(40 pts.)
Another proposed semantics for logic with negation is inflationary
semantics.
Here, we start out assuming no IDB facts, and we apply the rules to
infer some IDB facts.
Once inferred, an IDB fact can never become false.
More formally, the inflationary semantics for a Datalog program and EDB
is defined by:

Begin with all IDB predicates empty.

Apply the rules of the program once, using as true whatever EDB facts were
given and whatever IDB facts have already been inferred on previous
rounds.
Every other ground atom is assumed false.
Note this process differs from the GL transform or alternating fixpoint
in two ways:

The rules are only applied once, not repeatedly in this round.

A nonnegated IDB subgoal can be assumed true if it was true on the
previous round, rather than having to be inferred on this round, as in
the GL transform.

If at least one new fact has been inferred:

Add all new facts to the set of inferred IDB facts.

Repeat step (2).
Here is a simple, propositional example, that may help clarify the definition:
p : q
q : NOT r
s : NOT p
Round 0: None of p, q, r, and s are true.
Round 1: The falsehood of r lets us infer q, and the falsehood of p lets us infer s.
The inflationary model so far is {q,s}.
Round 2: The truth of q lets us infer p.
The inflationary model is now {p,q,s}.
Note that even though p is now true, the use of NOT p to infer s stands.
Round 3: No new inferences are possible; the inflationary model is {p,q,s}.
Your problems:
 a)

What is the inflationary semantics of the red/green bus line rules and
EDB given in class (second set of slides)?
Recall the rules are:
greenPath(X,Y) : green(X,Y)
greenPath(X,Z) : greenPath(X,Z) & greenPath(Z,Y)
monopoly(X,Y) : red(X,Y) & NOT greenPath(X,Y)
and the EDB is {red(1,2), red(2,3), green(1,2)}.
 b)

What is the inflationary semantics of the win program.
Interestingly, the inflationary semantics hardly depends on the EDB.
If we know the set of boards mentioned in the EDB, and we know one
simple property of the move relation (you must figure out the
property), then you can describe succinctly the winning and losing boards
according to the inflationary semanics.
 c)

One of the problems with inflationary semantics, as in (a) and (b), is
that if we don't write the rules carefully, we wind up inferring certain
IDB facts ``too soon,'' that is, using a negated subgoal that eventually
turns out not to be true (i.e., it is of the form NOT p, where
p eventually becomes true).
Show how to rewrite the red/green bus line rules so that, at least for
this EDB, the inflationary model of your rules
is the same as the stratified model of the rules given in class (i.e.,
greenPath(1,2) and monopoly(2,3) are the only IDB facts).
You can actually rewrite the rules so that the inflationary model is the
stratified model of the original rules on every EDB, and it would
be great if you can discover how.
But we'll accept something that works on only this EDB.