Propnets are an alternative to GDL for
expressing the dynamics of games.
With a few additional provisions, it's
possible to convert any
GDL game description into a propositional
net, with the same dynamics.
As an example of this, consider the simple
game shown here.
It's just one role, white.
There's just one base proposition, and
there are two actions.
The two actions are always legal.
The player gets 100 points if S is true,
otherwise the player gets zero points, and
the game ends if Q ever becomes true.
Notice the dynamics: if A is performed
and S is true then P becomes true.
If P is false Q becomes true.
If Q is true R becomes true and if Y
performs B then R becomes true as well.
Now, let's build a prop net for this game.
We start with the game's physics.
The base proposition's in the prop net
consist of the propositions
defined in the base relation in the game
description, namely S.
The input propositions correspond to the
actions defined by the
input relation in the game description,
namely A and B.
We use the next relation to capture the
dynamics of
the game, starting with the base and input
propositions, where we
add links for each rule, using inverters
for the negations,
and gates for multiple conditions, and or
gates for multiple rules.
And so doing, we augment the propos-,
prop net with additional new propositions
as necessary.
The result is the prop net shown here
which is
exactly the same as the one introduced in
the previous segment.
Okay now, we have more things to model to
complete our game description.
First of all termination.
Model terminate in the prop net by adding
a special node for termination.
In this case, terminal corresponds exactly
to q.
Remember, the game is over if q ever
becomes true.
So, we could just use q as our terminal
node.
However, for the sake of clarity we can
also add a new
node, t, as shown here, and insert a
connection from q to t.
Note that I've used an one-input and gate.
I could equally have used a two-input and
gate with both inputs supplied
by q, but this is simpler and the behavior
is exactly the same.
Rewards are handled analogously.
We create a new node for each reward value
and we
use the definitions for these values to
extend the prop net further.
In this case, Goal of white being one
hundred corresponds exactly to S,
so we don't really need to add a new node
for that case.
However for clarity in our example again
we have added
a new node for this case and labeled it
100.
And then we've also added a node for zero
which is the true
in exactly the opp, the rema, other cases
from the case where the goal is
100.
Legality is by far the trickiest part.
Of for input in, in encoding GDL in prop
nets.
There are various ways we can do this.
The simplest method conceptually is to
simply
add one legality proposition for each
possible
action, and extend the prop net to say
that when these nodes are true.
In this case, we would add two new
propositions, la
and lb, for the legality of a and legality
of b.
In this case there always true, actions
can always be performed, to model
this we add a self loop for each of these
legality propositions, and
then we initialize that proposition to
true, of course because of the dynamics
of transitions the propositions remain
true indefinitely
meaning that those actions are always
legal.
And that's it.
Once this is done, we have a prop net
that reflects the game described in the
given GDL.
In the next section, we discuss how to use
this prop net to play games.