In the introduction, we saw that it's possible to think of the
dynamics of the game as a state grpah, like the one shown here.
The game is characterized by a finitie number of states and also
a finite number of players and there's a finite number of actions for each player.
At each point in time the graph is in one of the possible states.
The players choose from their possible actions and
as the players perform their chosen actions
the game changes from one state to another.
In the vast majority of games, states and actions, however, are not monolithic.
They can be defined in terms of more fundamental entities.
In chess, for example, states can be thought of
in terms of locations of individual pieces on the chessboard.
For example, the state shown on the left here can be thought of in terms of
the locations of the white king, the white
queen, the black king, and the black queen.
Also, in the vast majority of games, the effects of individual actions are local.
As actions are performed, some of these
conditions become true and others become false.
However the truth values of the majority of these conditions remain the same.
The ideas of state decomposition and limited influence of actions suggest
the conceptualization of games in terms of individual propositions and actions
rather than states, together with the representation of the effects of
these actions on these propositions rather than on the entire states.
Results is an alternative representation of dynamics called a Propositional Net.
Unlike a state
machine, in which the nodes represent states,
in a Propositional Net, the nodes denote
propositions and actions together with connectives that
capture their behavior, as we shall see.
One of the benefits of formalizing games' propositional nets is compactness.
A set of end propositions corresponds to a set of 2 to the n
possible states, all of the different combinations
of the truth values for the n propositions.
Thus it's often possible to characterize the dynamics of games
with graphs that are much smaller than the corresponding state machines.
For example, a prop net with just three propositions corresponds
to a state machine with eight possible states.
The second benefit of propositional nets is ease of analysis.
It's sometimes possible to use a prop net to discover independence, or game factors,
dead states, and other features that can
dramatically reduce the cost of game tree search.
In the next segment, we formalize propositional nets.
And in the section thereafter, we show how to describe games in this way.
And in the segment after that, we see how to play
games using game descriptions encoded as propositional nets rather than in GDL.
And we then see how to use propositional
nets in restructuring games and discovering game-specific heuristics.