Let $latex \Sigma=\{0,1\}$, $latex {\Sigma}^n$ be the set of words $latex \sigma_1\dots\sigma_n$ with each $latex \sigma_i$ being $latex 0$ or $latex 1$, and let $latex \Sigma^*=\bigcup\limits_{n=0}^\infty \Sigma^n$. Let $latex |x|$ denote the length of any word $latex x\in \Sigma^*$. Let $latex A:\Sigma^*\to \Sigma$ be an arbitrary function which we call oracle. Imagine a computer which can do everything a regular computer can do, but, in addition, can instantly compute $latex A(x)$ for any $latex x\in \Sigma^*$. Let $latex P_k^A$ be the set of functions $latex F(x_1, \dots, x_k)$ in k variables $latex x_i\in \Sigma^*$, for which there is a polynomial P and a program on such a computer which computes $latex F$ in at most $latex P\left(\sum\limits_{i=1}^k|x_i|\right)$ operations. Let $latex \text{PH}^A$ be the set of functions $latex f:\Sigma^*\to \Sigma$ for which there exists integer $latex k\geq 0$, polynomial P, and function $latex F\in P_{2k+1}^A$ such that $latex f(x)=1$ if and only if $latex \forall y_1 \in \Sigma^{P(|x|)}\,\exists z_1 \in \Sigma^{P(|x|)} \dots \forall y_k \in \Sigma^{P(|x|)}\,\exists z_k \in \Sigma^{P(|x|)}$ such that $latex F(y_1,z_1,\dots,f_k,z_k,x)=1$. Let $latex \text{BQP}^A$ be the set of all functions $latex g:\Sigma^*\to \Sigma$ for which there is a polynomial P and a program on a quantum computer (again with extra ability to instantly compute A) which computes $latex g(x)$ in at most $latex P(|x|)$ operations. The Theorem proves the existence of oracle $latex A:\Sigma^*\to \Sigma$ such that $latex \text{BQP}^A \not\subseteq \text{PH}^A$.

]]>Something which I couldn’t understand at first (and to which neither you nor Lindbergh refers) is the room with the strange metal arms hovering over a table, on which we see a dead mouse, a clock, and various other artifacts. Somehow the data collected from that room is aiding the computer.

And I think those scenes are intended to solve one of the problems you point out: how is the quantum computer getting enough input?

The computer (or the program it’s running) is *extrapolating* the (block) universe from that collection of objects. Ie, there is much, much less Information in the universe than we think (or calculate according to our understanding of the laws of physics). Physical extrapolation is possible in some ways. For example, if we have a perfect crystal of salt, we can extrapolate its orientation from the positions of a handful of atoms.

I’m not arguing that the universe really *can* be extrapolated in this way, just that it’s a reasonable premise upon which to base the computer’s ability to have complete knowledge of past and future.

And what the members of Devs are doing when they conjure up a view on their Big Screen is asking the computer (program) to do the calculations to extrapolate to that particular time and place. Ie, the computer’s memory never has a *complete* picture of the universe in its state (impossible anyway because it is embedded within it), but can ‘look’ at one time and place, and present what it finds (using aesthetically pleasing camera angles and movements).

Similarly, in the final simulation, it doesn’t have to simulate Forest’s and Lily’s universe completely, but just enough for them to experience it as real. The moon really isn’t there if no one’s looking at it. And that falling tree makes no sound.

Caveat: I may have fallen for the intentional fallacy here, of course.

Cheers, Paul

]]>Ok, oracle A is just an arbitrary language, easy! BQP^A is easy as well: this is the set of languages recognised by quantum computer which can also ask extra questions in the form “does x belong to A”? P^A would be easy as well – same definition without the word “quantum”. Similarly, we can easily define P^(PH,A) – standard computer with 2 oracles. But what is PH^A? The “PH-powerful device” with oracle A? But what is the “PH-powerful device?” Low-depth alternating circuits? Too complicated!

So, my questions are: Is P^PH = PH ? Is, for any oracle A, P^(PH,A) = PH^A? Is the main result in the “Oracle Separation of BQP and PH” paper is equivalent to the statement that there exists an oracle A such that BQP^A is not a subset of P^(PH,A)?

]]>He proposes that we may have a chance to know the exact configuration of this graph in an earlier moment and its update rule, so basically we have a chance to know everything about everything. I really hope, that this is the case.

(Okay, I’m shy, I don’t like the idea that future civilizations may be able to reconstruct my life and my thoughts, but that is a sacrifice I am willing to make for the idea of Knowing All.)

Besides that the whole idea is reassuring, likely to be provable, and it could solve every question about free will and epistemology forever.

Basically like what Laplace did with his daemon, but this finite-but-evergrowing world behaves much better computationally.

arch1 #42: Sounds somewhat similar yes! Except it would be interesting to consider an unambiguously “magical” mechanism. I guess the point is that even if such a thing exists, it would be amenable to the same scientific process as everything else, and would cease to be magic. Unless it completely defied logic itself. ]]>

Erwin Schrodinger noted:

What puts one off when examining what are called objective, historical accounts of ancient or modern philosophy, is that one keeps finding such statements as: A or B was a ‘representative’ of this or that view; so-and-so was an X-ian or a Y-ian, holding allegiance to this system or that, or partly to one and partly to another.

Different views are almost always opposed to each other as though they really were different views of the same object. But this kind of account practically forces us to regard one or other of these thinkers, or both of them, as crazy, or at the very least as totally lacking in judgement. One is then very apt to start wondering how posterity, including oneself, can possibly think the ill-considered babblings of such blockheads worth any closer attention. But in fact one is dealing, at least in very many cases, with well-founded convictions of highly competent minds, and hence one can be sure that differences in their judgement correspond to differences in the object of it, at least in so far as very different aspects of that object were given prominence in their reflective consciousness. A critical account of their thought should, instead of stressing the contradictions between them, as is usually done, aim at combining these different aspects into one total picture—needless to say, without compromise, which can only lead to confused and hence a priori untrue statements.

The real trouble is this: giving expression to thought by the observable medium of words is like the work of the silkworm. In being made into silk, the material achieves its value. But in the light of day it stiffens; it becomes something alien, no longer malleable. True, we can then more easily and freely recall the same thought, but perhaps we can never experience it again in its original freshness. Hence it is always our latest and deepest insights that are voce meliora.

It was said by Epicurus, and he was probably right, that all philosophy takes its origin from philosophical wonder. The man who has never at anytime felt consciously struck by the extreme strangeness and oddity of the situation in which we are involved, we know not how, is a man with no affinity for philosophy — and has, by the way, little cause to worry. The unphilosophical and philosophical attitudes can be very sharply distinguished (with scarcely any intermediate forms) by the fact that the first accepts everything that happens as regards its general form, and finds occasion for surprise only in that special content by which something that happens here today differs from what happened there yesterday; whereas for the second, it is precisely the common features of all experience, such as characterise everything we encounter, which are the primary and most profound occasion for astonishment; indeed, one might almost say that it is the fact that anything is experienced and encountered at all. It seems to me that this second type of astonishment—and there is no doubt that it does occur—is itself something very astonishing. Surely astonishment and wonder are what we feel on encountering something that differs from what is normal, or at least from what is for some reason or other expected. But this whole world is something we encounter only once. We have nothing with which to compare it, and it is impossible to see how we can approach it with any particular expectation. And yet we are astonished; we are puzzled by what we find, yet are unable to say what we should have to have found in order not to be surprised, or how the world would have to have been constructed in order not to constitute a riddle!

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