On Dec 28, 2013, at 6:09 AM, Bruno Marchal <[email protected]> wrote:
On 28 Dec 2013, at 04:56, Jason Resch wrote:
On Fri, Dec 27, 2013 at 10:42 PM, Stephen Paul King <[email protected]
> 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.
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.
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.
There was a joke cartoon of some computer code:
int getRandomNumber()
{
return 4; // this number was determined by a random die roll
}
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?
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
<[email protected]> wrote:
On Fri, Dec 27, 2013 at 9:31 PM, Stephen Paul King <[email protected]
> 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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