arly in the film "A Beautiful Mind," the mathematician John Nash is seen sitting in a Princeton courtyard, hunched over a playing board covered with small black and white pieces that look like pebbles. He was playing Go, an ancient Asian game. Frustration at losing that game inspired the real Mr. Nash to pursue the mathematics of game theory, research for which he eventually won a Nobel Prize.
In recent years, computer experts, particularly those specializing in artificial intelligence, have felt the same fascination — and frustration.
Programming other board games has been a relative snap. Even chess has succumbed to the power of the processor. Five years ago, a chess-playing computer called Deep Blue not only beat but thoroughly humbled Garry Kasparov, the world champion at the time. That is because chess, while highly complex, can be reduced to a matter of brute force computation.
Go is different. Deceptively easy to learn, either for a computer or a human, it is a game of such depth and complexity that it can take years for a person to become a strong player. To date, no computer has been able to achieve a skill level beyond that of the casual player.
The game is played on a board divided into a grid of 19 horizontal and 19 vertical lines. Black and white pieces called stones are placed one at a time on the grid's intersections. The object is to acquire and defend territory by surrounding it with stones.
Programmers working on Go see it as more accurate than chess in reflecting the ineffable ways in which the human mind works. The challenge of programming a computer to mimic that process goes to the core of artificial intelligence, which involves the study of learning and decision-making, strategic thinking, knowledge representation, pattern recognition and, perhaps most intriguingly, intuition.
"A good Go player could make a move and other players say, `Yes, that's a good move,' but they can't explain to you why it's a good move, or how they even know it's a good move," said Dr. John McCarthy, a professor emeritus at Stanford University and a pioneer in artificial intelligence.
Dr. Danny Hillis, a computer designer and chairman of the technology company Applied Minds, said that the depth of Go made it ripe for the kind of scientific progress that comes from studying one example in great detail. "We want the equivalent of a fruit fly to study," Dr. Hillis said. "Chess was the fruit fly for studying logic. Go may be the fruit fly for studying intuition."
Along with intuition, pattern recognition is a large part of the game. While computers are good at crunching numbers, people are naturally good at matching patterns. Humans can recognize an acquaintance at a glance, even from the back. "Every Go book is filled with advice on patterns of different kinds," Dr. McCarthy said.
Dr. Daniel Bump, a mathematics professor at Stanford, works on a program called GNU Go in his spare time. "You can very quickly look at a chess game and see if there's some major issue," he said. But to make a decision in Go, he said, players must learn to combine their pattern-matching abilities with the logic and knowledge they have accrued in years of playing.
"If you watch really strong players," Dr. Bump said, "some seem to make fairly mundane moves, but at the end of the game they're ahead. Others do spectacular things."
One measure of the challenge the game poses is the performance of Go computer programs. The last five years have yielded incremental improvements but no breakthroughs, said David Fotland, a programmer and chip designer in San Jose, Calif., who created and sells The Many Faces of Go, one of the few commercial Go programs.
Mr. Fotland's program was the winner of a tournament last weekend in Edmonton, Alberta, that pitted 14 Go-playing programs — including several from Japan — against one another. But even The Many Faces of Go is weak enough that most strong players could beat it handily.
Part of the challenge has to do with processing speed. The typical chess program can evaluate about 300,000 positions per second, and Deep Blue was able to evaluate some 200 million positions per second. By midgame, most Go programs can evaluate only a couple of dozen positions each second, said Anders Kierulf, who wrote a program called SmartGo.
In the course of a chess game, a player has an average of 25 to 35 moves available. In Go, on the other hand, a player can choose from an average of 240 moves. A Go-playing computer would take about 30,000 years to look as far ahead as Deep Blue can with chess in three seconds, said Michael Reiss, a computer scientist in London.
If processing power were all there was to it, the solution would be simply a matter of time, since computers are growing ever faster. But the obstacles go much deeper. Not only do Go programs have trouble evaluating positions quickly, they have trouble evaluating them correctly.
Nonetheless, the allure of computer Go increases as the difficulties it poses encourage programmers to advance basic work in artificial intelligence. Graduate students produce dissertations on the topic, and a handful of researchers around the world devote much or all of their attention to it.
The game attracts people from all fields. For example, Chen Zhixing, a retired chemistry professor in Guangzhou, China, wrote a program called Handtalk, which dominated the computer Go field for several years. Dr. Bump, 50, whose field is number theory, has been playing Go for 35 years and taught himself the C programming language four years ago so he could write Go software. Mr. Fotland, 44, the creator of The Many Faces of Go has been working on computer Go for 20 years and is chief technology officer at Ubicom, a small semiconductor company in Silicon Valley.
All are very strong Go players, and it takes a strong Go player to write even a weak Go program. Mr. Fotland, for instance, said he had written programs for checkers, Othello and chess. The algorithms are all very similar, and it is not difficult to write a reasonably strong program, he said. Each of the games took him a year or two to finish. "But when I started on Go," he said, "there was no end to it."
Mr. Fotland said that his Go programming was especially weak when he was a beginning player. "A lot of the stuff I wrote was just plain wrong because I didn't understand the game well enough," he said.
Even when skill develops, however, translating it into a program is not an obvious task. "There's a certain stream of consciousness when you're looking at positions," Dr. Bump said. "You might look at 10 variations, but you don't really know what's going on in the back of your mind. Even a strong player doesn't know how his mind works when he looks at a position."
"We think we have the basics of what we do as humans down pat," Dr. Bump said. "We get up in the morning and make breakfast, but if you tried to program a computer to do that, you'd quickly find that what's simple to you is incredibly difficult for a computer."
The same is true for Go. "When you're deciding what variations to consider, your subconscious mind is pruning," he said. "It's hard to say how much is going on in your mind to accomplish this pruning, but in a position on the board where I'd look at 10 variations, the computer has to look at thousands, maybe a million positions to come to the same conclusions, or to wrong conclusions."
Dr. Reiss, who is the author of Go4++, a previous champion that placed second in last weekend's playoff, agrees with Dr. Bump. Dr. Reiss, who is an expert in neural networks, compares a human being's ability to recognize a strong or weak position in Go with the ability to distinguish between an image of a chair and one of a bicycle. Both tasks, he said, are hugely difficult for a computer.
For that reason, Mr. Fotland said, "writing a strong Go program will teach us more about making computers think like people than writing a strong chess program."
Dr. Reiss, who works on Go full time, said he would not think of devoting his time to any other problem. "It's a fundamentally interesting problem, but also it's just the right level of difficulty," he said. "If it was too easy it would have been solved already. If it was fantastically difficult, people might give up in frustration."
"I think in the long run the only way to write a strong Go program is to have it learn from its own mistakes, which is classic A.I., and no one knows how to do that yet," Mr. Fotland said. A few programs have some learning capabilities built into them.
Mr. Fotland's program, for instance, refers to a database of games played by strong players in deciding its moves, and Dr. Reiss's program employs a learning scheme for deciding which moves are interesting to look at.
Dr. Reiss said he had come up with an idea for a new Go program that would learn by analyzing professional games. But to pursue his idea would require too much work, he said, depriving him of time to continue making updates to his current program.
It seems unlikely that a computer will be programmed to drub a strong human player any time soon, Dr. Reiss said. "But it's possible to make an interesting amount of progress, and the problem stays interesting," he said. "I imagine it will be a juicy problem that people talk about for many decades to come."
