Hi Cliff,
So "Solomonoff induction", whatever that precisely is, depends on a somehow compressible universe. Do the AIXI theorems *prove* something along those lines about our universe,
AIXI and related work does not prove that our universe is compressible. Nor do they need to. The sun seems to come up most days, the text in this email is clearly compressible, laws of chemistry, biology, physics, economics and so on seem to work. So in short: our universe is VASTLY compressible.
or do they *assume* a compressible universe (i.e. do they state "IF the universe is somehow compressible, these algorithms (given infinite resources) can figure out how)?
They assume that the environment (or universe) that they have to deal with is compressible. If it wasn't they (and indeed any computer based AI system) would be stuffed. However that's not a problem as the real world is clearly compressible...
Assuming the latter, does that mean that there is a "mathematical definition of 'pattern'"? As I stated I'm not a math head, but with what little knowledge I have I find it hard to imagine "pattern" as a definable entity, somehow.
Yes, there is a mathematical definition of 'pattern' (in fact there are a few but I'll just talk about the one that is important here). It comes from Kolmogorov complexity theory and is actually quite simple. Essentially it says that something is a pattern if it has an effective description (i.e. computer program for a Turing machine) that is significantly shorter than just describing the thing in full bit by bit. So for example: For x = 1 to 1,000,000,000,000 print "1" Next Describes a string of a trillion 1's. The description (i.e. the lenght of this program above in bits) is vastly shorter than a trillion and so a string of a trillion 1's is highly compressible and has a strong pattern. On the other hand if I flipped a coin a trillion times and used that to generate a string of 0's and 1's, it would be exceedingly unlikely that the resulting string would have any description much shorter than just listing the whole thing "010011110010010010101..." Thus this is not compressible and has no pattern -- it's random. There is a bit more to the story than that but not a lot more.
OK, let's say you reward it for winning during the first 100 games, then punish it for winning / reward it for losing during the next 100, reward it for winning the next 100, etc. Can it perceive that pattern?
Clearly this pattern is computationally expressible as so it's no problem at all. Of course it will take the AI a while to work out the rules of the game and on game 101 it will be surprised to be punished for winning. And probably for games 102 and a few more. After a while it will lose a game and realise that it needs to start losing games. At game 201 is will probably again get a surprise when it's punished for losing and will take a few games to realise that it needs to start winning again. By game 301 is will suspect that it need to start losing again and will switch over very quickly. By game 401 it would probably switch automatically as it will see the pattern. Essentially this is just another rule in the game. Of course these are not exact numbers, I'm just giving you an idea of what would in fact happen if you had an AIXI system.
Given infinite resources, could it determine that I am deciding to punish or reward a win based on a pseudo-random (65536-cyclic or whatever it's called) random number generator?
Yes. It's "pseudo-random" and thus computationally expressible and so again it's no problem for AIXI. In fact AIXItl would solve this just fine with only finite resources.
And if the "compressibility of the Universe" is an assumption, is there a way we might want to clarify such an assumption, i.e., aren't there numerical values that attach to the *likelihood* of gravity suddenly reversing direction; numerical values attaching to the likelihood of physical phenomena which spontaneously negate like the chess-reward pattern; etc.?
This depends on your view of statistics and probability. I'm a Bayesian and so I'd say that these things depend on your prior and how much evidence you have. Clearly the evidence that gravity stays the same is rather large and so the probability that it's going to flip is extremely super hyper low and the prior doesn't matter to much...
I don't see what you are getting at here. You might need to explainIn fact -- would the chess-reward pattern's unpredictability *itself* be an indication of life? I.e., doesn't Ockham's razor fail in the case of, and possibly *only* in the case of, conscious beings*?
some more. (I understand Ockham's razor, you don't need to explain
that part; actually it comes up a lot in the theory behind Solomonoff
induction and AIXI...)
Thanks for your comments.
Cheers
Shane
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