Games solved:Now and in the future H.jaap van den Herik, Jos W.H.M. Uiterwijk, Jack van Rijswijck...

Games solved:Now and in the Games solved:Now and in the futurefuture

H.jaap van den Herik, Jos W.H.M. Uiterwijk, Jack van Rijswijck

Summarized by Seung-joon Yi

(c) 2002 SNU CSE Biointelligence Lab and Center for Bioinformation Techonology (CBIT)

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IntroductionIntroduction

The domain of strategic games To witness the ‘intelligence’ of computers To establish the game-theoretic value of a game, i.e., the

outcome when all participants play optimally.


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ConventionsConventions

Two-person zero-sum games with perfect information. Connection games Mancala games Endgame problems of chess-like games A subclass of other games.

At least three different definitions of a solution can be used. Ultra-weakly solved

game-theoretic value of the initial position has been determined Weakly solved

For the initial position a strategy has been determined to achieve the game-theoretic value against any opposition.

Strongly solved Such a strategy has been determined for all legal positions.


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Classification by complexityClassification by complexity

Connect-four,Qubic,Nine men’s morris, Go-moku is now solved.

We can say that the predictions were rather accurate.


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Classification by complexityClassification by complexity

State-space complexity No. of legal game positions reachable from the initial position of the game.

Game-tree complexity No. of leaf nodes in the solution search tree of the initial position of the game.

Categoy-1 is most suitible to being solved. Solving category-2 games is dependent on the computer power. Solving category-3 games is dependent on the development of new AI techniques. Solving category-4 games is practically impossible.


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QuestionsQuestions

Can perfect knowledge obtained from solved games be translated into rules and strategies which human beings can assimilate?

Are such rules generic, or do they constitute a multitude of ad hoc recipes?

Can methods be transferred between games? More specifically, are there generic methods for all category-n games, or is each game in a specific category a law unto itself?


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Methods developed for solving gamesMethods developed for solving games

Brute-force methods Retrograde analysis

For each position of some specific game or endgame the number of moves towards the best reachable goal is stored.

Database is constructed by starting in terminal positions and then working backwards.

Does not guarantee the best performance in a position which is game-theoretically drawn or lost.

Enhanced transposition-table methods The ‘traditional’ transposition table used in game-playing programs

normally exploit the DEEP replacement scheme, i.e., when two different positions compete for the same entry in the table, the old position is overwritten by the newer one provided that the latter is searched at least as deep as the former.

Research on this and other replacement schemes showed that two-level replacement schemes are often more appropriate.


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Methods developed for solving gamesMethods developed for solving games

Knowledge-based methods Provides an appropriate move ordering or selection in the search

trees. Threat-space search and λ-search

– Investigates whether by a sequence of threats, to which the opponent at any time has only a limited set of replies, a win can be forced.

Proof-number search Depth-first proof-number search Pattern search


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Classification by game typeClassification by game type

Convergence We consider a game to be convergent when the size of the state

space decreases as the game prograsses. If the size of the state increases, the game is said to be

divergent.

Informally, convergent games start with many pieces on the board and pieces are gradually removed during the course of the game, while divergent games start with an empty or almost empty board and pieces are added during the game.


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Convergent gamesConvergent games

Unlike divergent games, convergent games by their definition allow for the construction of endgame databases.

These databases are a powerful tool for solving the games by retrograde analysis, starting at the goal states and working backward.

Nine men’s morris Mancala family of games Checkers and Chess endgames


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Nine men’s morrisNine men’s morris

Gasser solved the game in 1995 Total number of states:7,673,759,269. He built all 28 w-b middle-game and endgame

databases, and an 18-ply search from the start position using the DB was performed.

After exloiting the DB, it was estabilished that the game-theoretic value of the game is a draw.

Rules Opening phase Middle game phase Endgame phase


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Mancala gamesMancala games

All use a playing board with a number of holes, called pits. The pits are distributed over 2 or more rows;in addition there

are 0,1,2 large holes, called stores. Usually starts with an equal number of stones in each pit, and with

no stone in the stores. A turn consists of

– Taking all stones out of a pit

– Distributing them one after another over the subsequent pits.

Hundreads of mancala games Awari Kalah Dakon


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AwariAwari

At present, database for up to 38 stones are calculated and the 39-stone DB is under construction.

36-stone DB shows that Awari with 3 stones per pit in the initial position is a draw.

Few more years at most is expected to solve the awari weakly.


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KalahKalah

Kalah(m,n), m being the number of pits per side and n the number of stones per pit, is weakly solved up to (6,5) using a combination of endgame databases and search.

Standard game, Kalah(6,4), is a first-player win.


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DakonDakon

‘Winning opening’ If the starting player captures n(2n-1) stones, the game is over in

a single move. Winning opening for Dakon-8 was found ‘by hand’

Except for Dakon-3, many winning openings was found for n up to 18. Thus these games are weakly solved.


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CheckersCheckers

CHINOOK officially became the first man-machine world champion in any game in 1994.

Next goal:weakly solving the game of checkers for all 144 valid three-move opening sequences played in tournaments.

– Endgame databases:the program has perfect information about all positions involving eight or fewer pieces on the board, a total of 443,748,401,247 positions, compressed into 6GB.

– Middlegame databases:Whenever the programe is able to determine the game-theoretic value of a position during a game, the position is added to the middle-game database. In practice, position with as many as 22 pieces on the board have been solved.

– Verification of opening analysis:starting with published lines of play for each opening, deep searches are used to try and solve positions as close to the start of the game as possible. These solved positions are added to the middle-game database.

CHINOOK proved the ‘100 year position’ for less than a second.


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Chess endgamesChess endgames

Endgames play a most prominent role in chess. Thompson built all important 5-piece endgames and made them

accessible to the public. However it is unlikely that important endgames with more than

6 (non-blocked) pieces will be built in near future.


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Summary-convergent gamesSummary-convergent games

Nine men’s Morris, the mancala games, and endgames in checkers and chess all belong to category 2. Endgame databases combined with search algorithm have been

successful in attacking these games. Nine men’s morris has been solved, as well as the smaller

versions of Kalah and Dakon. Increasing computing power and storage capacity are expected

to be instrumental in solving Awari within a few years, and Checkers within a decade.

Full game of chess belongs to category 4.


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Divergent gamesDivergent games

Divergent games are immune to the retrograde-analysis methods that were prevalent in section 2. Connection games Polymino games The games of Othello, Shogi, and Go.


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Connection gamesConnection games

Connect-four Weakly solved by allen using a brute-force

approach, and by Allis using a knowledge-based approach.

First-player win. Qubic

Weakly solved in 1980, combining depth-first search and expert knowledge for ordering the moves.

First-player win. Go-moku

Allis estibalished that the game-theoretic value is a first-player win, using combination of threat-space search, proof-number search and database construction.


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Connection gamesConnection games

Renju Weakly solved as first-player

win by Wagner and Virag in 2000, using transposition tables and threat-sequence search, and expert knowledge for no-threat moves.

General k-in-a-row games K-in-a row in m by n boards Many game-theoretic values

of many mnk-games have been published, based on using knowledge-based rules.


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Polymino gamesPolymino games

Pentomino Two-player version has been weakly

solved using a straightforward search program based on opening-move suggestions by the user.

Domineering Many weakly-solved instances.


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Othello and ShogiOthello and Shogi

Othello Buro’s LOGISTELLO playes

stronger than the human World Champion.

6 by 6 game is weakly solved. Shogi

Not convergent, unlike western and chinese chess.

Tsume-Shogi solving program SEO weakly solved a well-known problem with remarkable solution length of 1525 steps, setting a world record.


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GoGo

19 by 19 Go is much too complex to solve with the current means. Localized endgame positions may be subject to exhaustivie analysis. Largest square board solved is 4 by 4, a first-player win.


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Summary-nonconvergent gamesSummary-nonconvergent games

Connect-Four and Qubic belong to category 1 Category 1 games are solved by both approaches.

Go-moku, Renju,mnk-games to category 3. Category 3 games are solved by a combination of expert

knowledge, threat-space search, threat-sequence search, proof-number search.

The polyomino games, Othello, and Go on small boards belong to category 2. Category 2 games are solved by brute-force methods. Instances on larger boards will typicaly belong to category 4.


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Game characteristics detemining Game characteristics detemining solverbilitysolverbility Conclusion

Low state-space complexity is more important than a low game-tree complexity as a determining factor in solving games.

State-space complexity provides a bound on the complexity of games solvable by complete enumeration.

The game-tree complexities form a real challenge for intelligent search methods.

Brute-force versus Knowledge-based methods Games with relatively low state-space

complexity have mainly been solved with brute-force emthods.

Games with a relatively low game-tree complexity have mainly been solved with knowledge-based methods.

Games with a relatively low complexity in both measure have been solved by both methods.


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ConclusionConclusion

The use of retrograde analysis in building DB has had the greatest impact on solving (end)games or almost-solving games.

The knowledge-based methods mostly inform us the structure of the games.

Questions revisited Can perfect knowledge obtained from solved games be translated into rules and

strategies which human beings can assimilate? The most difficult solved (end)games are still a mystery for human experts. The best we can hope for is that perfect knowledge is translated into a correct strategy.

Are such rules generic, or do they constitute a multitude of ad hoc recipes? In fact, the database itself is a long list of ad hoc recipes. Hence,the research question is how to combine them into tractable cluster of analogue

positions and then to formulate a human-understandable rule. Can methods be transferred between games?

Data mining attempt Understanding many intricacies of a game is a prerequisite to applying one of the proven

methods out of the data-mining areas succesfully.

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Games solved:Now and in the future H.jaap van den Herik, Jos W.H.M. Uiterwijk, Jack van Rijswijck...

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