Re: computer-go: Which Maths Is OT?

[rbrown@xxxxxxxxxxxxxxx]

> What does the majority think?

I have utterly no idea.

As for me, I can't think of any mathematics that's really off-topic for
computer go.

Topology, number theory (lots of _counting_ in go), probability and statistics,
geometry, chaos theory, and even the now-much-maligned catastrophe theory of
Rene Thom...  All these certainly have some bearing on programming go...  In my
opinion.  And of course game theory and combinatorics are quite on-topic.

If one generalizes go to boards of arbitrary size or dimensionality, indeed to
any arbitrary undirected graph, not to mention n-player go, then of course it
is necessary to discuss graph theory as well.  In any event, results that
are discovered for such generalizations may well give insights also into the
specific case of 19x19 two-player go.  Just adding one more line all around, to
get a 21x21 board, increases the difficulty of the problem tremendously.  (An
estimate in a _Scientific_American_ article some years ago said that each line
added -- they only considered square boards with an odd number of intersections
-- increases the difficulty of go as a triple exponent of the previous size.
In other words, this function skyrockets!

Given the right notation, one might even begin to speak of a go algebra, in
which operators and operands are manipulated according to algorithms that
arise from observations of their axiomatic properties.  Er, um...  What I am
trying to say, in my clumsy way, is that one may consider the placing of a
stone to be an operation, the board state to be an operand, and successive
placing of stones as sequences or series.  Placing a stone on the optimum
points, of course, is the goal, so one would naturally like to find the optimal
sequence.  This all may seem tautological, but a good notation will facilitate
the manipulation of these operators (stones) and operands (board states).  It
can't hurt to formalize such a notion into a go algebra.  Stay tuned.

As for the discrete vs. continuous debate...  Just because go is discrete,
doesn't mean you can't map it onto the reals.  Consider a board state.  Now
consider the pairing of that board state with each legal move (under your
favorite rule set :)  If it is desirable to order the moves from best to worst
(and it is) then it may be that the function which so ranks each legal move
has real values.  That is, for each pair {state, legal move} there is some
real number associated with it, and the relative values of each pair determine
their rank.  An appropriate notation for the board state is essential for
assigning such values.  (Discovery of the ranking function is left as an
exercise for the reader.  :)

Rich