On 28 Dec 2013, at 16:51, Jason Resch wrote:



On Dec 28, 2013, at 6:09 AM, Bruno Marchal <marc...@ulb.ac.be> wrote:


On 28 Dec 2013, at 04:56, Jason Resch wrote:




On Fri, Dec 27, 2013 at 10:42 PM, Stephen Paul King <stephe...@provensecure.com > wrote:
Hi Jason,

"Any program, and whether or not it ever terminates can be translated to a statement concerning numbers in arithmetic. Thus mathematical truth captures the facts concerning whether or not any program executes forever, and what all of its intermediate states are. "

this also captures every instance of random numbers as well.

It is not clear to me what "random" means in arithmetical truth.

Randomness can appear from the perspectives of observers, but I don't see how it can arise in arithmetic.

?

It appears in all numbers written in any base. Most numbers are already random (even incompressible).
I guess you know that.

I agree most numbers are incompressible, but I was using random in a different sense than the unpredictability of the next digits of the number given previous ones.

OK. But in the iterated self-duplication, both form of indeterminacy can be mixed.




In the phi_i(j) in the UD, randomness can appear in the many j used as input, as we usually dovetail on the function of one variable. (but such input can easily be internalized in 0-variable programs).

For a long time I got opponent saying that we cannot generate computationally a random number, and that is right, if we want generate only that numbers.

Right, all the random numbers are there, the question is how to throw the dart so that it lands on one.

Of course. And here the 1-indetermlinacy provides one answer.




but a simple counting algorithm generating all numbers, 0, 1, 2, .... 6999500235148668, ... generates all random finite incompressible strings, and even all the infinite one (for the 1p view, notably).

I think we are using the term in a slightly different sense. Certainly any number in the range 1 - N can be considered as a random number in that range (as it is a candidate to be output by some RNG), but the problem is selecting it in a random (in the sense of not-predictable) way.

Yes. here I just point on the fact that random number (with random digits) exists.




There was a joke cartoon of some computer code:

int getRandomNumber()
{
  return 4; // this number was determined by a random die roll
}


Lol
Close to my favorite infinite binary random sequence:
00000000000000000000000000000000000000000000000000000000...

The term "random" is very large.




While a number can be interpreted as random once, it might not be the second time.

While selecting and using all possibilities is arguably a way to achieve randomness (unpredictibilty), (from some points of view) it is often not practical nor useful. Consider encrypting a message with all possible keys and sending the recipient all possible messages.

Not only might you need to send 2^256 possible ciphertexts but any eavesdropper could use the first possible key to decrypt it. This achieves randomness from the POV of the cipher, but not for the user or the attackers.

In quantum cryptography this is essentially what is done, but it requires that the sender and reciever (and attackers) be duplicated for each possible key. So they need to be embedded in that larger program that provides all possible inputs for it to seem random. This is just FPI though, is it not?

Yes.

Bruno




Jason


In that (trivial) sense, arithmetic contains a lot of 3p randomness, even perhaps too much. Then 1p randomeness appears too, by the 1p indeterminacy (and that one is in the eyes of the machine).

Chaitin's results can also explain why we cannot filter out that 3p randomness from arithmetic.

Bruno



What method is deployed to ensure that a program is not just a "regular" random number and not some random number prefixed on a "real" halting program?

It don't see how it makes a difference.


Truth is not a measure zero set, or is it?

I don't understand this question..  Could you clarify?

Jason




On Fri, Dec 27, 2013 at 10:09 PM, Jason Resch <jasonre...@gmail.com> wrote:



On Fri, Dec 27, 2013 at 9:31 PM, Stephen Paul King <stephe...@provensecure.com > wrote:
Hi Jason,

Could you discuss the "trace of the UD" that LizR mentioned? How is it computed? Could you write an explicit example? I have never been able to grok it.


Bruno has written an actual UD in the LISP programming language. I will write a simple one in pseudo-code below:

List listOfPrograms = new List[]; # Empty list
int i = 0;
while (true)
{
# Create a program corresponding to the binary expansion of the integer i
   Program P = createProgramFromInteger(i);

   # Add the program to a list of programs we have generated so far
   listOfPrograms.add(P);

# For each program we have generated that has not halted, execute one instruction of it
   for each (Program p in listOfPrograms)
   {
     if (p.hasHalted() == false)
     {
        executeOneInstruction(p);
     }
   }

# Finally, increment i so a new program is generated the next time through
   i = i + 1;
}


Any program, and whether or not it ever terminates can be translated to a statement concerning numbers in arithmetic. Thus mathematical truth captures the facts concerning whether or not any program executes forever, and what all of its intermediate states are. If these statements are true independently of you and me, then the executions of these programs are embedded in arithmetical truth and have a platonic existence. The first, second, 10th, 1,000,000th, and 10^100th, and 10^100^100th state of the UD's execution are mathematical facts which have definite values, and all the conscious beings that are instantiated and evolve and write books on consciousness, and talk about the UD on their Internet, etc. as part of the execution of the UD are there, in the math.

Jason



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