8. Re-run the 20th century with any of them being female and you get vastly different results. That thought experiment makes me suspect our intuitions for the inevitability of history are faulty.