MAE 552 – Heuristic Optimization Lecture 28 April 5, 2002 Topic:Chess Programs Utilizing Tree Searches

The material in this lecture is from

Tree Searches as Applied to Chess Programs How can a computer play chess? For many people that is a mind-boggling concept. Chess seems like a distinctly human activity requiring intelligence and thought, so how can a computer possibly do it? Today we will talk about how efficient tree searches are vital to a successful chess program

Tree Searches as Applied to Chess Programs How does a computer play chess??? What you will find is that computers don't really "play" chess like people do. A computer that is playing chess is not "thinking.” It is calculating through a set of formulas that cause it to make good moves. As computers have gotten faster and faster, the quality of these calculated moves has gotten better and better.

Tree Searches as Applied to Chess Programs In 1997 a computer built by IBM, deep blue defeated Gary Kasparov the world champion is chess. Computers chess calculators are now the best chess players on the planet, even though they do it totally blindly!

How People Play Chess Computers chess calculators are now the best chess players on the planet, even though they do it totally blindly! If you have ever watched a person first learning to play chess, you know that a human chess player starts with very limited abilities. Once a player understands the basic rules that control each piece, he or she can "play" chess. However, the new player is not very good. Each early defeat comes as something of a surprise -- "Oh, I didn't think about that!" or "I didn't see that coming!" are common exclamations.

Tree Searches as Applied to Chess Programs The human mind absorbs these experiences, stores away different board configurations, discovers certain tricks and ploys, and generally soaks up the nuances of the game one move at a time. For a human being, therefore, the game of chess involves a great deal of high-level abstract thought -- visual pattern matching to recall board positions, rules and guidelines, conscious thought and even psychology! Computers do none of this... So how does a computer win at chess against the best human players???

Tree Searches as Applied to Chess Programs The current state-of-the-art in computer chess is fairly intricate, but all of it involves blind computation that is very simple at the core. Consider the start of game of chess….. Each player starts with 16 pieces Let's say that white starts. White has 20 possible moves: The white player can move any pawn forward one or two positions. (16 moves). The white player can move either knight in two different ways. (4 moves).

Initial Moves Possible for White Player RKBKiQBKR PPPPPPPP PPPPPPPP RKB QBKR

Tree Searches as Applied to Chess Programs The white player chooses one of those 20 moves and makes it. For the black player, the options are the same: 20 possible moves. So black chooses a move. Now white can move again. This next move depends on the first move that white chose to make, but there are about 20 or so moves white can make given the current board position, and then black has 20 or so moves it can make, and so on.

Tree Searches as Applied to Chess Programs This is how a computer looks at chess. It thinks about it in a world of "all possible moves," and it makes a big tree for all of those moves, like this:

Tree Searches as Applied to Chess Programs

Let us determine how many possible moves are represented on each level. Initially white has 20 possible moves to make. Black has 20 moves possible for each moves that white could make - 20*20=400 possible moves. Then white has about 20 moves for each move black could make - 20*20*20=8000 Then black has about 20 moves for each move black could make - 20*20*20*20=160,000

Tree Searches as Applied to Chess Programs If you were to fully develop the entire tree for all possible chess moves, the total number of board positions is about: 1,000,000,000,000,000,000,000,000, 000,000,000,000,000,000,000,000,000,000,000,00 0,000,000,000,000,000,000,000,000,000,000,000, 000,000,000,000,000, 000,000,000,000, or , give or take a few.

Tree Searches as Applied to Chess Programs That's a very big number. –Consider there have only been nanoseconds since the Big Bang. –There are thought to be only atoms in the entire universe. –When you consider that the Milky Way galaxy contains billions of suns, and there are billions of galaxies, you can see that that's a whole lot of atoms. –That number is dwarfed by the number of possible chess moves. Chess is a pretty intricate game!

Tree Searches as Applied to Chess Programs No computer is ever going to calculate the entire tree. What a chess computer tries to do is generate the board-position tree five or 10 or 20 moves into the future. Assuming that there are about 20 possible moves for any board position, a five-level tree contains 3,200,000 board positions. A 10-level tree contains about 10,000,000,000,000 (10 trillion) positions. The depth of the tree that a computer can calculate is controlled by the speed of the computer playing the game.

Tree Searches as Applied to Chess Programs Once it generates the tree, then the computer needs to "evaluate the board positions." That is, the computer has to look at the pieces on the board and decide whether that arrangement of pieces is "good" or "bad." The way it does this is by using an evaluation function. The simplest possible function might just count the number of pieces each side has. If the computer is playing white and a certain board position has 11 white pieces and nine black pieces, the simplest evaluation function might be: = 2

Tree Searches as Applied to Chess Programs Obviously, for chess that formula is way too simple, because some pieces are more valuable than others. So the formula might apply a weight to each type of piece Eval=w 1 *P 1 +w 2 *P 2 ……..etc.

Tree Searches as Applied to Chess Programs As the programmer thinks about it, he or she makes the evaluation function more and more complicated by adding things like: board position control of the center vulnerability of the king to check vulnerability of the opponent's queen and tons of other parameters. No matter how complicated the function gets, however, it is condensed down to a single number that represents the "goodness" of that board position.

Tree Searches as Applied to Chess Programs Consider the following situation. The diagram shows a three-level tree that looks three moves ahead and has evaluated the value of the final board positions.

Tree Searches as Applied to Chess Programs The values of each board position indicate the utility for the white player (the computer). In this tree, white can make three possible moves. From each of those three possible moves, black can make three possible moves. From each of those nine board positions, white can make two possible moves.

Tree Searches as Applied to Chess Programs To decide what to do, the computer looks at this tree and works upward from the bottom. Its calculations are set up so that it finds the best board positions from each of the possible positions black will take (it takes the maximum):

Tree Searches as Applied to Chess Programs One level up, it assumes that black will choose the worst possible position for white (it takes the minimum):

Tree Searches as Applied to Chess Programs Finally, it takes the maximum of the top three numbers: 7. That is the move the computer will make. Once black makes its move, the computer goes through this whole process again, generating a new tree and evaluating all of the board positions to figure out its next move.

Tree Searches as Applied to Chess Programs In theory, the minimax algorithm allows one to play "perfect" chess; that is, the player always makes a winning move in a won position or a drawing move in a drawn position. Unfortunately, it is impossible to search the entire tree in a reasonable amount of time The computer can generally on look 5-10 moves ahead.

Tree Searches as Applied to Chess Programs Normally a variant of the A * algorithm is used to search the tree - the best potential moves are examined first. A method call alpha-beta pruning is used to reduce the size of the tree by getting rid of branches that are sub-optimal This can speed the search to a particular depth by as much as 50%. This is increase the number of moves a computer can look ahead.