Hey Anders - the conference was a lot of fun. So, about the 'noise' structures - I've been worried about this for quite a few years now, but I think I have a resolution that I am maybe 90% happy with. And you're right, if you make randomly connected graphs, these are going to dominate the counting among the abstract graphs, both for a finite and infinite number of objects and relationships (and they grow quickly too - if you use a representation with N distinct objects and binary connections, then you get 2^(N*(N-1)/2)) different graphs...). If we are then trying to extract observers as the largest subset of the mathematical ensemble, it seems we are immediately foiled by this. It's not completely clear that this is a problem, perhaps the noise graphs do dominate the counting, but within the tiny residue of interesting graphs that we are actually interested in, the evolving observers dominate, which is why we are obervers - but if possible I like to go ahead and eliminate the noise structures. Generally I think this is valid because the noise structures contain very little 'real' information: as we just said, their implicit information is very slight - just random connections! Slightly more formally, this is where I like to use things like the concept of representation invariant information (imagine translating some program from one set of hardware and operating system to another - here we don't care about the nuts and bolts of how the program is realized on the millions of individual transistors, but rather the program itself) - and the noise structures have precious little representation independent information. To form a loose analogy, one could imagine in general relativity only counting over topologically distinct metrics, and not also over all the different coordinate systems that describe the same geometry (like cartesian and spherical and cylindrical and all the infinite other (and usually much less usefull) coordinates for flat space). I think Hawking is refering to something like this in his new paper, although I haven't actually read it yet. And there's another analogy in field theory - you get all sorts of infinities in the path integral due to gauge invariances in bosonic fields, so you use Faddeev and Popov's method to separate the infinities out in order to get what you want. Using this idea of only counting over the representation invariant information also allows us to count over the implicit representations of information, which makes the counting much easier since these have finite integers encodings. That said, I need to put all of this in a more formal mathematical langauge - I'm reading Jech's set theory right now, and that might help. Turing machine notation is also very tempting, but it is crucial that we don't limit ourselves to one finite machine, we need to keep on adding rules and states as time progresses... So yeah, have you read my semi-formal paper on my website? Any logical problems? (and even if not, it still needs to be experimentally verified!) I'd be interested to hear your thoughts.

Sincerely,
Travis

Just a quick note, since this really requires deep thoughts.

A noise majority is IMHO a problem, because it robs your hypothesis of the interesting property of explaining why we are observers - in a world dominated by observers one should not be surprised to be one, or expressed differently, consciousness isn't a strange thing. But in a world dominated by non-observers it becomes a very strange and unusual phenomenon.

I think representation independence is the clever idea that makes your model make sensee. I guess Chaitin's work might be relevant, since he shows that the Kolomogorov complexity is machine-equivalent up to a constant.