dear Mark,
> Well, although Dr. Tromp seems rather modest about this result, I haven't
> heard of anyone else doing similarly interesting work on the theoretical
> foundations of the game.
There is a lot of other interesting research beyond counting things.
Just to name a few there's rule explorations, where Robert Jasiek
has done a lot of work, and the theory of evaluating Go positions,
endgames, kos, etc. in the setting of Combinatorial Game Theory,
where Elwyn Berlekamp is a name that comes to mind. Apologies for
leaving out countless others who have made valuable contributions.
> 1. So, as you go up the chart, what is the percentage of all possible
> positions that are legal?
The asymptotics of legal positions is derived in our paper (under some
conjecture) as
L(m,n) ~ 0.850639925845714538 * 0.96553505933837387^{m+n} *
2.97573419204335724938^{m*n}
So the legal probability grows as that divided by 3^{m*n}, or
Prob_legal(m,n) ~ 0.850639925845714538 * 0.96553505933837387^{m+n}
*0.99191139734778574979 ^{m*n}
> And isn't that an interesting sequence? Perhaps more intuitively useful
> to a go-programmer than the raw numbers themselves?
No; to a programmer that much precision is not interesting.
And to those for whom a lot of precision is interesting, the exact number
of legal positions is more natural than the probability.
> Does this set of ratios make any
> intuitive sense to you
> ... or should I rephrase that as -- can you rationalize these results of
> the ratios?
Yes, the constant 2.97573419204335724938 is the effective freedom
per point under the legality constraint. That's why we call it the
"base of liberties".
> 2. One of the most frustrating things about writing a program to play go is
> that the rules are
> a bit blurry. Far too blurry to really satisfy a computer programmer.
> I think some of the
> work you've done over the years is in creating a rigorous and computable
> set of rules.
> Is this correct, or have I heard wrong on this count? Do you have a set
> of rules that
> could be profitably used for the basis of a go-playing program, that you
> like today?
> Is there a link to such a rule set somewhere?
Obviously I'm inclined to direct you to my personal Go page at
http://tromp.github.io/go.html
which states the Logical rules that I developed with Bill Taylor.
regards,
-John
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