From: Anders Sandberg (asa@nada.kth.se)
Date: Mon Jul 01 2002 - 00:51:15 MDT
On Sat, Jun 29, 2002 at 05:57:05PM -0700, Kenneth Hurst wrote:
> On Thursday, June 27, 2002 4:27 PM, Anders Sandberg wrote,
>
> > One of the stronger ways of defining randomness of a string of bits is
> > that having access to all previous bits give you no information about
> > the next. This can be expressed in algorithmic information theory by
> > saying that the shortest program that can generate the first N bits of
> > the string is N bits long. If they can be generated by a shorter
> > program, they are less random. In this sense pi and e are rather
> > nonrandom numbers, while the vast majority of numbers have truly random
> > bit sequences. The problem is that proving that a number is
> > algorithmically random is not possible in general...
>
> I fail to see why this would classify a number as "less random." If I
> generated a six bit number string randomly, 111111 could be generated. This
> has exactly the same probability as generating 847952. Since there is a
> pattern to 111111, it is classified as "less random," even though (as it
> seems to me) it is just as random because it was randomly generated. I
> suppose the essential question is: Is randomness the fact that a string was
> randomly generated, or is it the characteristic of having no pattern?
There is a difference between having a number whose bits are random and
having a number which is selected at random. If you select numbers at
random, that can be viewed as having a long bit string of these numbers
which you extend for each picked number. A good random number generator
would produce a bit string that is algorithmically random - but
individual numbers picked would sometimes have plenty of patterns.
This is the reverse of another example: a program that prints out the
bit strings of the natural numbers. The entire bit string has a low
randomness (a simple program can generate it), but some of these
individual numbers actually have a high algorithmic randomness, since
the simplest program that *only* prints them is as long as the number. A
bit counterintuitive.
-- ----------------------------------------------------------------------- Anders Sandberg Towards Ascension! asa@nada.kth.se http://www.nada.kth.se/~asa/ GCS/M/S/O d++ -p+ c++++ !l u+ e++ m++ s+/+ n--- h+/* f+ g+ w++ t+ r+ !y
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