Chapter 2
Search

Early AI research emphasized the optimization of search algorithms. This approach made a lot of sense because many AI tasks can be solved by effectively by defining state spaces and using search algorithms to define and explore search trees in this state space. Search programs were frequently made tractable by using heuristics to limit areas of search in these search trees. This use of heuristics converts intractable problems to solvable problems by compromising the quality of solutions; this trade off of less computational complexity for less than optimal solutions has become a standard design pattern for AI programming. We will see in this chapter that we trade off memory for faster computation time and better results; often, by storing extra data we can make search time faster, and make future searches in the same search space even more efficient.

What are the limitations of search? Early on, search applied to problems like checkers and chess mislead early researchers into underestimating the extreme difficulty of writing software that performs tasks in domains that require general world knowledge or deal with complex and changing environments. These types of problems usually require the understanding and then the implementation of domain specific knowledge.

In this chapter, we will use three search problem domains for studying search algorithms: path finding in a maze, path finding in a static graph, and alpha-beta search in the games: tic-tac-toe and chess. The examples in this book should be included in the examples ZIP file for this book. The examples for this chapter are found in the subdirectory src that contains:

• src
• src/search
• src/search/game contains alpha-beta search framework and tic-tac-toe and chess examples
• src/search/statespace
• src/search/statespace/graphexample graph search code
• src/search/statespace/mazeexample maze search code

2.1 Representation of State Space, Nodes in Search Trees and Search Operators

We will use a single search tree representation in graph search and maze search examples in this chapter. Search trees consist of nodes that define locations in state space and links to other nodes. For some small problems, the search tree can be easily specified statically; for example, when performing search in game mazes, we can precompute a search tree for the state space of the maze. For many problems, it is impossible to completely enumerate a search tree for a state space so we must define successor node search operators that for a given node produce all nodes that can reached from the current node in one step; for example, in the game of chess we can not possibly enumerate the search tree for all possible games of chess, so we define a successor node search operator that given a board position (represented by a node in the search tree) calculates all possible moves for either the white or black pieces. The possible chess moves are calculated by a successor node search operator and are represented by newly calculated nodes that are linked to the previous node. Note that even when it is simple to fully enumerate a search tree, as in the game maze example, we still might want to generate the search tree dynamically as we will do in this chapter).

For calculating a search tree we use a graph. We will represent graphs as node with links between some of the nodes. For solving puzzles and for game related search, we will represent positions in the search space with Java objects called nodes. Nodes contain arrays of references to both child and parent nodes. A search space using this node representation can be viewed as a directed graph or a tree. The node that has no parent nodes is the root node and all nodes that have no child nodes a called leaf nodes.

Search operators are used to move from one point in the search space to another. We deal with quantized search spaces in this chapter, but search spaces can also be continuous in some applications. Often search spaces are either very large or are infinite. In these cases, we implicitly define a search space using some algorithm for extending the space from our reference position in the space. Figure 2.1 shows representations of search space as both connected nodes in a graph and as a two-dimensional grid with arrows indicating possible movement from a reference point denoted by R.

Figure 2.1: A directed graph (or tree) representation is shown on the left and a two-dimensional grid (or maze) representation is shown on the right. In both representations, the letter R is used to represent the current position (or reference point) and the arrowheads indicate legal moves generated by a search operator. In the maze representation, the two grid cells are marked with an X indicate that a search operator cannot generate this grid location.

When we specify a search space as a two-dimensional array, search operators will move the point of reference in the search space from a specific grid location to an adjoining grid location. For some applications, search operators are limited to moving up/down/left/right and in other applications; operators can additionally move the reference location diagonally.

When we specify a search space using node representation, search operators can move the reference point down to any child node or up to the parent node. For search spaces that are represented implicitly, search operators are also responsible for determining legal child nodes, if any, from the reference point.

Note: I use slightly different libraries for the maze and graph search examples. I plan to clean up this code in the future and have a single abstract library to support both maze and graph search examples.

2.2 Finding paths in mazes

The example program used in this section is MazeSearch.java in the directory src/search/maze and I assume that the reader has downloaded the entire example ZIP file for this book and placed the source files for the examples in a convenient place. Figure 2.2 shows the UML class diagram for the maze search classes: depth first and breadth first search. The abstract base class AbstractSearchEngine contains common code and data that is required by both the classes DepthFirstSearch and BreadthFirstSearch. The class Maze is used to record the data for a two-dimensional maze, including which grid locations contain walls or obstacles. The class Maze defines three static short integer values used to indicate obstacles, the starting location, and the ending location.

Figure 2.2: UML class diagram for the maze search Java classes

The Java class Maze defines the search space. This class allocates a two-dimensional array of short integers to represent the state of any grid location in the maze. Whenever we need to store a pair of integers, we will use an instance of the standard Java class java.awt.Dimension, which has two integer data components: width and height. Whenever we need to store an x-y grid location, we create a new Dimension object (if required), and store the x coordinate in Dimension.width and the y coordinate in Dimension.height. As in the right hand side of Figure 2.1, the operator for moving through the search space from given x-y coordinates allows a transition to any adjacent grid location that is empty. The Maze class also contains the x-y location for the starting location (startLoc) and goal location (goalLoc). Note that for these examples, the class Maze sets the starting location to grid coordinates 0-0 (upper left corner of the maze in the figures to follow) and the goal node in (width 1)-(height 1) (lower right corner in the following figures).

The abstract class AbstractSearchEngine is the base class for both DepthFirstSearchEngine and BreadthFirstSearchEngine. We will start by looking at the common data and behavior defined in AbstractSearchEngine. The class constructor has two required arguments: the width and height of the maze, measured in grid cells. The constructor defines an instance of the Maze class of the desired size and then calls the utility method initSearch to allocate an array searchPath of Dimension objects, which will be used to record the path traversed through the maze. The abstract base class also defines other utility methods:

• equals(Dimension d1, Dimension d2) checks to see if two Dimension arguments are the same
• getPossibleMoves(Dimension location) returns an array of Dimension objects that can be moved to from the specified location. This implements the movement operator.

Now, we will look at the depth first search procedure. The constructor for the derived class DepthFirstSearchEngine calls the base class constructor and then solves the search problem by calling the method iterateSearch. We will look at this method in some detail. The arguments to iterateSearch specify the current location and the current search depth:

The class variable isSearching is used to halt search, avoiding more solutions, once one path to the goal is found.

We set the maze value to the depth for display purposes only:

Here, we use the super class getPossibleMoves method to get an array of possible neighboring squares that we could move to; we then loop over the four possible moves (a null value in the array indicates an illegal move):

Record the next move in the search path array and check to see if we are done:

If the next possible move is not the goal move, we recursively call the iterateSearch method again, but starting from this new location and increasing the depth counter by one:

Figure 2.3 shows how poor of a path a depth first search can find between the start and goal locations in the maze. The maze is a 10 by 10 grid. The letter S marks the starting location in the upper left corner and the goal position is marked with a G in the lower right hand corner of the grid. Blocked grid cells are painted light gray. The basic problem with the depth first search is that the search engine will often start searching in a bad direction, but still find a path eventually, even given a poor start. The advantage of a depth first search over a breadth first search is that the depth first search requires much less memory. We will see that possible moves for depth first search are stored on a stack (last in, first out data structure) and possible moves for a breadth first search are stored in a queue first in, first out data structure).

Figure 2.3: Using depth first search to find a path in a maze finds a non-optimal solution

The derived class BreadthFirstSearch is similar to the DepthFirstSearch procedure with one major difference: from a specified search location, we calculate all possible moves, and make one possible trial move at a time. We use a queue data structure for storing possible moves, placing possible moves on the back of the queue as they are calculated, and pulling test moves from the front of the queue. The effect of a breadth first search is that it fans out uniformly from the starting node until the goal node is found.

The class constructor for BreadthFirstSearch calls the super class constructor to initialize the maze, and then uses the auxiliary method doSearchOn2Dgrid for performing a breadth first search for the goal. We will look at the method BreadthFirstSearch in some detail. The class DimensionQueue implements a standard queue data structure that handles instances of the class Dimension.

The method doSearchOn2Dgrid is not recursive, it uses a loop to add new search positions to the end of an instance of class DimensionQueue and to remove and test new locations from the front of the queue. The two-dimensional array allReadyVisited keeps us from searching the same location twice. To calculate the shortest path after the goal is found, we use the predecessor array:

We start the search by setting the already visited flag for the starting location to true value and adding the starting location to the back of the queue:

This outer loop runs until either the queue is empty of the goal is found:

We peek at the Dimension object at the front of the queue (but do not remove it) and get the adjacent locations to the current position in the maze:

We loop over each possible move; if the possible move is valid (i.e., not null) and if we have not already visited the possible move location, then we add the possible move to the back of the queue and set the predecessor array for the new location to the last square visited (head is the value from the front of the queue). If we find the goal, break out of the loop:

We have processed the location at the front of the queue (in the variable head), so remove it:

Now that we are out of the main loop, we need to use the predecessor array to get the shortest path. Note that we fill in the searchPath array in reverse order, starting with the goal location:

Figure 2.4 shows a good path solution between starting and goal nodes. Starting from the initial position, the breadth first search engine adds all possible moves to the back of a queue data structure. For each possible move added to this queue in one search cycle, all possible moves are added to the queue for each new move recorded. Visually, think of possible moves added to the queue, as fanning out like a wave from the starting location. The breadth first search engine stops when this wave reaches the goal location. In general, I prefer breadth first search techniques to depth first search techniques when memory storage for the queue used in the search process is not an issue. In general, the memory requirements for performing depth first search is much less than breadth first search.

Figure 2.4: Using breadth first search in a maze to find an optimal solution

To run the two example programs from this section, change directory to src/search/maze and type:

Note that the classes MazeDepthFirstSearch and MazeBreadthFirstSearch are simple Java JFC applications that produced Figures 2.3 and 2.4. The interested reader can read through the source code for the GUI test programs, but we will only cover the core AI code in this book. If you are interested in the GUI test programs and you are not familiar with the Java JFC (or Swing) classes, there are several good tutorials on JFC programming at java.sun.com.

2.3 Finding Paths in Graphs

In the last section, we used both depth first and breadth first search techniques to find a path between a starting location and a goal location in a maze. Another common type of search space is represented by a graph. A graph is a set of nodes and links. We characterize nodes as containing the following data:

• A name and/or other data
• Zero or more links to other nodes
• A position in space (this is optional, usually for display or visualization purposes)

Links between nodes are often called edges. The algorithms used for finding paths in graph are very similar to finding paths in a two-dimensional maze. The primary difference is the operators that allow us to move from one node to another. In the last section, we saw that in a maze, an agent can move from one grid space to another if the target space is empty. For graph search, a movement operator allows movement to another node if there is a link to the target node.

Figure 2.5 shows the UML class diagram for the graph search Java classes that we will use in this section. The abstract class AbstractGraphSearch class is the base class for both DepthFirstSearch and BreadthFirstSearch. The classes GraphDepthFirstSearch and GraphBreadthFirstSearch and test programs that also provide a Java Foundation Class (JFC) or Swing based user interface. These two test programs produced Figures XXXX and XXXXX.

Figure 2.5: UML class diagram for the graph search classes

As seen in Figure 2.5, most of the data for the search operations (i.e., nodes, links, etc.) is defined in the abstract class AbstractGraphSearch. This abstract class is customized through inheritance to use a stack for storing possible moves (i.e., the array path) for depth first search and a queue for breadth first search.

The abstract class AbstractGraphSearch allocates data required by both derived classes:

The abstract base class also provides several common utility methods:

• addNode(String name, int x, int y) adds a new node
• addLink(int n1, int n2) adds a bidirectional link between nodes indexed by n1 and n2. Node indexes start at zero and are in the order of calling addNode.
• addLink(String n1, String n2) - adds a bidirectional link between nodes specified by their names
• getNumNodes() returns the number of nodes
• getNumLinks() returns the number of links
• getNodeName(int index) returns a nodes name
• getNodeX(), getNodeY() return the coordinates of a node
• getNodeIndex(String name) gets the index of a node, given its name

The abstract base class defines an abstract method that must be overridden:

We will start with the derived class DepthFirstSearch, looking at its implementation of findPath.

The findPath method returns an array of node indices indicating the calculated path:

The class variable path is an array that is used for temporary storage; we set the first element to the starting node index, and call the utility method findPathHelper:

The method findPathHelper is the interesting method in this class that actually performs the depth first search; we will look at it in some detail:

The path array is used as a stack to keep track of which nodes are being visited during the search. The argument num_path is the number of locations in the path, which is also the search depth:

First, re check to see if we have reached the goal node; if we have, make a new array of the current size and copy the path into it. This new array is returned as the value of the method:

We have not found the goal node, so call the method connected_nodes to find all nodes connected to the current node that are not already on the search path (see the source code for the implementation of connected_nodes):

If there are still connected nodes to search, add the next possible node to visit to the top of the stack (variable path) and recursively call findPathHelper again:

If we have not found the goal node, return null, instead of an array of node indices:

The derived class BreadthFirstSearch also must define the abstract method findPath. This method is very similar to the breadth first search method used for finding a path in a maze: a queue is used to store possible moves. For a maze, we used a queue class that stored instances of the class Dimension; so for this problem, the queue only needs to store integer node indices. The return value of findPath is an array of node indices that make up the path from the starting node to the goal.

We start by setting up a flag array alreadyVisited to prevent visiting the same node twice, and allocating a predecessors array that we will use to find the shortest path once the goal is reached:

The class IntQueue is a private class defined in the file BreadthFirstSearch.java; it implements a standard queue:

Before the main loop, we need to initialize the already visited and predecessor arrays, set the visited flag for the starting node to true, and add the starting node index to the back of the queue:

The main loop runs until either we find the goal node or the search queue is empty:

We will read (without removing) the node index at the front of the queue and calculate the nodes that are connected to the current node (but not already on the visited list) using the connected_nodes method (the interested reader can see the implementation in the source code for this class):

For each node connected by a link to the current node, if it has not already been visited set the predecessor array and add the new node index to the back of the search queue; we stop if the goal is found:

Now that the goal node has been found, we can build a new array of returned node indices for the calculated path using the predecessor array:

In order to run both the depth first and breadth first graph search examples, change directory to JavaAI2/src/search/graph and type the following commands:

Figure 2.6 shows the results of finding a route from node 1 to node 9 in the small test graph. Like the depth first results seen in the maze search, this path is not optimal.

Figure 2.6: Using depth first search in a sample graph

Figure ??shows an optimal path found using a breadth first search. As we saw in the maze search example, we find optimal solutions using breadth first search at the cost of extra memory required for the breadth first search.

Figure 2.7: Using breadth first search in a sample graph

2.4 Adding heuristics to Breadth First Search

We can usually make breadth first search more efficient by ordering the search order for all branches from a given position in the search space. For example, when adding new nodes, from a specified reference point in the search space, we might want to add nodes to the search queue first that are in the direction of the goal location: in a two-dimensional search like our maze search, we might want to search connected grid cells first that were closest to the goal grid space. In this case, pre-sorting nodes (in order of closest distance to the goal) added to the breadth first search queue could have a dramatic effect on search efficiency. In the next chapter, we will build a simple real-time planning system around our breadth first maze search program; this new program will use heuristics. The alpha beta additions to breadth first search are seen in in the next section.

2.5 Search and Game Playing

Now that a computer program has won a match against the human world champion, perhaps peoples expectations of AI systems will be prematurely optimistic. Game search techniques are not real AI, but rather, standard programming techniques. A better platform for doing AI research is the game of Go. There are so many possible moves in the game of Go, that brute force look ahead (as is used in Chess playing programs) simply does not work.

That said, min-max type search algorithms with alpha-beta cutoff optimizations are an important programming technique and will be covered in some detail in the remainder of this chapter. We will design an abstract Java class library for implementing alpha-beta enhanced min-max search, and then use this framework to write programs to play tic-tac-toe and chess.

2.5.1 Alpha-Beta search

The first game that we will implement will be tic-tac-toe, so we will use this simple game to explain how the min-max search (with alpha-beta cutoffs) works.

Figure 2.8 shows the possible moves generated from a tic-tac-toe position where X has made three moves and O has made 2 moves; it is Os turn to move. This is level 0 in Figure 2.8. At level 0, O has four possible moves. How do we assign a fitness values to each of Os possible moves at level 0? The basic min-max search algorithm provides a simple solution to this problem: for each possible move by O in level 1, make the move and store the resulting 4 board positions. Now, at level 1, it is Xs turn to move. How do we assign values to each of Xs possible three moves in Figure 2.8? Simple, we continue to search by making each of Xs possible moves and storing each possible board position for level 2. We keep recursively applying this algorithm until we either reach a maximum search depth, or there is a win, loss, or draw detected in a generated move. We assume that there is a fitness function available that rates a given board position relative to either side. Note that the value of any board position for X if the negative of the value for O.

Figure 2.8: Alpha-beta algorithm applied to part of a game of tic-tac-toe

To make the search more efficient, we maintain values for alpha and beta for each search level. Alpha and beta determine the best possible/worst possible move available at a given level. If we reach a situation, like the second position in level 2, where X has won, then we can immediately determine that Os last move in level 1 that produced this position (of allowing X an instant win) is a low valued move for O (but a high valued move for X). This allows us to immediately prune the search tree by ignoring all other possible positions arising from the first O move in level 1. This alpha-beta cutoff (or tree pruning) procedure can save a large percentage of search time, especially if we can set the search order at each level with probably best moves considered first.

While tree diagrams as seen in Figure 2.8 quickly get complicated, it is easy for a computer program to generate possible moves, calculate new possible board positions and temporarily store them, and recursively apply the same procedure to the next search level (but switching min-max sides in the board evaluation). We will see in the next section that it only requires about 100 lines of Java code to implement an abstract class framework for handling the details of performing an alpha-beta enhanced search. The additional game specific classes for tic-tac-toe require about an additional 150 lines of code to implement; chess requires an additional 450 lines of code.

2.5.2 A Java Framework for Search and Game Playing

The general interface for the Java classes that we will develop in this section was inspired by the Common LISP game-playing framework written by Kevin Knight and described in (Rich, Knight 1991). The abstract class GameSearch contains the code for running a two-player game and performing an alpha-beta search. This class needs to be sub classed to provide the eight methods:

The method drawnPosition should return a Boolean true value if the given position evaluates to a draw situation. The method wonPosition should return a true value if the input position is won for the indicated player. By convention, I use a Boolean true value to represent the computer and a Boolean false value to represent the human opponent. The method positionEvaluation returns a position evaluation for a specified board position and player. Note that if we call positionEvaluation switching the player for the same board position, then the value returned is the negative of the value calculated for the opposing player. The method possibleMoves returns an array of objects belonging to the class Position. In an actual game, like chess, the position objects will actually belong to a chess-specific refinement of the Position class (e.g., for the chess program developed later in this chapter, the method possibleMoves will return an array of ChessPosition objects). The method makeMove will return a new position object for a specified board position, side to move, and move. The method reachedMaxDepth returns a Boolean true value if the search process has reached a satisfactory depth. For the tic-tac-toe program, the method reachedMaxDepth does not return true unless either side has won the game or the board is full; for the chess program, the method reachedMaxDepth returns true is the search has reached a depth of 4 have moves deep (this is not the best strategy, but it has the advantage of making the example program short and easy to understand). The method getMove returns an object of a class derived from the class Move (e.g., TicTacToeMove or ChessMove).

The GameSearch class implements the following methods to perform game search:

The method alphaBeta is simple; it calls the helper method alphaBetaHelper with initial search conditions; the method alphaBetaHelper then calls itself recursively. The code for alphaBeta is:

It is important to understand what is in the vector returned by the methods alphaBeta and alphaBetaHelper. The first element is a floating point position evaluation for the point of view of the player whose turn it is to move; the remaining values are the best move for each side to the last search depth. As an example, if I let the tic-tac-toe program play first, it placed a marker at square index 0, then I placed my marker in the center of the board an index 4. At this point, to calculate the next computer move, alphaBeta is called and returns the following elements in a vector:

Here, the alpha-beta enhanced min-max search looked all the way to the end of the game and these board positions represent what the search procedure calculated as the best moves for each side. Note that the class TicTacToePosition (derived from the abstract class Position) has a toString method to print the board values to a string.

The same printout of the returned vector from alphaBeta for the chess program is:

Here, the search procedure assigned the side to move (the computer) a position evaluation score of 5.4; this is an artifact of searching to a fixed depth. Notice that the board representation is different for chess, but because the GameSearch class manipulates objects derived from the classes Position and Move, the GameSearch class does not need to have any knowledge of the rules for a specific game. We will discuss the format of the chess position class ChessPosition in more detail when we develop the chess program.

The classes Move and Position contain no data and methods at all. The classes Move and Position are used as placeholders for derived classes for specific games. The search methods in the abstract GameSearch class manipulate objects derived from the classes Move and Position.

Now that we understand the contents of the vector returned from the methods alphaBeta and alphaBetaHelper, it will be easier to understand how the method alphaBetaHelper works. The following text shows code fragments from the alphaBetaHelper method interspersed with book text:

Here, we notice that the method signature is the same as for alphaBeta, except that we pass floating point alpha and beta values. The important point in understanding min-max search is that most of the evaluation work is done while backing up the search tree; that is, the search proceeds to a leaf node (a node is a leaf if the method reachedMaxDepth return a Boolean true value), an then a return vector for the leaf node is created by making a new vector and setting its first element to the position evaluation of the position at the leaf node and setting the second element of the return vector to the board position at the leaf node:

If we have not reached the maximum search depth (i.e., we are not yet at a leaf node in the search tree), then we enumerate all possible moves from the current position using the method possibleMoves and recursively call alphaBetaHelper for each new generated board position. In terms of Figure 2.8, at this point we are moving down to another search level (e.g., from level 1 to level 2; the level in Figure 2.8 corresponds to depth argument in alphaBetaHelper):

Notice that when we recursively call alphaBetaHelper, that we are flipping the player argument to the opposite Boolean value. After calculating the best move at this depth (or level), we add it to the end of the return vector:

When the recursive calls back up and the first call to alphaBetaHelper returns a vector to the method alphaBeta, all of the best moves for each side are stored in the return vector, along with the evaluation of the board position for the side to move.

The GameSearch method playGame is fairly simple; the following code fragment is a partial listing of playGame showing how to call alphaBeta, getMove, and makeMove:

The debug printout of the vector returned from the method alphaBeta seen earlier in this section was printed using the following code immediately after the call to the method alphaBeta:

In the next few sections, we will implement a tic-tac-toe program and a chess-playing program using this Java class framework.

2.5.3 TicTacToe using the alpha beta search algorithm

Using the Java class framework of GameSearch, Position, and Move, it is simple to write a simple tic-tac-toe program by writing three new derived classes (see Figure 2.9) TicTacToe (derived from GameSearch), TicTacToeMove (derived from Move), and TicTacToePosition (derived from Position).

Figure 2.9: UML class diagrams for game search engine and tic-tac-toe

I assume that the reader has the code from my web site installed and available for viewing. In this section, I will only discuss the most interesting details of the tic-tac-toe class refinements; I assume that the reader can look at the source code. We will start by looking at the refinements for the position and move classes. The TicTacToeMove class is trivial, adding a single integer value to record the square index for the new move:

The board position indices are in the range of [0..8] and can be considered to be in the following order:

The class TicTacToePosition is also simple:

This class allocates an array of nine integers to represent the board, defines constant values for blank, human, and computer squares, and defines a toString method to print out the board representation to a string.

The TicTacToe class must define the following abstract methods from the base class GameSearch:

The methods that are overridden from the GameSearch base class must always cast arguments of type Position and Move to TicTacToePosition and TicTacToeMove. Note that in the method drawnPosition, the argument of class Position is cast to the class TicTacToePosition. A position is considered to be a draw if all of the squares are full. We will see that checks for a won position are always made before checks for a drawn position, to that the method drawnPosition does not need to make a redundant check for a won position. The method wonPosition is also simple; it uses a private helper method winCheck to test for all possible winning patterns in tic-tac-toe. The method positionEvaluation uses the following board features to assign a fitness value from the point of view of either player:

The method positionEvaluation is simple, and is a good place for the interested reader to start modifying both the tic-tac-toe and chess programs:

The only other method that we will look at here is possibleMoves; the interested reader can look at the implementation of the other (very simple) methods in the source code. The method possibleMoves is called with a current position, and the side to move (i.e., program or human):

It is very simple to generate possible moves: every blank square is a legal move. (This method will not be as simple in the example chess program!)

It is simple to compile and run the example tic-tac-toe program: change directory to src/search/game and type:

When asked to enter moves, enter an integer between 0 and 8 for a square that is currently blank (i.e., has a zero value). The following shows this labeling of squares on the tic-tac-toe board:

2.5.4 Chess using the alpha beta search algorithm

Using the Java class framework of GameSearch, Position, and Move, it is reasonably simple to write a simple chess program by writing three new derived classes (see Figure 2.10) Chess (derived from GameSearch), ChessMove (derived from Move), and ChessPosition (derived from Position). The chess program developed in this section is intended to be an easy to understand example of using alpha-beta min-max search; as such, it ignores several details that a fully implemented chess program would implement:

Allow the computer to play either side (computer always plays black in this example) Allow en-passant pawn captures. Allow the player to take back a move after making a mistake

The reader is assumed to have read the last section on implementing the tic-tac-toe game; details of refining the GameSearch, Move, and Position classes are not repeated in this section.

Figure 2.10 shows the UML class diagram for both the general purpose GameSearch framework and the classes derived to implement chess specific data and behavior.

Figure 2.10: UML class diagrams for game search engine and chess

The class ChessMove contains data for recording from and to square indices:

The board is represented as an integer array with 120 elements. A chessboard only has 64 squares; the remaining board values are set to a special value of 7, which indicates an off board square. The initial board setup is defined statically in the Chess class:

The class ChessPosition contains data for this representation and defines constant values for playing sides and piece types:

The class Chess also defines other static data. The following array is used to encode the values assigned to each piece type (e.g., pawns are worth one point, knights and bishops are worth 3 points, etc.):

The following array is used to codify the possible incremental moves for pieces:

The starting index into the pieceMovementTable array is calculated by indexing the following array with the piece type index (e.g., pawns are piece type 1, knights are piece type 2, bishops are piece type 3, rooks are piece type 4, etc.:

When we implement the method possibleMoves for the class Chess, we will see that, except for pawn moves, that all other possible piece type moves are very simple to calculate using this static data. The method possibleMoves is simple because it uses a private helper method calcPieceMoves to do the real work. The method possibleMoves calculates all possible moves for a given board position and side to move by calling calcPieceMove for each square index that references a piece for the side to move.

We need to perform similar actions for calculating possible moves and squares that are controlled by each side. In the first version of the class Chess that I wrote, I used a single method for calculating both possible move squares and controlled squares. However, the code was difficult to read, so I split this initial move generating method out into three methods:

• possibleMoves required because this was an abstract method in GameSearch. This method calls calcPieceMoves for all squares containing pieces for the side to move, and collects all possible moves.
• calcPieceMoves responsible to calculating pawn moves and other piece type moves for a specified square index.
• setControlData sets the global array computerControl and humanControl. This method is similar to a combination of possibleMoves and calcPieceMoves, but takes into effect moves onto squares that belong to the same side for calculating the effect of one piece guarding another. This control data is used in the board position evaluation method positionEvaluation.

We will discuss calcPieceMoves here, and leave it as an exercise to carefully read the similar method setControlData in the source code. This method places the calculated piece movement data in static storage (the array piece_moves) to avoid creating a new Java object whenever this method is called; method calcPieceMoves returns an integer count of the number of items placed in the static array piece_moves. The method calcPieceMoves is called with a position and a square index; first, the piece type and side are determined for the square index:

Then, a switch statement controls move generation for each type of chess piece (movement generation code is not shown):

The logic for pawn moves is a little complex but the implementation is simple. We start by checking for pawn captures of pieces of the opposite color. Then check for initial pawn moves of two squares forward, and finally, normal pawn moves of one square forward. Generated possible moves are placed in the static array piece_moves and a possible move count is incremented. The move logic for knights, bishops, rooks, queens, and kings is very simple since it is all table driven. First, we use the piece type as an index into the static array index; this value is then used as an index into the static array pieceMovementTable. There are two loops: an outer loop fetches the next piece movement delta from the pieceMovementTable array and the inner loop applies the piece movement delta set in the outer loop until the new square index is off the board or runs into a piece on the same side. Note that for kings and knights, the inner loop is only executed one time per iteration through the outer loop:

The method setControlData is very similar to this method; leave it as an exercise to the reader to read through the source code. Method setControlData differs in also considering moves that protect pieces of the same color; calculated square control data is stored in the static arrays computerControl and humanControl. This square control data is used in the method positionEvaluation that assigns a numerical rating to a specified chessboard position or either the computer or human side. The following aspects of a chessboard position are used for the evaluation:

Notice that the evaluation is calculated initially assuming the computers side to move; if the position if evaluated from the human players perspective, the evaluation value is multiplied by minus one. The implementation of positionEvaluation is: