To solve a problem when there is no clear computation for a valid solution, we turn to path finding. This chapter covers two related path-finding approaches: one using game trees for two-player games and the other using search trees for single-player games. These approaches rely on a common structure, namely a state tree whose root node represents the initial state and edges represent potential moves that transform the state into a new state. The searches are challenging because the underlying structure is not computed in its entirety due to the explosion of the number of states. In a game of checkers, for example, there are roughly 5*10^{20} different board configurations (Schaeffer, 2007). Thus, the trees over which the search proceeds are constructed on demand as needed. The two path-finding approaches are characterized as follows:
Two players take turns alternating moves that modify the game state from its initial state. There are many states in which either player can win the game. There may also be some “draw” states in which no one wins. A path-finding algorithm maximizes the chance that a player will win or force a draw.
A single player starts from an initial board state and makes valid moves until the desired goal state is reached. A path-finding algorithm identifies the exact sequence of moves that will transform the initial state into the goal state.
The game of tic-tac-toe is played on a 3×3 board where ...
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