Proving The Church-Turing-Deutsch Principle

[Demystifier]

Are you asserting that nature is continuous rather than discrete? Are not "physical laws" man made?

Note that I started my assertion with "If". By "physical laws", I meant the laws of Nature itself.

 

[Zafa Pi]

Note that I started my assertion with "If". By "physical laws", I meant the laws of Nature itself.

Help me out here, because I am not trying to nit pick, but to understand what you mean. It would help me if you told me whether Newton's laws are "physical laws" = "the laws of Nature itself". And if not what is an example of a physical law. I believe my questions are germane to this thread.

 

[n01]

I'm at a loss. If I have a program (computer) that when I input 2+3 it gives 5, but when I input 3+2 it gives 6, then is it inconsistent or merely a lousy addition program?
Can you verify if math is consistent?

Do you think that the concept of determinism can be programed? Is the result of flipping a coin in a wind determined?

Well, if a formal system is Turing complete and decidable, then it is consistent. If it is consistent then it is deterministic.

It's not that complicated.

And for the matter I envisioned a simple solution to the question posed in the OP.

Namely, if human consciousness can be simulated (due to it obeying the same physical laws as any other object in the universe), then so can the Church-Turing-Deutsch Principle be verified to be true through an indirect method of proof. Whether this is true or not is highly contested by physicists to this day (Penrose, et al).

 

[Demystifier]

Help me out here, because I am not trying to nit pick, but to understand what you mean. It would help me if you told me whether Newton's laws are "physical laws" = "the laws of Nature itself". And if not what is an example of a physical law. I believe my questions are germane to this thread.

I mean we don't know what are the "ultimate final fundamental" laws of nature, but the most fundamental man-made currently known laws of nature are based on real and complex numbers (not the integer ones).

 

[Zafa Pi]

I mean we don't know what are the "ultimate final fundamental" laws of nature, but the most fundamental man-made currently known laws of nature are based on real and complex numbers (not the integer ones).

Newton's laws (of nature) are continuous and were valid for hundreds of years, but turned out to be not in accord with experiment. The laws of E&M, GR, and QM are continuous and these theories are really great, but have problems with being consistent with one another. But what about the "ultimate final fundamental" laws of nature, what will they be? Many working in the field of Q-gravity think they will be discrete. A perusal of the internet gave me the impression that a majority of physicists tend to believe that nature itself is discrete rather than continuous, digital rather than analog (e.g. see http://fqxi.org/community/essay/winners/2011.1).

When you said, "If physical laws are based on real numbers (or some other continuous number system), isn't it obvious that a universal computing device, based on integer (or countable) number system, cannot simulate nature exactly?" in post #3, it seems as though you think otherwise. "Isn't it obvious"? No, it isn't obvious, in spite of your "If".

Continuous laws? I like your quote, "It's a thinking tool, you fool. :)"

 

[n01]

Linked below is the paper detailing Deutsche's Church-Turing-Deutsch principle.

http://www.daviddeutsch.org.uk/wp-content/deutsch85.pdf

In it Deutsch makes the assertion:

"For although a quantum computer has an infinite-dimensional state space, only a finite dimensional unitary transformation need be effected at every step to simulate its evolution."

Can anyone elucidate how this can be known to be true? This seems like a very complex statement that requires some complex justification

Thanks.

 

[Zafa Pi]

Linked below is the paper detailing Deutsche's Church-Turing-Deutsch principle.

http://www.daviddeutsch.org.uk/wp-content/deutsch85.pdf

In it Deutsch makes the assertion:

"For although a quantum computer has an infinite-dimensional state space, only a finite dimensional unitary transformation need be effected at every step to simulate its evolution."

Can anyone elucidate how this can be known to be true? This seems like a very complex statement that requires some complex justification

Thanks.

I'm confused, a recurring affliction. I thought the input to a q-computer was a finite number of qbits, each of which is 2D. And a finite number of gates.

 

[n01]

I'm confused, a recurring affliction. I thought the input to a q-computer was a finite number of qbits, each of which is 2D. And a finite number of gates.

Yes, but if a Q computer occupies an infinite state space doesn't that mean that equally an infinite state space must be simulated? Same confusion on this end.

 

[secur]

If you incorrectly treat "infinity' as something that exists in the real world you will get confused. Paradoxes are inevitable. That's why mathematicians worked out methods to deal with the concept of infinity, such as epsilon-delta definition of limit (Cauchy, Weierstrass). Math deals with the concept of infinity, not "infinity" itself, which has no meaning apart from finite formal limit definitions. The simplest example, we say a sequence "goes to infinity" if, for any N, we can produce a member of the sequence which is greater than N. When dealing with infinity, if you can't express what you mean by a finite limit algorithm like this, you literally don't know what you're talking about.

This is what we mean by an "infinite-dimensional" space: given any finite number N, I can demonstrate that the number of axes or dimensions in the space is greater than N. You might think otherwise. The position operator, for instance, has "infinite" eigenvalues simply because it's defined on the real number line. That doesn't seem to involve a finite definition like epsilon-delta. But, in fact, the mathematical underpinnings of all "infinities" are expressible in a finite algorithm. In this case, Cantor's diagonal proof, which demonstrates the real line's "order of infinity" (uncountable).

So when simulating (or calculating, or even just considering) evolution of a system, Deutsch is correct that "only a finite dimensional unitary transformation need be effected" - because nothing else can possibly be effected by a real-world simulation.

 

[n01]

Disregard.