Re: FW: GoSpace

[Henrik Rydberg <rydberg@xxxxxxxxxxxxxx>]

Hi Pieter,

OK, that's seems more alright. So my current interpretation
of your data is the following:

I randomly pick a board state. I then check whether it can be
reduced to some other state, by taking away stones (which may
be in a semai situation, by the way). If it can be reduced,
it is not regarded as a valid state, and is not counted.

Now I make a map over all possible combinations of black and
white stones, for each such combination checking a large number
of random states, counting the percentage of valid states in 
accordance with the above. The result is shown in the plot.
Hope I got you right this time, Pieter. :)

Then, the interpretation seem to be that over a wide range
of gaps between the # white stones and the # black stones,
the percentage of legal moves is the same. For # stones around
100, a round estimate of a middle game, it seems to be around
60-70% legal moves. Correct? This is however not true at the very
extremes, where there is only black or white stones on the board,
in which case all states a legal, and hence a strong upshoot along
the borders of the plot.

Now I have a question and a comment. According to these findings,
the distribution of states with +-x stones is more or less gaussian,
at least if we go to large board sizes. This does not seem too
uplifting, since the number of reasonable, interesting states always
will be an appricable share of all the states, which we simple cannot
afford to check.

There is however one consideration that may change that. The non-legal
states will after Pieters treat transform into another legal state,
which logically should be counted somewhere among the more
differentiated
ones. Say for instance a +-1 state is illegal. Then it should of course
be counted as a +-x state, with x>1. At least if one is interested in
a kind of effective evaluation of the board states, in accordance with
what I was looking for in a previous mail. Now, if you repeatedly
shuffle
the illegal states into more diffentiated ones, assuming for simplicity
and for the sake of the argument that they are moved from x to x+1, then
we'll get the relation (q is the fraction of illegal states)

n(x+1) -> n(x+1) + q n(x),

which yields an exponential modulation of the original distribution.
The fraction of states within a certain window, say +-3 stones,
will in this simple model then decrease *exponentially* with
increasing fraction of illegal states. Of course, the effect may
be small in reality, but then again, it could be large.

Best regards,
Henrik

-- 
Henrik Rydberg (http://fy.chalmers.se/~rydberg),
Department of Applied Physics, Chalmers University of Technology.