Synthesis of Strategies for Games
Finding winning strategies of some combinatorial games, such as the NIM game, tic-tae-toe, etc., can be formulated as solving the unknown component problem. Therefore, BALM can be used to synthesize winning strategies of these combinatorial games. The strategy we take is to describe the dynamics and the state of the game in the fixed component. The unknown component represents one of the players of a two person game. Generally, we want to input the state of the game to the unknown component, which will represent the strategy to be used by Player 2; otherwise the unknown component would have to have many states just to remember what state the game is in. The other player, Player 1, can be modeled as a random player. The reason for this is that it is simpler to not have to describe the strategy for making moves; we will allow it to make any move. If it makes an illegal move, it loses the game immediately. In this way, Player 2 only has to have a strategy for when Player 1 makes a legal move. A winning strategy is such that whatever move Player 1 makes, Player 2 has the ability to make a move to continue the possibility of winning.