This is a number beyond imagination and renders any thought of exhaustively evaluating all possible moves utterly and completely unrealistic. With its breadth of 250 possible moves each turn (go is played on a 19 by 19 board compared to the much smaller eight by eight chess field) and a typical game depth of 150 moves, there are about 250 150, or 10 360 possible moves. But go’s complexity is bigger, much bigger. Shannon’s number, at 10 123, is huge, in particular considering there are only about 1080 atoms in the entire observable universe of galaxies, stars, planets, dogs, trees and people. Given that a typical chess game has a branching factor of about 35 and lasts 80 moves, the number of possible moves is vast, about 35 80 (or 10 123), aka the “Shannon number” after the Bell Laboratory pioneer Claude Shannon who not only invented information theory but also wrote the first paper on how to program a machine to play chess, back in 1950. Assuming that a game of chess lasts on average d moves (called the game’s depth), the complete game tree from any one starting position-the list of all moves, countermoves, counter-countermoves and so on until one side or the other wins-contains about b times b times b., d times in a row, or b d end positions (so-called terminal nodes or leaves of the search tree). That is, after one turn, there are already b times b or b 2 moves that White needs to consider in her strategizing. If it’s White’s turn, she needs to pick one of b possible moves Black can respond to each of these with b countermoves of his own. Depending on the phase of the game, players must pick one out of a small number of possible moves, called the game’s breadth or branching factor b. Strictly logical games, such as chess and go, can be characterized by how many possible positions can arise, bounding their complexity. Out of this sheer simplicity great beauty arises-complex battles between Black and White armies that span from the corners to the center of the board. Such encircled stones are considered captured and are removed from the board. The intent of the game, originating in China more than 2,500 years ago, is to completely surround opposite stones. Black and White sides each have access to a bowl of black and white stones and each place one in turn on a 19 by 19 grid. Once placed, stones don’t move. The rules of go are considerably simpler than those of chess. Unlike card games where participants only see their own as well as everybody’s discarded cards, players have full access to all relevant information with chance playing no role. In a six-game match played in 1996, Kasparov prevailed against Deep Blue by three wins, two draws and one lossbut lost a year later in a historic rematch 2-1/2 to 3-1/2.Ĭhess is a classic game of strategy, similar to tic-tac-toe (noughts and crosses), checkers (draughts), reversi (Othello), backgammon and go, in which players take turns placing or moving pieces. In a widely expected development, the IBM team challenged then reigning world chess champion Garry Kasparov. Over a period of 10 years a dedicated team of hardware and software engineers, eventually hired by IBM, built and programmed a special-purpose supercomputer, named Deep Blue, that did one thing and one thing only-play chess by evaluating 200 million board positions per second. The ascent of AlphaGo to the top of the go world has been stunning and quite distinct from the trajectory of machines playing chess. The methods underlying AlphaGo, and its recent victory, have startling implications for the future of machine intelligence. An era is over and a new one is beginning. With this defeat, computers have bettered people in the last of the classical board games, a game known for both depth and simplicity. The victory over one of humanity’s best representatives of this very old and traditional board game is a crushing 3 to 1, with one more game to come. Sedol is among the top three players in the world, having attained the highest rank of nine dan. But what god beyond God begins the round Of dust and time and sleep and agony?Īs I write this column, a computer program called AlphaGo is beating the professional go player Lee Sedol at a highly publicized tournament in Seoul. God moves the player, he in turn the piece.
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