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.