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.