Tapani Raiko wrote:
Matt Gokey wrote:
I'm not sure I agree with this. I hypothesize that 2d, 3d, 4d, torus,
or any other shape is completely irrelevant with regard to game play.
The only thing that matters is the graph topology. A corollary is that
on any board that is completely balanced at the beginning with
identical number of neighbors for all nodes, any 1st play is
equivalent and therefore optimal.
It is true that the graph topology is the thing that matters, but having
an identical number of neighbors for all nodes does not mean that the
graphs are similar (isomorphic). For instance in the 3D diamond graph,
each node (disregarding the edges) has 4 neighbors as usual, but there
are 12 neighbor's neighbors, whereas normal Go board has only 8 (4
diagonals and 4 one-point jumps). I'd say there is a huge difference.
OK, I see. That is of course a big and very significant difference.
-So the graph topology is crucial not really the shape.
-And if a board is empty and balanced with equal numbers of neighbors
and uniform (e.g. same connectivity properties throughout) then any 1st
play should be equivalent and therefore optimal. Right?
A toroid board (defined as a normal square board with the top-bottom and
left-right edges connected) would fit this condition, however, the round
board does not appear to. Therefore they can't be isomorphic or
structurally identical.
When describing boards as mapped onto different shapes it would be
helpful to describe the graph connectivity properties being imagined
since so many different boards can be constructed on the same geometries.
As for Don's original question about the toroid board, I think it does
change the game significantly since it will be harder to make territory
without edges and perhaps less interesting, but the first play is easy
since any move is optimal and would not require running even one
simulation ;-)
_______________________________________________
computer-go mailing list
computer-go@computer-go.org
http://www.computer-go.org/mailman/listinfo/computer-go/
