Talk about a straw man argument! He builds up an entire argument, then shoots it down, and manages to generate pages of text, all based on a flawed premise. I don't agree that you can ignore the size of the working set when defining complexity.
Quite frankly it's idiotic to do so. If you allow that, then why not just create every single possible permutation of every single possible thing? (Not randomly, start at 1 and start counting.)
The complexity of that is 1 bit.
So, basically everything, no matter what, has a complexity of 1 bit.
What a pointless way of defining complexity. (Pointless for talking about the universe anyway, maybe it's useful in more specific math.)
Everything does not have a complexity of one bit. The object you describe with a complexity of 1 bit doesn't specify the object you want. In the permutation, there may be a string that specifies some universe, but then you have to specify where in the permutation that string is and how long it is, thus increasing the complexity required to describe that particular universe object.
The way he defined for most of that essay, it is more proper to speak of the "Kolmogorov complexity of the multiverse". The infinite possible starting conditions of your brand of physics merely adds the K-complexity of the TM necessary to describe the possible initial states, likely quite small. (Eliezer believes that many-worlds is the only possibility for the universe we live in, I believe.)
The K-complexity of this specific universe is much higher, because it would necessarily require a description of the multiverse and some sort of identifier of how to pick out the universe of interest.
I don't understand this. If the multiverse is a system of a certain (appearantly low) complexity, it should not be allowed to have sub-systems like the "universe of interest" which have a higher complexity. Also, I don't see why the information which identifies a certain subsystem/universe should only be available from the point of view of that subsystem, increasing its complexity, but not the complexity of the system it's a part of.
It's the nature of K-complexity. If you don't understand, I'd suggest (in all humility and seriousness) pondering the nature of TMs and K-complexity further.
Some time spent surfing through the Mandelbrot set can be enlightening, as a more concrete example. Remember, no matter what you see in Fractint (or your choice of fractal generator), the K-complexity of the Mandelbrot set is very, very low. (Not quite as small as the Mandelbrot-generation process alone because you also need to describe the coloring algorithm, but that's not that much either.) But, to uniquely identify the image you are looking at to someone else, you must transmit not just the Mandelbrot generation routines, but also the coordinates you are looking at, or a description of the routine you used to get there. This can easily be larger than the rather small Mandelbrot algorithm. (It's easy to lose track of how deep you are in the M-set, without realizing that your computer is chugging away on computations involving thousands of significant digits...)
This is exponentially (super-exponentially?) more true to pick out a particular piece of a particular universe from a TM simulating all possible string theory universes. String theory may be simple (or may not be), by the time you're done identifying which of the 10^120 vacuum states you want to deal with (~400 bits right there), which initial conditions you want to deal with (no idea what that would take), and where in space and time you wish to point at (many thousands of bits minimum, no known upper limit), you can easily exceed the size of the part of the TM that describes the physics itself.
It might be helpful to try to forget everything you know about "complexity"; the English meaning of the word misleads your mathematical intuition. K-complexity is really something completely different (as is part of Elizier's point), and, frankly, it's much less useful than it seems at first blush. It's part of the wild world of Turing Machines, which can not be tamed or understood by any finite being. (And it doesn't really help that you can't prove if you have the optimal TM for a given result.)
It may also help to intuitively consider the difficulty of "pointing" at something, as in the essay. It is easy to gesticulate wildly at the Earth, from where you sit now. It is far, far harder to unambiguously specify which protozoan you are talking about right now. The part of the description that filters through the near-infinite possibilities to uniquely identify the topic of interest can be very, very large, and can easily exceed the size of the specification of "all possible topics of interest".
I think I do understand most of the technicalities, using Gregory Chaitin's ideas about the properties of self-delimiting programs as a bridge. What I don't get is what I percieve of as the insistence on "surprises" in the "universe of interest" that can't be seen "from outside". Zooming through a Mandelbrot set visualization, while it can be a great experience, ultimately offers you few surprises, and because there are few surprises, you know that the complexity - represented by the original equation - must be low. And Eliezer seems to be saying the opposite, namely that the are lots of surprises in a particular universe that cannot be known from the system as a whole, because the system as a whole only contains 400 bits of information. If you would write this multiverse program as self-delimiting, the way Chaitin does, where would all the extra complexity/information/surprises found in the "universe of interest" come from?
You keep saying "complexity". There's a reason I keep writing "K-complexity". The extra K-complexity comes from the extra bits needed to narrow down the results. The whole multiverse does have English-complexity (the conventional meaning of the word, not the measure of how many words it takes to describe something which would just be K-complexity again) greater than the part... but English-complexity is ill-defined.
Look at a word-processing document. A real one, sitting on your hard drive. The program to output all possible documents is very, very simple. The specification of how to get to the exact document you are looking it is the (compressed) size of the document itself. The English-complexity of "the set of all word processing documents" is high, but the K-complexity is low. The English-complexity of "one particular document" is low, but the K-complexity is quite high.
You might say, "Well, I simply tell you to simulate the multiverse, then hand you instructions on how to get to that document", but the instructions will be of a very non-trivial size. I think you intuitively see the instructions as very small, but they are actually huge. Starting with just "Simulate the multiverse" leaves me with, quite literally, a multiverse in hand. Now what? Now how do I find what you are talking about? I'm worse off than when I had nothing at all!
When you have a gigantic set, simply the act of indicating a member within it takes bits. K-complexity measures those bits. English-complexity says you're lowering the complexity. Neither is wrong... it's a definitional matter.
If that's what he's talking about, then we'd need many less then 500 bits: we might as well run the machine that enumerates every Turing machine, and then runs one step for the first n machines, then the first n+1 machines, etc. This is as much a multiverse as any.
Yes, it is, but since you've radically increased the size of your space, you'll need yet more bits to describe a specific thing than if you start from a TM that only describes multiverses like ours.
Due to the way K-complexity is defined, there is no escape. As you increase the size of the space you are considering, the bits needed to cut the space back down to something specific grow in proportion.
Isn't many worlds worse? The universe forks at every step! At least with the reduction to the generating turn machine we have only a bounded number of bits the description truncates.
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[ 3.4 ms ] story [ 35.5 ms ] threadQuite frankly it's idiotic to do so. If you allow that, then why not just create every single possible permutation of every single possible thing? (Not randomly, start at 1 and start counting.)
The complexity of that is 1 bit.
So, basically everything, no matter what, has a complexity of 1 bit.
What a pointless way of defining complexity. (Pointless for talking about the universe anyway, maybe it's useful in more specific math.)
The K-complexity of this specific universe is much higher, because it would necessarily require a description of the multiverse and some sort of identifier of how to pick out the universe of interest.
Some time spent surfing through the Mandelbrot set can be enlightening, as a more concrete example. Remember, no matter what you see in Fractint (or your choice of fractal generator), the K-complexity of the Mandelbrot set is very, very low. (Not quite as small as the Mandelbrot-generation process alone because you also need to describe the coloring algorithm, but that's not that much either.) But, to uniquely identify the image you are looking at to someone else, you must transmit not just the Mandelbrot generation routines, but also the coordinates you are looking at, or a description of the routine you used to get there. This can easily be larger than the rather small Mandelbrot algorithm. (It's easy to lose track of how deep you are in the M-set, without realizing that your computer is chugging away on computations involving thousands of significant digits...)
This is exponentially (super-exponentially?) more true to pick out a particular piece of a particular universe from a TM simulating all possible string theory universes. String theory may be simple (or may not be), by the time you're done identifying which of the 10^120 vacuum states you want to deal with (~400 bits right there), which initial conditions you want to deal with (no idea what that would take), and where in space and time you wish to point at (many thousands of bits minimum, no known upper limit), you can easily exceed the size of the part of the TM that describes the physics itself.
It might be helpful to try to forget everything you know about "complexity"; the English meaning of the word misleads your mathematical intuition. K-complexity is really something completely different (as is part of Elizier's point), and, frankly, it's much less useful than it seems at first blush. It's part of the wild world of Turing Machines, which can not be tamed or understood by any finite being. (And it doesn't really help that you can't prove if you have the optimal TM for a given result.)
It may also help to intuitively consider the difficulty of "pointing" at something, as in the essay. It is easy to gesticulate wildly at the Earth, from where you sit now. It is far, far harder to unambiguously specify which protozoan you are talking about right now. The part of the description that filters through the near-infinite possibilities to uniquely identify the topic of interest can be very, very large, and can easily exceed the size of the specification of "all possible topics of interest".
Look at a word-processing document. A real one, sitting on your hard drive. The program to output all possible documents is very, very simple. The specification of how to get to the exact document you are looking it is the (compressed) size of the document itself. The English-complexity of "the set of all word processing documents" is high, but the K-complexity is low. The English-complexity of "one particular document" is low, but the K-complexity is quite high.
You might say, "Well, I simply tell you to simulate the multiverse, then hand you instructions on how to get to that document", but the instructions will be of a very non-trivial size. I think you intuitively see the instructions as very small, but they are actually huge. Starting with just "Simulate the multiverse" leaves me with, quite literally, a multiverse in hand. Now what? Now how do I find what you are talking about? I'm worse off than when I had nothing at all!
When you have a gigantic set, simply the act of indicating a member within it takes bits. K-complexity measures those bits. English-complexity says you're lowering the complexity. Neither is wrong... it's a definitional matter.
Due to the way K-complexity is defined, there is no escape. As you increase the size of the space you are considering, the bits needed to cut the space back down to something specific grow in proportion.