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Comment by Steuard | Hacker News Reader

The really neat physics argument limiting Moore's law (to my eye) is the ultimate limit on the information content in a given region of space. It turns out to be impossible to pack entropy (and thus information) more densely than in a black hole. Because the entropy of a black hole scales with surface area rather than volume, Moore's law runs up against the maximum number of bits containable within the solar system (or even the galaxy or the entire observable universe) surprisingly quickly. I haven't looked at the numbers lately, but I seem to recall that those absolute limits show up within a few centuries, tops.

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Here's some napkin math:

Moore's law says that the number of transistors on a chip doubles approximately every two years. The Core i7 Sandy Bridge has ~2.3 billion transistors [1], and measures 434 mm^2 [1]. Silicon has a density of 2.4g/cm^3 [2] or .0024g/mm^3. Silicon's atomic weight is 28 --- meaning ~610^23 atoms of Silicon weigh 28 grams --- which means a single atom weighs 4.710^-23g.

Let us pretend that Si crystals are simple-cubic (they're actually diamond-cubic [3]). In that case, a 1x1x1mm cube of crystal weighs 2.410^-3g == 5.210^19 atoms. The cube root of this number, 3.710^6, is the number of atoms on a side, so the number of atoms on a mm^2 face is 1.410^13.

The chip is 434 mm^2, so the face has about 6.1*10^15 atoms on it, otherwise known as 6 million billion.

For roundness, say we have 2 billion transistors on the chip today. Log_2(6,000,000) == 22.

If Moore's law continues until the year 2053, there would be more transistors than atoms on the surface of the chip :-)

[1] http://www.tomshardware.com/reviews/core-i7-3960x-x79-sandy-...

[2] http://en.wikipedia.org/wiki/Silicon

[3] http://en.wikipedia.org/wiki/Diamond_cubic

That is for what is essentially a 2D chip surface. Certainly layering chips has not been a viable option so far but look at the various "nano" machines that have been in science news, today a micron-sized modle truck, perhaps the next/next gen chips will have fractal surfaces that extend/configure/grow dynamically.

Do away with your heat dissipation problem by going to reversible computing, and you aren't restricted to surface atoms anymore.

The black hole argument goes way beyond that, though. It's the ultimate physical limit of the universe, rather than what's potentially practical or imaginable. Even a computer built of neutronium might not reach that limit.

Long live the UL-class computations!

I googled but didn't find anything I understood. Can you explain without giving a course in computation theory?

You can think about decision problems (such that you answer "yes" or "no" to a question) as trying to find a path in (very specific) a graph. UL consists (in a hand-waving fashion) of problems that can be stated in such way by undirected graphs of polynomial size. Roughly it means "if I'm in state s and can go to s' I can also go back from s' to s", which is a reversible computation -- reversible processes are not required by thermodynamics to create heat.

In my experience, UL is "unambiguous log", decision problems that require no more than logarithmic space, and also have a single unique proof (if it exists).

It seems like UL-computers would be optimized for finding the path through the maze to the cheese, but log space is a pretty severe restriction...

Undirected st-connectivity is UL complete (and by showing that it's in L it was shown that UL = L). Logarithmic space is just an arbitrary constraint. It could be polynomial or exp-space.

Could you explain a bit more about the black hole argument?

Wish I could, I don't really know any more than what Steuard posted. Perhaps he could expand on it a bit, if he sees this.

Here's some napkin math on the black hole argument. Information is essentially equivalent to entropy: the entropy S = k * log M, where k is Boltzmann's constant and log M is the number of bits encoding the information. Now, it turns out to be fundamentally impossible to store more entropy in any region of space than would be contained in a black hole of the same size: S_max = (k A c^3)/(4 G hbar), where A is the "surface area" of the region, c is the speed of light, G is Newton's gravitational constant, and hbar is Planck's constant. That means the maximum number of bits of information in a region of space is (A c^3)/(4 G hbar), or about A * (10^69 1/m^2). Our solar system is maybe 1 light year in size: about 10^16 m, so its "surface area" is 4pi r^2 ~ 10^33 m^2. That means that the solar system can hold at most 10^102 bits of information.

If Moore's law says that computer memory doubles in capacity every 2 years, then they'll go up by a factor of 1000 about every 20 years. We currently have computers that hold about 10^12 bits of information, so just 90*20 = 1800 years would get up to that absolute limit, regardless of what technology we used, unless our single computer was bigger than a light year in radius.

Whoops! I had a feeling I was being careless at the end there. It should have been 30x20 years rather than 90x20, so just 600 years until Moore's law is absolutely finished regardless of technology. (That shouldn't feel all that much closer, but it does.)

Its not actually obvious that you can't have more transistors than atoms, but I'm not sure it isn't possible in theory. I can imagine electrons in different orbitals having different interactions, or having for different energy levels in an interaction to effectively double the number of transistors.

There are theoretical limits to the density of useful computation, but they don't correspond neatly to atoms or such.

Well, for example say you are using light based computing and transistors. Light doesn't interfere with itself for the most part so you can reuse the same transistors to add more cores by sending more channels of light through. You just need to add a gateway/filter at the end to let them communicate. Viola 1 million channels at maximum density gives a few hundred thousand times as many transistors as atoms.

Thanks for reminding me of the totally mind-blowing fact that information density of a black hole is proportional to its surface area instead of volume.

Some have argued this makes the universe analogous to a hologram: a 3d projection out of 2d.

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