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Computer Go is the field of artificial intelligence (AI) dedicated to creating a computer program that plays the traditional board game Go. The game of Go has been a fertile subject of artificial intelligence research for decades, culminating in 2017 with AlphaGo Master winning three of three games against Ke Jie, who at the time continuously held the world No. 1 ranking for two years.
- 1 Performance
- 2 Obstacles to high-level performance
- 3 Tactical search
- 4 State representation
- 5 System design
- 6 Datasets
- 7 List of Go-playing computer programs
- 8 Competitions among computer Go programs
- 9 See also
- 10 References
- 11 Further reading
- 12 External links
Go is a complex board game that requires intuition, creative and strategic thinking. It has long been considered a difficult challenge in the field of artificial intelligence (AI) and is considerably more difficult to solve than chess. Many in the field of artificial intelligence consider Go to require more elements that mimic human thought than chess. Mathematician I. J. Good wrote in 1965:
Go on a computer? – In order to programme a computer to play a reasonable game of Go, rather than merely a legal game – it is necessary to formalise the principles of good strategy, or to design a learning programme. The principles are more qualitative and mysterious than in chess, and depend more on judgment. So I think it will be even more difficult to programme a computer to play a reasonable game of Go than of chess.
Prior to 2015, the best Go programs only managed to reach amateur dan level. On the small 9×9 board, the computer fared better, and some programs managed to win a fraction of their 9×9 games against professional players. Prior to AlphaGo, some researchers had claimed that computers would never defeat top humans at Go.
The first Go program was written by Albert Lindsey Zobrist in 1968 as part of his thesis on pattern recognition. It introduced an influence function to estimate territory and Zobrist hashing to detect ko.
In April 1981, Jonathan K Millen published an article in Byte discussing Wally, a Go program with a 15x15 board that fit within the KIM-1 microcomputer's 1K RAM. Bruce F. Webster published an article in the magazine in November 1984 discussing a Go program he had written for the Apple Macintosh, including the MacFORTH source.
In 1998, very strong players were able to beat computer programs while giving handicaps of 25–30 stones, an enormous handicap that few human players would ever take. There was a case in the 1994 World Computer Go Championship where the winning program, Go Intellect, lost all three games against the youth players while receiving a 15-stone handicap. In general, players who understood and exploited a program's weaknesses could win with much larger handicaps than typical players.
Developments in Monte Carlo tree search and machine learning brought the best programs to high dan level on the small 9x9 board. In 2009, the first such programs appeared which could reach and hold low dan-level ranks on the KGS Go Server on the 19x19 board as well.
The 2014 Codecentric Go Challenge, a best-of-five match in an even 19x19 game, was played between Crazy Stone and Franz-Jozef Dickhut (6d). No stronger player had ever before agreed to play a serious competition against a go program on even terms. Franz-Jozef Dickhut won, though Crazy Stone won the first match by 1.5 points.
2015 onwards: The deep learning era
In March 2016, AlphaGo beat Lee Sedol in the first three of five matches. This was the first time that a 9-dan master had played a professional game against a computer without handicap. Lee won the fourth match, describing his win as "invaluable". AlphaGo won the final match two days later.
In October 2017, DeepMind revealed a new version of AlphaGo, trained only through self play, that had surpassed all previous versions, beating the Ke Jie version in 89 out of 100 games.
Since the basic principles of AlphaGo had been published in the journal Nature, other teams were able to produce high-level programs. By 2017, both Zen and Tencent's project Fine Art were capable of defeating very high-level professionals some of the time and the open source Leela Zero engine was released.
Obstacles to high-level performance
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For a long time, it was a widely held opinion that computer Go posed a problem fundamentally different from computer chess. It was believed that methods relying on fast global search with relatively little domain knowledge would not be effective against human experts. Therefore, a large part of the computer Go development effort was during these times focused on ways of representing human-like expert knowledge and combining this with local search to answer questions of a tactical nature. The result of this were programs that handled many situations well but which had very pronounced weaknesses compared to their overall handling of the game. Also, these classical programs gained almost nothing from increases in available computing power per se and progress in the field was generally slow.
A few researchers grasped the potential of probabilistic methods and predicted that they would come to dominate computer game-playing, but many others considered a strong Go-playing program something that could be achieved only in the far future, as a result of fundamental advances in general artificial intelligence technology. Even writing a program capable of automatically determining the winner of a finished game was seen as no trivial matter.
The advent of programs based on Monte Carlo search (started in 2006) changed this situation in many ways with the first 9-dan professional Go players being defeated in 2013 by multicore computers, albeit with four-stone handicap.
Size of board
The large board (19×19, 361 intersections) is often noted as one of the primary reasons why a strong program is hard to create. The large board size prevents an alpha-beta searcher from achieving deep look-ahead without significant search extensions or pruning heuristics.
In 2002, a computer program called MIGOS (MIni GO Solver) completely solved the game of Go for the 5×5 board. Black wins, taking the whole board.
Number of move options
Continuing the comparison to chess, Go moves are not as limited by the rules of the game. For the first move in chess, the player has twenty choices. Go players begin with a choice of 55 distinct legal moves, accounting for symmetry. This number rises quickly as symmetry is broken, and soon almost all of the 361 points of the board must be evaluated. Some moves are much more popular than others and some are almost never played, but all are possible.
While a simple material counting evaluation is not sufficient for decent play in chess, it is often the backbone of a chess evaluation function, when combined with more subtle considerations like isolated/doubled pawns, rooks on open files (columns), pawns in the center of the board and so on. These rules can be formalized easily, providing a reasonably good evaluation function that can run quickly.
These types of positional evaluation rules cannot efficiently be applied to Go. The value of a Go position depends on a complex analysis to determine whether or not the group is alive, which stones can be connected to one another, and heuristics around the extent to which a strong position has influence, or the extent to which a weak position can be attacked.
More than one move can be regarded as the best depending on which strategy is used. In order to choose a move, the computer must evaluate different possible outcomes and decide which is best. This is difficult due to the delicate trade-offs present in Go. For example, it may be possible to capture some enemy stones at the cost of strengthening the opponent's stones elsewhere. Whether this is a good trade or not can be a difficult decision, even for human players. The computational complexity also shows here as a move might not be immediately important, but after many moves could become highly important as other areas of the board take shape.
Sometimes it is mentioned in this context that various difficult combinatorial problems (in fact, any NP-hard problem) can be converted to Go-like problems on a sufficiently large board; however, the same is true for other abstract board games, including chess and minesweeper, when suitably generalized to a board of arbitrary size. NP-complete problems do not tend in their general case to be easier for unaided humans than for suitably programmed computers: it is doubtful that unaided humans would be able to compete successfully against computers in solving, for example, instances of the subset sum problem.
Given that the endgame contains fewer possible moves than the opening (fuseki) or middle game, one might suppose that it is easier to play, and thus that a computer should be able to easily tackle it. In chess, computer programs perform worse in chess endgames because the ideas are long-term, unless the number of pieces is reduced to an extent that allows taking advantage of solved endgame tablebases.
The application of surreal numbers to the endgame in Go, a general game analysis pioneered by John H. Conway, has been further developed by Elwyn R. Berlekamp and David Wolfe and outlined in their book, Mathematical Go (ISBN 978-1-56881-032-4). While not of general utility in most playing circumstances, it greatly aids the analysis of certain classes of positions.
Nonetheless, although elaborate study has been conducted, Go endgames have been proven to be PSPACE-hard. There are many reasons why they are so hard:
- Even if a computer can play each local endgame area flawlessly, we cannot conclude that its plays would be flawless in regard to the entire board. Additional areas of consideration in endgames include sente and gote relationships, prioritization of different local endgames, territory counting and estimation, and so on.
- The endgame may involve many other aspects of Go, including 'life and death', which are also known to be NP-hard.
- Each of the local endgame areas may affect one another. In other words, they are dynamic in nature although visually isolated. This makes it much more difficult for computers to deal with. This nature leads to some very complex situations like Triple Ko, Quadruple Ko, Molasses Ko and Moonshine Life.
Thus, it is very unlikely that it will be possible to program a reasonably fast algorithm for playing the Go endgame flawlessly, let alone the whole Go game.
Order of play
Current Monte-Carlo-based Go engines can have difficulty solving problems when the order of moves is important.
One of the main concerns for a Go player is which groups of stones can be kept alive and which can be captured. This general class of problems is known as life and death. The most direct strategy for calculating life and death is to perform a tree search on the moves which potentially affect the stones in question, and then to record the status of the stones at the end of the main line of play.
However, within time and memory constraints, it is not generally possible to determine with complete accuracy which moves could affect the 'life' of a group of stones. This implies that some heuristic must be applied to select which moves to consider. The net effect is that for any given program, there is a trade-off between playing speed and life and death reading abilities.
With Benson's algorithm, it is possible to determine the chains which are unconditionally alive and therefore would not need to be checked in the future for safety.
An issue that all Go programs must tackle is how to represent the current state of the game. For programs that use extensive searching techniques, this representation needs to be copied and/or modified for each new hypothetical move considered. This need places the additional constraint that the representation should either be small enough to be copied quickly or flexible enough that a move can be made and undone easily.
The most direct way of representing a board is as a one- or two-dimensional array, where elements in the array represent points on the board, and can take on a value corresponding to a white stone, a black stone, or an empty intersection. Additional data is needed to store how many stones have been captured, whose turn it is, and which intersections are illegal due to the Ko rule.
Most programs, however, use more than just the raw board information to evaluate positions. Data such as which stones are connected in strings, which strings are associated with each other, which groups of stones are in risk of capture and which groups of stones are effectively dead are necessary to make an accurate evaluation of the position. While this information can be extracted from just the stone positions, much of it can be computed more quickly if it is updated in an incremental, per-move basis. This incremental updating requires more information to be stored as the state of the board, which in turn can make copying the board take longer. This kind of trade-off is indicative of the problems involved in making fast computer Go programs.
An alternative method is to have a single board and make and take back moves so as to minimize the demands on computer memory and have the results of the evaluation of the board stored. This avoids having to copy the information over and over again.
New approaches to problems
Historically, GOFAI (Good Old Fashioned AI) techniques have been used to approach the problem of Go AI. More recently, neural networks have been used as an alternative approach. One example of a program which uses neural networks is WinHonte.
These approaches attempt to mitigate the problems of the game of Go having a high branching factor and numerous other difficulties.
Computer Go research results are being applied to other similar fields such as cognitive science, pattern recognition and machine learning. Combinatorial Game Theory, a branch of applied mathematics, is a topic relevant to computer Go.
The only choice a program needs to make is where to place its next stone. However, this decision is made difficult by the wide range of impacts a single stone can have across the entire board, and the complex interactions various stones' groups can have with each other. Various architectures have arisen for handling this problem. The most popular use:
- some form of tree search,
- the application of Monte Carlo methods,
- the application of pattern matching,
- the creation of knowledge-based systems, and
- the use of machine learning.
Few programs use only one of these techniques exclusively; most combine portions of each into one synthetic system.
Minimax tree search
One traditional AI technique for creating game playing software is to use a minimax tree search. This involves playing out all hypothetical moves on the board up to a certain point, then using an evaluation function to estimate the value of that position for the current player. The move which leads to the best hypothetical board is selected, and the process is repeated each turn. While tree searches have been very effective in computer chess, they have seen less success in Computer Go programs. This is partly because it has traditionally been difficult to create an effective evaluation function for a Go board, and partly because the large number of possible moves each side can make each leads to a high branching factor. This makes this technique very computationally expensive. Because of this, many programs which use search trees extensively can only play on the smaller 9×9 board, rather than full 19×19 ones.
There are several techniques, which can greatly improve the performance of search trees in terms of both speed and memory. Pruning techniques such as alpha–beta pruning, Principal Variation Search, and MTD-f can reduce the effective branching factor without loss of strength. In tactical areas such as life and death, Go is particularly amenable to caching techniques such as transposition tables. These can reduce the amount of repeated effort, especially when combined with an iterative deepening approach. In order to quickly store a full-sized Go board in a transposition table, a hashing technique for mathematically summarizing is generally necessary. Zobrist hashing is very popular in Go programs because it has low collision rates, and can be iteratively updated at each move with just two XORs, rather than being calculated from scratch. Even using these performance-enhancing techniques, full tree searches on a full-sized board are still prohibitively slow. Searches can be sped up by using large amounts of domain specific pruning techniques, such as not considering moves where your opponent is already strong, and selective extensions like always considering moves next to groups of stones which are about to be captured. However, both of these options introduce a significant risk of not considering a vital move which would have changed the course of the game.
Results of computer competitions show that pattern matching techniques for choosing a handful of appropriate moves combined with fast localized tactical searches (explained above) were once sufficient to produce a competitive program. For example, GNU Go was competitive until 2008.
Novices often learn a lot from the game records of old games played by master players. There is a strong hypothesis that suggests that acquiring Go knowledge is a key to making a strong computer Go. For example, Tim Kinger and David Mechner argue that "it is our belief that with better tools for representing and maintaining Go knowledge, it will be possible to develop stronger Go programs." They propose two ways: recognizing common configurations of stones and their positions and concentrating on local battles. "Go programs are still lacking in both quality and quantity of knowledge."
After implementation, the use of expert knowledge has been proved very effective in programming Go software. Hundreds of guidelines and rules of thumb for strong play have been formulated by both high-level amateurs and professionals. The programmer's task is to take these heuristics, formalize them into computer code, and utilize pattern matching and pattern recognition algorithms to recognize when these rules apply. It is also important to have a system for determining what to do in the event that two conflicting guidelines are applicable.
Most of the relatively successful results come from programmers' individual skills at Go and their personal conjectures about Go, but not from formal mathematical assertions; they are trying to make the computer mimic the way they play Go. "Most competitive programs have required 5–15 person-years of effort, and contain 50–100 modules dealing with different aspects of the game."
This method has until recently been the most successful technique in generating competitive Go programs on a full-sized board. Some example of programs which have relied heavily on expert knowledge are Handtalk (later known as Goemate), The Many Faces of Go, Go Intellect, and Go++, each of which has at some point been considered the world's best Go program.
Nevertheless, adding knowledge of Go sometimes weakens the program because some superficial knowledge might bring mistakes: "the best programs usually play good, master level moves. However, as every games player knows, just one bad move can ruin a good game. Program performance over a full game can be much lower than master level."
One major alternative to using hand-coded knowledge and searches is the use of Monte Carlo methods. This is done by generating a list of potential moves, and for each move playing out thousands of games at random on the resulting board. The move which leads to the best set of random games for the current player is chosen as the best move. The advantage of this technique is that it requires very little domain knowledge or expert input, the trade-off being increased memory and processor requirements. However, because the moves used for evaluation are generated at random it is possible that a move which would be excellent except for one specific opponent response would be mistakenly evaluated as a good move. The result of this are programs which are strong in an overall strategic sense, but are imperfect tactically. This problem can be mitigated by adding some domain knowledge in the move generation and a greater level of search depth on top of the random evolution. Some programs which use Monte-Carlo techniques are Fuego, The Many Faces of Go v12, Leela, MoGo, Crazy Stone, MyGoFriend, and Zen.
In 2006, a new search technique, upper confidence bounds applied to trees (UCT), was developed and applied to many 9x9 Monte-Carlo Go programs with excellent results. UCT uses the results of the play outs collected so far to guide the search along the more successful lines of play, while still allowing alternative lines to be explored. The UCT technique along with many other optimizations for playing on the larger 19x19 board has led MoGo to become one of the strongest research programs. Successful early applications of UCT methods to 19x19 Go include MoGo, Crazy Stone, and Mango. MoGo won the 2007 Computer Olympiad and won one (out of three) blitz game against Guo Juan, 5th Dan Pro, in the much less complex 9x9 Go. The Many Faces of Go won the 2008 Computer Olympiad after adding UCT search to its traditional knowledge-based engine.
While knowledge-based systems have been very effective at Go, their skill level is closely linked to the knowledge of their programmers and associated domain experts. One way to break this limitation is to use machine learning techniques in order to allow the software to automatically generate rules, patterns, and/or rule conflict resolution strategies.
This is generally done by allowing a neural network or genetic algorithm to either review a large database of professional games, or play many games against itself or other people or programs. These algorithms are then able to utilize this data as a means of improving their performance. AlphaGo used this to great effect. Other programs using neural nets previously have been NeuroGo and WinHonte.
Machine learning techniques can also be used in a less ambitious context to tune specific parameters of programs which rely mainly on other techniques. For example, Crazy Stone learns move generation patterns from several hundred sample games, using a generalization of the Elo rating system.
AlphaGo, developed by Google DeepMind, made a significant advance by beating a professional human player in October 2015, using techniques that combined deep learning and Monte-Carlo tree search. AlphaGo is significantly more powerful than other previous Go programs, and the first to beat a 9 dan human professional in a game without handicaps on a full-sized board.
List of Go-playing computer programs
- AlphaGo, the first computer program to win in even matches against a professional human Go player
- AYA by Hiroshi Yamashita
- Crazy Stone by Rémi Coulom (sold as Saikyo no Igo in Japan)
- Darkforest by Facebook
- Fine Art by Tencent
- Fuego, an open source Monte Carlo program
- Goban, Macintosh OS X Go program by Sen:te (requires free Goban Extensions)
- GNU Go, an open source classical Go program
- Go++ by Michael Reiss (sold as Strongest Go or Tuyoi Igo in Japan)
- Leela, the first Monte Carlo program for sale to the public
- Leela Zero, a reimplementation of the system described in the AlphaGo Zero paper
- The Many Faces of Go by David Fotland (sold as AI Igo in Japan)
- MyGoFriend by Frank Karger
- MoGo by Sylvain Gelly; parallel version by many people.
- Pachi open source Monte Carlo program by Petr Baudiš, online version Peepo by Jonathan Chetwynd, with maps and comments as you play
- Smart Go by Anders Kierulf, inventor of the Smart Game Format
- Steenvreter by Erik van der Werf
- Zen by Yoji Ojima aka Yamato (sold as Tencho no Igo in Japan); parallel version by Hideki Kato.
Competitions among computer Go programs
Several annual competitions take place between Go computer programs, the most prominent being the Go events at the Computer Olympiad. Regular, less formal, competitions between programs used to occur on the KGS Go Server (monthly) and the Computer Go Server (continuous).
Prominent go-playing programs include Crazy Stone, Zen, Aya, Mogo, The Many Faces of Go, pachi and Fuego, all listed above; and Taiwanese-authored coldmilk, Dutch-authored Steenvreter, and Korean-authored DolBaram.
The first computer Go competition was sponsored by Acornsoft, and the first regular ones by USENIX. They ran from 1984 to 1988. These competitions introduced Nemesis, the first competitive Go program from Bruce Wilcox, and G2.5 by David Fotland, which would later evolve into Cosmos and The Many Faces of Go.
One of the early drivers of computer Go research was the Ing Prize, a relatively large money award sponsored by Taiwanese banker Ing Chang-ki, offered annually between 1985 and 2000 at the World Computer Go Congress (or Ing Cup). The winner of this tournament was allowed to challenge young players at a handicap in a short match. If the computer won the match, the prize was awarded and a new prize announced: a larger prize for beating the players at a lesser handicap. The series of Ing prizes was set to expire either 1) in the year 2000 or 2) when a program could beat a 1-dan professional at no handicap for 40,000,000 NT dollars. The last winner was Handtalk in 1997, claiming 250,000 NT dollars for winning an 11-stone handicap match against three 11–13 year old amateur 2–6 dans. At the time the prize expired in 2000, the unclaimed prize was 400,000 NT dollars for winning a nine-stone handicap match.
Many other large regional Go tournaments ("congresses") had an attached computer Go event. The European Go Congress has sponsored a computer tournament since 1987, and the USENIX event evolved into the US/North American Computer Go Championship, held annually from 1988–2000 at the US Go Congress.
Japan started sponsoring computer Go competitions in 1995. The FOST Cup was held annually from 1995 to 1999 in Tokyo. That tournament was supplanted by the Gifu Challenge, which was held annually from 2003 to 2006 in Ogaki, Gifu. The Computer Go UEC Cup has been held annually since 2007.
Rule Formalization Problems in computer-computer games
When two computers play a game of Go against each other, the ideal is to treat the game in a manner identical to two humans playing while avoiding any intervention from actual humans. However, this can be difficult during end game scoring. The main problem is that Go playing software, which usually communicates using the standardized Go Text Protocol (GTP), will not always agree with respect to the alive or dead status of stones.
While there is no general way for two different programs to "talk it out" and resolve the conflict, this problem is avoided for the most part by using Chinese, Tromp-Taylor, or American Go Association (AGA) rules in which continued play (without penalty) is required until there is no more disagreement on the status of any stones on the board. In practice, such as on the KGS Go Server, the server can mediate a dispute by sending a special GTP command to the two client programs indicating they should continue placing stones until there is no question about the status of any particular group (all dead stones have been captured). The CGOS Go Server usually sees programs resign before a game has even reached the scoring phase, but nevertheless supports a modified version of Tromp-Taylor rules requiring a full play out.
It should be noted that these rule sets mean that a program which was in a winning position at the end of the game under Japanese rules (when both players have passed) could lose because of poor play in the resolution phase, but this is not a common occurrence and is considered a normal part of the game under all of the area rule sets.
The main drawback to the above system is that some rule sets (such as the traditional Japanese rules) penalize the players for making these extra moves, precluding the use of additional playout for two computers. Nevertheless, most modern Go Programs support Japanese rules against humans and are competent in both play and scoring (Fuego, Many Faces of Go, SmartGo, etc.).
Historically, another method for resolving this problem was to have an expert human judge the final board. However, this introduces subjectivity into the results and the risk that the expert would miss something the program saw.
Many programs are available that allow computer Go engines to play against each other and they almost always communicate via the Go Text Protocol (GTP).
GoGUI and its addon gogui-twogtp can be used to play two engines against each other on a single computer system. SmartGo and Many Faces of Go also provide this feature.
To play as wide a variety of opponents as possible, the KGS Go Server allows Go engine vs. Go engine play as well as Go engine vs. human in both ranked and unranked matches. CGOS is a dedicated computer vs. computer Go server.
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- "Archived copy". Archived from the original on 2006-11-28. Retrieved 2007-02-21.CS1 maint: Archived copy as title (link)
- "Pachi – Board Game of Go / Weiqi / Baduk".
- http://www.peepo.com Archived 2011-09-04 at the Wayback Machine
- Anders Kierulf. "SmartGo".
- "Zen (go program)".
- "Computer Go Tournaments on KGS".
- "9x9 Go Server". Archived from the original on 2007-01-19. Retrieved 2007-03-25.
- "Acorn 1984 The First Computer Go Tournament". computer-go.info.
- David Fotland. "World Computer Go Championships". Retrieved 28 January 2016.
- Using GoGUI to play go computers against each other Archived 2011-03-09 at the Wayback Machine
- Co-Evolving a Go-Playing Neural Network, written by Alex Lubberts & Risto Miikkulainen, 2001
- Computer Game Playing: Theory and Practice, edited by M.A. Brauner (The Ellis Horwood Series in Artificial Intelligence), Halstead Press, 1983. A collection of computer Go articles. The American Go Journal, vol. 18, No 4. page 6. [ISSN 0148-0243]
- A Machine-Learning Approach to Computer Go, Jeffrey Bagdis, 2007.
- Minimalism in Ubiquitous Interface Design Wren, C. and Reynolds, C. (2004) Personal and Ubiquitous Computing, 8(5), pages 370–374. Video of computer Go vision system in operation shows interaction and users exploring Joseki and Fuseki.
- Monte-Carlo Go, presented by Markus Enzenberger, Computer Go Seminar, University of Alberta, April 2004
- Monte-Carlo Go, written by B. Bouzy and B. Helmstetter from Scientific Literature Digital Library
- Static analysis of life and death in the game of Go, written by Ken Chen & Zhixing Chen, 20 February 1999
- article describing the techniques underlying Mogo
|Wikibooks has a book on the topic of: Computer Go|
- video: computer Go to come
- Mick's Computer Go Page
- Extensive list of computer Go events
- All systems Go by David A. Mechner (1998), discusses the game where professional Go player Janice Kim won a game against program Handtalk after giving a 25-stone handicap.
- Kinger, Tim and Mechner, David. An Architecture for Computer Go (1996)
- Computer Go and Computer Go Programming pages at Sensei's Library
- Computer Go bibliography
- Another Computer Go Bibliography
- Computer Go mailing list
- Published articles about computer Go on Ideosphere gives current estimate of whether a Go program will be best player in the world
- Information on the Go Text Protocol commonly used for interfacing Go playing engines with graphical clients and internet servers
- The Computer Go Room on the K Go Server (KGS) for online discussion and running "bots"
- Two Representative Computer Go Games, an article about two computer Go games played in 1999, one with two computers players, and the other a 29-stone handicap human-computer game
- What A Way to Go describes work at Microsoft Research on building a computer Go player.
- Cracking Go by Feng-hsiung Hsu, IEEE Spectrum magazine (October 2007) – Why it should be possible to build a Go machine stronger than any human player