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The following archive was created by hippie-mail 7.98617-22 on Fri Aug 21 1998 - 17:20:42 ADT
Enzo Michelangeli (em@who.net)
Tue, 21 Jul 1998 11:14:30 +0800
From: Mok-Kong Shen <mok-kong.shen@stud.uni-muenchen.de>
Date: Tuesday, July 21, 1998 2:13 AM
>I understand this to mean Borel has a certain definition of true
>'randomness'.
In fact, Borel was only referring to the statistics, not to the entropy
(which, IMHO, is the qualifying point of "true randomness"). A normal number
is one whose decimal digits would be _guaranteed_ to pass DIEHARD or other
statistical test suites. Apparently, based on the measured frequencies for
digits and blocks of digits, numbers like "pi" or "e" or sqrt(2) fit the
bill, but I don't think they have been _proven_ normal. Some normal numbers
have been constructed, though.
> Would you please supply a reference to literature?
Unfortunately not, but searching the Web for it I have found an interesting
article linking the works of Borel, Hilbert, Goedel and Turing: "Randomness
& Complexity in Pure Mathematics", by G. J. Chaitin of IBM. The URL is:
http://www.cs.auckland.ac.nz/CDMTCS/chaitin/ijbc.html
Chaitin argues that there IS intrinsic randomness in pure mathematics after
all, and true random numbers can be defined: but not computed. As an
example, he constructs a Diophantine equation (one whose variables are
natural numbers) with seventeen thousand variables and one parameter. Then,
he defines a number called "Omega" whose N-th bit is zero if and only if
that equation admits a finite number of solutions when the parameter has
value N. It can be shown that Omega is a true random, but unfortunately it's
not algorithmically computable under Church-Turing.
Cheers --
Enzo
The following archive was created by hippie-mail 7.98617-22 on Fri Aug 21 1998 - 17:20:42 ADT