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Propnets are an alternative to GDL for
expressing the dynamics of games.

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With a few additional provisions, it's
possible to convert any

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GDL game description into a propositional
net, with the same dynamics.

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As an example of this, consider the simple
game shown here.

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It's just one role, white.

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There's just one base proposition, and
there are two actions.

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The two actions are always legal.

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The player gets 100 points if S is true,
otherwise the player gets zero points, and

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the game ends if Q ever becomes true.
Notice the dynamics: if A is performed

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and S is true then P becomes true.
If P is false Q becomes true.

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If Q is true R becomes true and if Y
performs B then R becomes true as well.

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Now, let's build a prop net for this game.
We start with the game's physics.

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The base proposition's in the prop net
consist of the propositions

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defined in the base relation in the game
description, namely S.

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The input propositions correspond to the
actions defined by the

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input relation in the game description,
namely A and B.

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We use the next relation to capture the
dynamics of

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the game, starting with the base and input
propositions, where we

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add links for each rule, using inverters
for the negations,

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and gates for multiple conditions, and or
gates for multiple rules.

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And so doing, we augment the propos-,

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prop net with additional new propositions
as necessary.

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The result is the prop net shown here
which is

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exactly the same as the one introduced in
the previous segment.

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Okay now, we have more things to model to
complete our game description.

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First of all termination.

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Model terminate in the prop net by adding
a special node for termination.

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In this case, terminal corresponds exactly
to q.

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Remember, the game is over if q ever
becomes true.

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So, we could just use q as our terminal
node.

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However, for the sake of clarity we can
also add a new

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node, t, as shown here, and insert a
connection from q to t.

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Note that I've used an one-input and gate.

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I could equally have used a two-input and
gate with both inputs supplied

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by q, but this is simpler and the behavior
is exactly the same.

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Rewards are handled analogously.

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We create a new node for each reward value
and we

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use the definitions for these values to
extend the prop net further.

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In this case, Goal of white being one
hundred corresponds exactly to S,

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so we don't really need to add a new node
for that case.

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However for clarity in our example again
we have added

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a new node for this case and labeled it
100.

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And then we've also added a node for zero
which is the true

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in exactly the opp, the rema, other cases
from the case where the goal is

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100.
Legality is by far the trickiest part.

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Of for input in, in encoding GDL in prop
nets.

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There are various ways we can do this.

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The simplest method conceptually is to
simply

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add one legality proposition for each
possible

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action, and extend the prop net to say
that when these nodes are true.

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In this case, we would add two new
propositions, la

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and lb, for the legality of a and legality
of b.

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In this case there always true, actions
can always be performed, to model

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this we add a self loop for each of these
legality propositions, and

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then we initialize that proposition to
true, of course because of the dynamics

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of transitions the propositions remain
true indefinitely

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meaning that those actions are always
legal.

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And that's it.

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Once this is done, we have a prop net

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that reflects the game described in the
given GDL.

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In the next section, we discuss how to use
this prop net to play games.