Re: Monte Carlo computer go methods


Denis Feldmann <feldmann.denis.asupprimer@xxxxxxx> wrote:
Do you honestly believe you could figure for yourself that the
Shusaku kosumi is a good idea ? Go has a strong "historical"
character, and is for the most part cumulated wisdom, almost
impossible to rediscover buy onself (as is mathematics, by the way)


Apparently Shusaku had figured it out, so it can't be
terribly difficult. As for "historical" character in mathematics
we are still learning more and more about it every day. See
for example the inset to an article in _Science_ magazine,
vol. 319, 15 February 2008, p.899, "A Woman Who Counted."

Sophie Germain was one of the great mathematicians
of the early 19th century. Number theorists laud her
for "Sophie Germain's theorem," an insight into Fermat's
famous equation x^n + y^n = z^n aimed at establishing
its lack of solutions (in positive integers) for certain
exponents. Oddly, Germain's fame for her theorem
stems not from anything she herself published but from a
footnote in a treatise by her fellow Parisian Andrien-Marie
Legendre, in which he proved Fermat's Last Theorem
for exponent n=5. Now, two mathematicians have
found that Germain did far more work in number theory
than she has ever been given credit for.

Poring over long-neglected manuscripts and
correspondence, David Pengelly of New Mexico State
University in Las Cruces and Reinhard Laudenbacher
of Virginia Polytechnic Institute and State University in
Blackburg have discovered that Germain had an
ambitious strategy and many results aimed at proving
not just special cases of Fermat's Last Theorem but
the whole enchilada. "What we thought we knew
[of her work in number theory] is actually only the tip
of the iceberg," Pengelly said at a session on the
history of mathematics.

The theorem in Legendre's footnote asserts that if
the exponent `n' is a prime number satisfying certain
properties, then any solution to x^n + y^n = z^n
must have one of the numbers x, y, or z divisible by
`n.' Pengelly and Laudenbacher report this is just
the first of numerous theorems in a long manuscript
by Germain now housed at the Bibliotheque Nationale
in Paris. Germain also wrote another bulky manuscript
on the subject and covered it at length in a letter to
the German mathematician Carl Friedrich Gauss. In
the three sources, Germain elaborated programs for
proving, first, that all primes satisfy the necessary
properties; and second, that exponents `n' cannot
divide any of the numbers x, y, or z. Together,
those two results would have proved Fermat's Last
Theorem. Along the way, she showed that any
counterexample would involve numbers "whose
size frightens the imagination," as she put it to
Gauss.

A proper Hollywood ending would have Germain's
proof of Fermat's Last Theorem take precedence
over Andrew Wiles's monumental accomplishment
15 years ago. However, Pengelly notes, that's not
how the story goes. Much of Germain's approach
was rediscovered later by others and found to fall
short of its goal. Nonetheless, Pengelley says, the
fact that she had developed far more than a single
footnoted theorem "calls for a reexamination of
the scope and depth of her work."
- Barry Cipra



Which then leads to rekindled interest in John Conway's
number game "Sylver Coinage" a name itself coined from
J.J. Sylvester who established some crucial upper bounds.
Simply explained, players take turns naming a natural number
such that no number named can be a previous number named
or the sum of any combination of previous numbers named.
Then, the player who names "1" (remaining after all others
have been covered) is the loser. John Conway introduces
a brief introduction to "Sylver Coinage" in his game theory
talk at MSRI (previously referenced on this newsgroup). It
may require a bit of mental juggling to interpret his sketchy
remarks, so please bear with the topic for the time being.
Game "Sylver Coinage" is 'unboundedly bounded' so is
potentially much more difficult to play than any other game.
Obviously the pair (2,3) or (3,2) cover the set of natural
numbers, less "1", so form a "P" position, i.e. "previous
player win." This means that, by itself, a "2" or "3" forms
an "N" position, i.e. "next player win" because the next
player can move it to a "P" position by adding "3" or "2"
respectively. Therefore both players seek to avoid naming
"2" or "3" from the outset. So they start with "4" but soon
discover that "4" is also an "N" position. However "5" is
a "P" position, as well starting with any prime number,
because of the proof that if GCD(x,y)=1 then {x, y} is an
"N" position, even though we might not have the explicit
strategy which establishes the win in a specific case.
Any number {Prime, z} will be an "N" position and the
first player whose first move was into a prime can win.
If a non-prime is named, however, then the second
player will not wish to name GCD(x,y)=1 because that
leaves an "N" position for the opponent. An even parity
of remaining numbers can indicate a losing position,
so one seeks not to play on even parity or, at least,
not leave an odd parity position for one's opponent.

John Conway will also remembered for the remark
that his discovery of "surreal numbers" was motivated
by his investigations with the game of Go.

Considerable research has transpired on the game
"Sylver Coinage" though it has not generated a lot of
interest recently, until these brief articles summarizing
activity at the Joint Mathematics Meeting, 6-9 Jan 2008,
in San Diego, California, (in the _Science_ Magazine
article already cited): "Number Theorists' Big Cover-Up
Proves Harder Than It Looks" and "Exact-Postage Poser
Still Not Licked." Names such as Carl Pomerance, Paul
Erdos, Jeffrey Lagarias, John Selfridge, Pace Nielsen,
Waclaw Sierpinski, Sturle Sunde, Jason Gibson, Matthias
Beck, Ravi Kannan, Bjarke Roune, Stan Wagon, James
Sylvester, appear in connection with a famous Frobenius
Diophantine problem ...

"Any algorithm powerful enough to give an
efficient general solution would automatically
solve an entire class of problems, including
everything that currently underlies the
cryptographic security of Internet transactions."


And, all of this, to reaffirm how mathematical re-discovery
is surely the difficult task Denis Feldmann makes it up
to be. Why would they desire to build a moon-base
which radios the human legacy back to us, just in case
we destroy ourselves in a planetary holocaust? History
touches mathematics just as it touches everything else.



Nick Wedd:
I just had to accept it as an arbitrary fact. If I had been
taught Go like that, "in this position you must play here" with
no explanation offered, I would very soon have abandoned it.

But the explanations have to be correct ones, else bad ideas
and habits will be learned.


We neglect to remember some "bad move" made by
Shindo Hikaru which caused a tournament game to evolve
askew, and then Touya Akira to wonder how that position
came about, obsess and fret over it, even during nightmares.



Nick Wedd:
And in math, I did once believe that 0/1=1, it seemed to work
well enough at the time. I only accepted a different view once
I knew enough to understand it.


Perhaps you had misread 0!=1 into your form somewhere.
Do you find no distinction between 0/0 and x/0, x<>0 ?



- regards
- jb

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