Formally we define a propositional net, or prop net, as a directed
bipartite graph, consisting of propositions alternating
with connectives. In this case there are six propositions.
The round circles labeled a, b, p, q, r, and s.
And there are four connectives.
The grey and black nodes in the graph shown here.
Basic prop nets, there are four types of
connectives and they're all present in this case.
There is an and gate on the upper left, an inverter in the upper right,
an or gate on the lower right, and a transition on the lower left.
Propositions are typically partitioned into three classes.
Input propositions, those with, are those with no inputs.
Base propositions are those with incoming arcs from transitions.
And view propositions are those with
incoming arcs from connectives other than transitions.
In our example here, nodes a and b are input propositions.
Note s is a base proposition, and notes p, q, and r are view propositions.
An input marking, is a function from the
input propositions of a propositional net to boolean values.
a base marking is a function from the
base propositions of a propositional net boolean values.
And a view marking is a function from the
view propositions of a propositional net to boolean values.
And we use the word marking to refer to
a combination of an input base and view marking.
Now given a prop net, an input marking, and a
base marking, determine a unique view marking for that prop net.
This is based on the types of
the connectives feeding into the view propositions.
The output of an inverter is true, if and only if its input is false, for example.
The output of an and gate is true if and only if all of its inputs are true.
That's suggested by the second table.
And the output of an or gate is true if
and only if at least one of its inputs is true.
Transitions behave just like and gates, except,
that there's a one step time delay.
The output occurs one step after the inputs.
Here's an example.
Suppose that we had an input marking that assigned the node
A, the value one, and the node B, the value zero.
That's the node in the upper left and the down node in the lower right.
And suppose we had a base marking that assigned s, the value one.
That's the one in the lower left.
Then, the output of the and gate would be a one.
The output of the inverter would be a zero.
The output of the or gate would also be a zero.
And at this point, we then have values for
all of the view propositions in the prop net.
Now let's move on to the next step.
Supposing once again the input is, is, has A being one and B being zero.
What's the value of our base proposition on this step?
Since it's the output of a transition, its value on this step is
the same as the value of that transitions input on the preceding step.
In this case the transition input was zero on that preceding step as
we just saw, and so the value is zero on this new step.
As before, we can compute the view marking corresponding
to this new input marking and this new base marking.
In this case, since the second input to the and gate is now 0, the output is 0.
The output of the inverter becomes 1, and the output of the or gate is 1 as well.
Now, if we leave the inputs the same, for subsequent
steps, the prop net will go on, alternating this way.
If input A ever becomes false, it will stop
alternating, however the alternation will begin again, as soon as it's set to true.