Question
Proving that G matches L: [.5pts per prompt = 5pts] Finish the Inductive Proof L = {w | w has an equal number of as
Proving that G matches L: [.5pts per prompt = 5pts]
Finish the Inductive Proof L = {w | w has an equal number of as and bs}
// we have 3 non-terminals
1 S -> aB // starts with a
2 S -> bA // starts with b
// comes to A after starting with b. A means balance that b with a.
3 A -> a
4 A -> aS
5 A -> bAA
// comes to B after starting with a. B means balance that a with b.
6 B -> b
7 B -> bS
8 B -> aBB
We were doing induction to prove that G can generate all strings of L.
GIVEN ASSUMPTIONS when |w| <= k-1
T1. S can =*=> w if w consists of an equal number of a's and b's
T2. A can =*=> w if w has one more a than b's
T3. B can =*=> w if w has one more b than a`s
You must now prove the properties (T2 and T3) of A and B
for the |w| = k step.
Please fill in the template below. For Why questions, refer to the above given assumptions. E.g. Using the given assumption T1.
T2 step for A:
Prove that if w has one more a than b's, A can =*=> w where |w| = k
(describe w in terms of shorter strings z, y1 and y2)
w looks like: a< z > or b
Thus z and y1 and y2 have properties such that:
z: ?> ?>
each y1 and y2: ??> ??>
What non-terminal can derive z? ??
Why???
What non-terminal can derive each y???
Why???
Thus A =*=> w:
Starting with these rules from A: **
T3 step for B:
Prove that if w has one more b than a's, B can =*=> w where |w| = k
Has to be done for the proof to be complete. But it is not required for this HW.
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