Board Representation
The chessboard is represented in the simplest possible manner - as an 8 by 8 matrix, each containing a Piece (with a "blank" piece representing empty board spaces). Furthermore, flag variables keep track of whether queen/king side castling is allowed for each player, and whether an en-passant capture move is allowed at a given point in time.
In order to save space and time during the min-max search, its optimal not to have separate board instance at each branch. After all, they differ only by the position of one piece. Hence each move contains information not only about which piece was moved from where to where, but also whether it affected castling, en passant, and whether it captured an enemy piece in the process. Thus reversing a move on a board is very simple.
The algorithm thus only needs one board object, on which it makes and reverses all the moves it considers during its search.
Advanced chess playing programs have far more clever board representations, which operate on bits. Separate instances are kept to keep track of individual pieces, and often bit-wise operations can reveal a lot of information about board positions very quickly (particularly with respect to pawns).
Years of research have been spent trying to optimize these representations for speed.
Min-max Searching
The core of the chess playing algorithm is a local min-max search of the gamespace.
The algorithm attempts to MINimize the opponent's score, and MAXimize its own. At each depth
(or "ply" as it's as its referred to in computer chess terminology), all possible moves are examined,
and the static board evaluation function is used to determine the score at the leafs of the search tree.
These scores propagate up the tree and are used to select the optimal move at each depth. The bigger
the ply, the better the chosen move will be (as the algorithm is able to look ahead more moves). The
branching factor is typically between 25 and 40 moves per ply (the average is around 35).
Alpha-beta Pruning
This common pruning fuction is used to considerably decrease the min-max search space. It essentially keeps track of the worst and best moves for each player so far, and using those can completely avoid searching branches which are guaranteed to yield worse results. Using this pruning will return the same exact moves as using min-max (i.e. there is no loss of accuracy). Ideally, it can double the depth of the search tree without increasing search time. To get close to this optimum, the available moves at each branch should be appropriately sorted.The sorting is done by the looking at the scores of each possible move, looking only 1 ply ahead. The intuitive sort would be to arrange them from best to worst, but that's not always best. Most moves in chess end up being ones with small gains and losses (ones that improve position, not necessarily capturing pieces), so it makes sense to order the "stand pat" moves first. So the sorting is based on the absolute value of the move scores (with the smaller ones coming first). The average branching factor for min-max in chess is about 35, but with the alpha-beta pruning and sorting, the program acheives a branching factor of around 25.
Null Move Heuristic
This simple heuristic improves the beginning of the alpha-beta search. Initially, there are no values for what
the worst and best possible moves are, so they default to negative and positive infinity respectively. But using
this heuristic the algorithm can find an initial lower bound on the best moves. It lets the opposing player play
two moves in sequence (choosing them based on a small-ply min-max search), and computes the score after that.
Certainly, any move made by the current player should beat a score obtainable by the opponent getting two chances
to move.
Quiescence Searching
Since the depth of the min-max search is limited, problems can occur at the frontier. A move that may seem great may actually be a disaster because of something that could happen on the very next move. Looking at all these possibilites would mean increasing the ply by 1, which is not the solution, as we would need to extend it to arbitrarily large depths. The goal is thus to search the tree until "quiescent" positions are found - i.e ones that don't affect the current positions too much (most maneuvers in chess result in only slight advantages or disadvantages to each player, not big ones at once). Hence, looking at higher depths is important only for significant moves - such as captures. Consider for example a move in which you capture the opponent's knight with your queen. If that is the limit of your min-max search, it seems to be a great move - you receive points for capturing the opponent's knight. But suppose that in the very next move your opponent can capture your queen. Then the move is clearly seen as bad, as trading a queen for a knight is to your disadvantage. Quiescence searching will be able to detect that by looking at the next move. Again, it doesn't need to do this for every move - just for ones that affect the score a lot (like captures).
One important caveat in the quiescence searching algorithm is that it should only look at moves that became available because of the current move being made. Consider the following situation. Your bishop is threatened by an opponent's pawn, and you have the ability to capture the opponent's knight with a different pawn. Suppose the algorithm is looking only 1 ply ahead, and is examining some non-capturing move. It won't notice that the bishop can be captured in the next turn. But what happens when it's examining the knight-capturing move with quiescence. It will see that the opponent can take your bishop, which will even out the piece possession, making the move not seem as good. So it's highly likely that the algorithm would pick a move other than capturing the knight, thus needlessly losing the bishop in the next turn. To prevent this, the algorithm must look at ONLY those moves available because of its own move. Since the opponent's "pawn captures bishop" was available regardless of whether you capture the knight or not, it should be ignored.
Static Board Evaluation Function
When the min-max algorithm gets down to the leaves of its search, it's unlikely that it reached a goal state (i.e. a check-mate).
Therefore, it needs some way to determine whether the given board position is "good" or "bad" for it, and to what degree.
A numerical answer is needed so that it can be compared to other board positions in a quantifiable way. Advanced chess
playing programs can look at hundreds features of the board to evalaute it. The simplest, and perhaps most intuitive, look
at only piece possession. Clearly, having a piece is better than not having one (in most cases at least). Furthermore, the
pieces have different values. A pawn is worth the least; the bishop and knight are next, then the rook, and finally: the queen.
The king is obviously priceless, as losing it means losing the game.
The additional features that my chess program examines are:
pawn advancement
How far up the board has each pawn advanced. Reaching the opposite end is important because it promotes the pawn to a different piece.piece mobility (separate for each type of piece)
How many different spaces can the given piece move to?piece threats (separate for each type of piece)
How many of the opponent's pieces are threatened by attack? This includes checks (which is a threat on the king)piece protects (separate for each type of piece)
How many of your own piece are protecting the given piece to prevent it from being captured without reprecussion?
The total number of unique numbers that sum up to give the total board score is thus 25. The allowable weights for each
feature are forced to be integers. This allows for faster computations (as opposed to using real numbers). This isn't a real
loss of generality though, as the scale of numbers used can be large (very large integers to represent piece possession, and
small ones to represent the other features).
Optimizing board evaluation functions via genetic algorithms while certain aspects of evaluating a board are obvious (such as piece values - a queen is clearly worth more than a pawn), other factors are not as easily determined purely by intuition. How much is a bishop's mobility worth? How important is it to check the opponent? Is threatening an enemy's piece better than protecting your own? One can make relatively good educated guesses to such questions, and thus develop a decent static board evaluation function, but I was hoping for a more analytical method. My goal was to attempt to optimize
the board evaluation function by utilizing genetic algorithms to determine it. One module of the program is capable of running chess tournaments, where the computer plays against itself with different evaluation functions. It generates random evaluation functions, which then get mutated or preserved based on how well they perform in the tournaments. The core of the tournament algorithm does the following. It has a set of 10 evaluation functions, and pits them all against each other. Each side gets to play both black and white for fairness. Subsequently, it selects the best five, and generates 5 new ones to replace the worst 5. This continues for any desirable number of iterations (the default was set to 10). There are two version of the algorithm that were run. One was a "preservation" one, which kept the best 5 "as is" in between iterations. The other algorithm was a "mutation" one, which kept 1 of the 5, and mutated the other 4. Each mutation was between a pairing of some 2 of the best 5 functions.
Determining the winner of a given game is not always trivial. For time constraints, each game in the tournament is limited to 50 moves, which won't necessarily yield an outright check-mate. Also, draws are possible. Furthermore, for low plys (a ply of 2 was used), it is unlikely for the computer to ever reach check-mate when playing deterministically against itself (since there is not end-game database). But the genetic algorithm requires that there be a "winner" for each game played. The way this done is by scoring the board position from the perspective of each of the functions. Most likely they will both has a consensus as to which side has more points (and hence is winning); however, since obviously each side has a different evaluation function, there is a small probability in a close game that each side will think it's winning. The starting functions weren't completely random. For instance, the piece possession values were always preset to fixed values, as those are well known to be good.
Since possession is more important than any other factors, the randomized weights generated for the other were allowed only to be integers between 0 and 5.However, this still allowed for relatively large weights overall - for instance, a rook could theoretically have a mobility of 14 spaces (7 horizontal and 7 vertical), so even if it's mobility factor was only 3, and there were two rooks, this was worth a whopping 14*3*2 = 84.
Unfortunately, the results of the tournaments weren't as productive as one would expect. This is because the static board evaluation function often seem to be circular in nature. By this, I mean the following: suppose you have three different functions, A, B, and C. It is possible that A beats B, B beats C, and C beats A. Hence it's impossible to tell which one is "better." Clearly, some functions in extreme cases are always worse than others - for instance, if we make protecting bishops and knights worthless, but protecting pawns worth a lot, then the AI using this function is likely to lose key pieces quickly. But for functions that are deemed "reasonable," the genetic algorithms in their current form will fail to determine which ones are better overall. Another problem is that only a very small subset of all possible functions can be examined. There are 19 factors in each function, each of which can take on 5 different values. This yields 5^19 possible functions, even with those limitations. But in each round of a tournament, only 10 functions are examined, by running 10^2 = 100 games, which takes hours even at low ply levels.
Some general observations, however, both from the tournaments and from observations of individual matches, can be made. The pieces with higher values ought to have higher mobility/threats/ weights as well. It makes sense that threatening a queen is more valuable than threatening a bishop or a knight. The opposite is true for the "protects" weights. It doesn't make much sense in protecting a queen too much, because if it gets killed with anything other than the opponent's queen, killing the capturing piece is little consolation. Protecting knights and bishops is very valuable, however. In the current scheme, assigning weights to the pawns' parameters is often detrimental, as there are 8 of them (multiplying all the weights by 8), and it may cause an unecessary overuse of the piece by the computer. Pawn advancement seems to be a sufficent parameter for dictating pawn maneuvers. Checking (threatening) a king is also valuable, as it can be considered a "local goal" of the ultimate goal, which is a check-mate. With all these factors in mind, the default static board evaluation has been set to:
With a pawn advancement weight of 1.
This is by no means the only decent board evaluation function - many others work just as well, or better in certain games.
More on the speed efficency of this is discussed in the performace section.
You can also look at the raw data for a run of the preservation tournament and the mutation tournament.
Opening Move Database
Many combinations of the first few initial moves on the board are known to establish a good position on the board. Thus, over
the years, chess masters have developed what is essentially an encyclopedia of these openings. The program can use this database
at the beginning of play - instead of performing the usual local search, it simply checks whether the moves performed so far fit into a given opening strategy, and then playes the appropriate response. The database contains over 2000 openings, stored in a binary, allowing for constant time access. Furthermore, a text-based version of the database is available, which can easily be expanded by a user to include more openings. The program is then capable of parsing the text and creating a new binary tree of moves out of it. For many of the very first moves, there are many possible responses. However, some are deemed more common (or better) than others, and the program nondeterministically selects the more common responses with a higher probability than the less common ones. You can look at the text version of the database used by the program. You can also look how to extend the database.
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