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Complexity of Go
    Keywords: Theory
  • HarryWang OK, I cannot help it to notice there are no page in sensei here to discuss the mathematical nature of GO?.

I copied this following part from [ext] http://en.wikipedia.org/wiki/Go_(board_game). I have no clue where they come up with these numbers. I hope someone can actually illustrate various math natures of Go on this page.

personally, I know that a un-pruned GO game tree is 361! = xxx × 10 ^ 71. Most GO games seems to end with less than 300 moves. so I guess the average GO game tree is 300! But how to prune the GA grame tree. I am facinated to know

from wiki: "It is commonly said that no game has ever been played twice. This may be true: On a 19×19 board, there are about 3^361×0.012 = 2.1×10^170 possible positions, most of which are the end result of about (120!)^2 = 4.5×10^397 different (no-capture) games, for a total of about 9.3×10^567 games. Allowing captures gives as many as 10^(7.49×10^48) possible games, most of which last for over 1.6×10^49 moves! (By contrast, the number of legal positions in chess is estimated to be between 10^43 and 10^50, and physicists estimate that there are not more than 10^90 protons in the entire universe.)"

unkx80: I am not sure which aspect of mathematics you are interested in. Perhaps you want to see the CGT pages, there is extensive discussion there.


unkx80: Since the page title is also a common term in computer science, I shall also mention something about the computational power required for perfect computer Go players.

The game of Go on a general n by n board is a computationally difficult problem. It is known to be at least PSPACE-hard, and possibly also EXPTIME-complete, which means that it is very computationally expensive for a computer to solve any position perfectly. In fact, Go endgames is already known to be PSPACE-hard.

The implication is that it is completely infeasible for computer Go programs to solve any Go problem perfectly, particularly by using brute force searching. Hence, programs for Go have to rely on heuristics, which gives imperfect answers to problems.

Even on the normal 19 by 19 board, which has 361 points, current computer Go players are still very weak compared to current computer chess (both Chinese and International) players. While International chess programs like X3D Fritz can draw with grandmasters like Garry Kasparov, current Go programs are probably not even amateur Dan strength.

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This is a copy of the living page "Complexity of Go" at Sensei's Library.
(OC) 2004 the Authors, published under the OpenContent License V1.0.