d. By putting the Zeno effect inside of another Zeno effect, you can work it so that even if you are looking to exclude a particular database element, and the answer *is* that element, then the computer doesn’t run (but you still get the answer). Therefore, you can now check each of the elements one by one, to find the answer without the computer running. This was the first main theoretical point of the paper. Contrary to some popular press descriptions, we did not implement this experimentally (nor do we intend to, as it’s likely to be inconveniently difficult).
I'll end with the clearest account of counterfactual computing I've seen, courtesy of one Homer J. Simpson.Dear Lord, the gods have been good to me. As an offering, I present these milk and cookies. If you wish me to eat them instead, please give me no sign whatsoever.
Dear Lord, the gods have been good to me. As an offering, I present these milk and cookies. If you wish me to eat them instead, please give me no sign whatsoever.
Turing proved that this problem, called the Halting Problem, is unsolvable by Turing machines. The proof is a beautiful example of self-reference. It formalizes an old argument about why you can never have perfect introspection: because if you could, then you could determine what you were going to do ten seconds from now, and then do something else. Turing imagined that there was a special machine that could solve the Halting Problem. Then he showed how we could have this machine analyze itself, in such a way that it has to halt if it runs forever, and run forever if it halts.
To exceed higher-level Busy Beavers, we’d presumably need some new computational model surpassing even Turing machines. I can’t imagine what such a model would look like.
You might also wonder why we can’t use infinity in the contest. The answer is, for the same reason why we can’t use a rocket car in a bike race. Infinity is fascinating and elegant, but it’s not a whole number. Nor can we ‘subtract from infinity’ to yield a whole number. Infinity minus 17 is still infinity, whereas infinity minus infinity is undefined: it could be 0, 38, or even infinity again. Actually I should speak of infinities, plural. For in the late nineteenth century, Georg Cantor proved that there are different levels of infinity:
If Dehaene et al.’s hypothesis is correct, then which representation do we use for big numbers? Surely the symbolic one—for nobody’s mental number line could be long enough to contain 99^9, 5 pentated to the 5, or BB(1000). And here, I suspect, is the problem. When thinking about 3, 4, or 7, we’re guided by our spatial intuition, honed over millions of years of perceiving 3 gazelles, 4 mates, 7 members of a hostile clan. But when thinking about BB(1000), we have only language, that evolutionary neophyte, to rely upon.
In this case, the catch is simple. Say you've got two programs, Dif and Doof, running in the Windows taskbar. Dif is performing some enormous calculation, while Doof (being a Doof) is doing nothing. If Dif's calculation returns any answer other than 5, then Dif closes Doof. You come back to your computer and find that Doof is still running. Even though Doof didn't calculate anything, and even though Dif never did anything to Doof, you can immediately conclude -- from Doof alone -- that the answer you wanted was 5. Mindblowing! Unbelievable!
Now let Dif and Doof run, not in different windows, but in different branches of the wavefunction -- that is, in quantum superposition. And instead of Dif using an operating system to close Doof, have Dif's branch of the wavefunction interfere destructively with Doof's branch, thereby preventing Doof's branch from being observed. That's the idea of counterfactual quantum computing.
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