pumping lemma problem: –let b be the language of all palindromes over {0,1} containing an equal...
TRANSCRIPT
![Page 1: Pumping Lemma Problem: –Let B be the language of all palindromes over {0,1} containing an equal number of 0s and 1s. Show that B is not CFL Solution:](https://reader036.vdocuments.site/reader036/viewer/2022062320/56649c735503460f94926242/html5/thumbnails/1.jpg)
![Page 2: Pumping Lemma Problem: –Let B be the language of all palindromes over {0,1} containing an equal number of 0s and 1s. Show that B is not CFL Solution:](https://reader036.vdocuments.site/reader036/viewer/2022062320/56649c735503460f94926242/html5/thumbnails/2.jpg)
Pumping Lemma
Problem: – Let B be the language of all palindromes over {0,1} containing an
equal number of 0s and 1s. Show that B is not CFL
Solution:– We assume that B is a CFL and obtain a contradiction.
– Let p be the pumping constant of B that is guaranteed to exist by the pumping lemma. Select the string s = 0p1p1p0p .
– Clearly s is a member of B and of length at least p . We can show that no matter how we divide s into uvxyz, one of the conditions of the lemma is violated.
• Pumping lemma, | vxy |<= p , we can only place vxy in the following ways:
![Page 3: Pumping Lemma Problem: –Let B be the language of all palindromes over {0,1} containing an equal number of 0s and 1s. Show that B is not CFL Solution:](https://reader036.vdocuments.site/reader036/viewer/2022062320/56649c735503460f94926242/html5/thumbnails/3.jpg)
1. vxy completely falls in the first 0p
– If we pump v and y , then the new string is no longer a palindrome, and the number of 0s will be greater than the number of 1s, which is a contradiction.
2. vxy falls between 0p and 1p
– In this case, v will only contain 0s while y will only contain 1s. So if we pump s , the new string is no longer palindrome, which is a contradiction.
3. vxy completely falls in the first 1p
– Similar to (1), after pumping s , the number of 1s will be greater than the number of 0s, which is a contradiction.
4. vxy falls between 1p and 1p
– After pumping s , the number of 1s will be greater than the number of 0s, which is a contradiction.
5. vxy completely falls in the second 1p
– This case is same with (3). 6. vxy falls between 1p and 0p
– Similar to (2), v will only contain 1s while y will only contain 0s. So if we pump s , the new string is no longer palindrome, which is a contradiction.
7. vxy completely falls in the second 0p– This case is same with (1).
Hence, B is not context free.
![Page 4: Pumping Lemma Problem: –Let B be the language of all palindromes over {0,1} containing an equal number of 0s and 1s. Show that B is not CFL Solution:](https://reader036.vdocuments.site/reader036/viewer/2022062320/56649c735503460f94926242/html5/thumbnails/4.jpg)
• Let L = {w {0; 1}* : w = wR}.
• (a) Show that L is context-free by giving a context-free grammar for L.
• (b) Show that L is context-free by giving a pushdown automaton for L.
• (c) Show that L is not regular.
![Page 5: Pumping Lemma Problem: –Let B be the language of all palindromes over {0,1} containing an equal number of 0s and 1s. Show that B is not CFL Solution:](https://reader036.vdocuments.site/reader036/viewer/2022062320/56649c735503460f94926242/html5/thumbnails/5.jpg)
CFG
• L is a symmetric language. Consider a context-free grammar for L : (V, Σ, R, S) , where
i. V = {S}
ii. Σ = {0,1}
iii. Rules:
S 0S0 |1S1| 0 |1|ε
iv. S = S V
![Page 6: Pumping Lemma Problem: –Let B be the language of all palindromes over {0,1} containing an equal number of 0s and 1s. Show that B is not CFL Solution:](https://reader036.vdocuments.site/reader036/viewer/2022062320/56649c735503460f94926242/html5/thumbnails/6.jpg)
PDA
![Page 7: Pumping Lemma Problem: –Let B be the language of all palindromes over {0,1} containing an equal number of 0s and 1s. Show that B is not CFL Solution:](https://reader036.vdocuments.site/reader036/viewer/2022062320/56649c735503460f94926242/html5/thumbnails/7.jpg)
Pumping Lemma
• Assume that L regular. Let p be the pumping constant given by the pumping lemma.
• Let s be the string 1p-1101p. • s can be split into three pieces, s = xyz , where for any i
>= 0 the string xyiz is in L .• According to the pumping lemma condition, we must
have | xy |<= p . If this is the case, then y must consist only of 1’s, so xyyz L . Therefore s cannot be pumped.
• This is a contradiction.
![Page 8: Pumping Lemma Problem: –Let B be the language of all palindromes over {0,1} containing an equal number of 0s and 1s. Show that B is not CFL Solution:](https://reader036.vdocuments.site/reader036/viewer/2022062320/56649c735503460f94926242/html5/thumbnails/8.jpg)
![Page 9: Pumping Lemma Problem: –Let B be the language of all palindromes over {0,1} containing an equal number of 0s and 1s. Show that B is not CFL Solution:](https://reader036.vdocuments.site/reader036/viewer/2022062320/56649c735503460f94926242/html5/thumbnails/9.jpg)
Pushdown Automaton -- PDAInput String
Stack
States
![Page 10: Pumping Lemma Problem: –Let B be the language of all palindromes over {0,1} containing an equal number of 0s and 1s. Show that B is not CFL Solution:](https://reader036.vdocuments.site/reader036/viewer/2022062320/56649c735503460f94926242/html5/thumbnails/10.jpg)
Initial Stack Symbol
Stack
$
Stack
z
bottomspecial symbol
stackhead
top
Appears at time 0
![Page 11: Pumping Lemma Problem: –Let B be the language of all palindromes over {0,1} containing an equal number of 0s and 1s. Show that B is not CFL Solution:](https://reader036.vdocuments.site/reader036/viewer/2022062320/56649c735503460f94926242/html5/thumbnails/11.jpg)
The States
q1 q2a, b c
Inputsymbol
Popsymbol
Pushsymbol
![Page 12: Pumping Lemma Problem: –Let B be the language of all palindromes over {0,1} containing an equal number of 0s and 1s. Show that B is not CFL Solution:](https://reader036.vdocuments.site/reader036/viewer/2022062320/56649c735503460f94926242/html5/thumbnails/12.jpg)
q1 q2a, b c
a
b top
input
stack
a
Replaceeh
$eh
$
c
![Page 13: Pumping Lemma Problem: –Let B be the language of all palindromes over {0,1} containing an equal number of 0s and 1s. Show that B is not CFL Solution:](https://reader036.vdocuments.site/reader036/viewer/2022062320/56649c735503460f94926242/html5/thumbnails/13.jpg)
q1 q2ca ,
a a
Pushb
eh
$eh
$
bc
top
input
stack
![Page 14: Pumping Lemma Problem: –Let B be the language of all palindromes over {0,1} containing an equal number of 0s and 1s. Show that B is not CFL Solution:](https://reader036.vdocuments.site/reader036/viewer/2022062320/56649c735503460f94926242/html5/thumbnails/14.jpg)
q1 q2ba,
a a
Popb
eh
$eh
$
top
input
stack
![Page 15: Pumping Lemma Problem: –Let B be the language of all palindromes over {0,1} containing an equal number of 0s and 1s. Show that B is not CFL Solution:](https://reader036.vdocuments.site/reader036/viewer/2022062320/56649c735503460f94926242/html5/thumbnails/15.jpg)
q1 q2 ,a
a a
No Changeb
eh
$eh
$
btop
input
stack
![Page 16: Pumping Lemma Problem: –Let B be the language of all palindromes over {0,1} containing an equal number of 0s and 1s. Show that B is not CFL Solution:](https://reader036.vdocuments.site/reader036/viewer/2022062320/56649c735503460f94926242/html5/thumbnails/16.jpg)
q1 q2$,a
a a
Pop$ top
input
stack
Empty Stack
empty
The automaton HALTS
No possible transition after q2
![Page 17: Pumping Lemma Problem: –Let B be the language of all palindromes over {0,1} containing an equal number of 0s and 1s. Show that B is not CFL Solution:](https://reader036.vdocuments.site/reader036/viewer/2022062320/56649c735503460f94926242/html5/thumbnails/17.jpg)
q1 q2ba $,
a a
Pop$ top
input
stack
A Possible Transition
b
![Page 18: Pumping Lemma Problem: –Let B be the language of all palindromes over {0,1} containing an equal number of 0s and 1s. Show that B is not CFL Solution:](https://reader036.vdocuments.site/reader036/viewer/2022062320/56649c735503460f94926242/html5/thumbnails/18.jpg)
Non-Determinism
q1
q2a, b c
q3a, b c
q1 q2cb,
transition
PDAs are non-deterministic
Allowed non-deterministic transitions
![Page 19: Pumping Lemma Problem: –Let B be the language of all palindromes over {0,1} containing an equal number of 0s and 1s. Show that B is not CFL Solution:](https://reader036.vdocuments.site/reader036/viewer/2022062320/56649c735503460f94926242/html5/thumbnails/19.jpg)
Example PDA
,
aa ,
ab,q0 q1 q2 q3
ab,
$$,
PDA M
}0:{)( nbaML nn
![Page 20: Pumping Lemma Problem: –Let B be the language of all palindromes over {0,1} containing an equal number of 0s and 1s. Show that B is not CFL Solution:](https://reader036.vdocuments.site/reader036/viewer/2022062320/56649c735503460f94926242/html5/thumbnails/20.jpg)
}0:{)( nbaML nn
Basic Idea:
1. Push the a’s
on the stack2. Match the b’s on input
with a’s on stack
3. Match
found
,
aa ,
ab,q0 q1 q2 q3
ab,
$$,
![Page 21: Pumping Lemma Problem: –Let B be the language of all palindromes over {0,1} containing an equal number of 0s and 1s. Show that B is not CFL Solution:](https://reader036.vdocuments.site/reader036/viewer/2022062320/56649c735503460f94926242/html5/thumbnails/21.jpg)
0q q1 q2 q3
Input
a a a b b b
Stack
$
aa , ab,
$$, , ab,
![Page 22: Pumping Lemma Problem: –Let B be the language of all palindromes over {0,1} containing an equal number of 0s and 1s. Show that B is not CFL Solution:](https://reader036.vdocuments.site/reader036/viewer/2022062320/56649c735503460f94926242/html5/thumbnails/22.jpg)
q0 q1 q2 q3
a a a b b b$
$$, ,
aa ,
ab,
ab,
Input
Stack
![Page 23: Pumping Lemma Problem: –Let B be the language of all palindromes over {0,1} containing an equal number of 0s and 1s. Show that B is not CFL Solution:](https://reader036.vdocuments.site/reader036/viewer/2022062320/56649c735503460f94926242/html5/thumbnails/23.jpg)
q0 q1 q2 q3
a a a b b b
$
a
$$, ,
aa ,
ab,
ab,
Input
Stack
![Page 24: Pumping Lemma Problem: –Let B be the language of all palindromes over {0,1} containing an equal number of 0s and 1s. Show that B is not CFL Solution:](https://reader036.vdocuments.site/reader036/viewer/2022062320/56649c735503460f94926242/html5/thumbnails/24.jpg)
q0 q1 q2 q3
a a a b b b$
aa
$$, ,
aa ,
ab,
ab,
Input
Stack
![Page 25: Pumping Lemma Problem: –Let B be the language of all palindromes over {0,1} containing an equal number of 0s and 1s. Show that B is not CFL Solution:](https://reader036.vdocuments.site/reader036/viewer/2022062320/56649c735503460f94926242/html5/thumbnails/25.jpg)
q0 q1 q2 q3
a a a b b b
$
aaa
$$, ,
aa ,
ab,
ab,
Input
Stack
![Page 26: Pumping Lemma Problem: –Let B be the language of all palindromes over {0,1} containing an equal number of 0s and 1s. Show that B is not CFL Solution:](https://reader036.vdocuments.site/reader036/viewer/2022062320/56649c735503460f94926242/html5/thumbnails/26.jpg)
q0 q1 q2 q3
a a a b b b
$
aaa
$$, ,
aa ,
ab,
ab,
Input
Stack
![Page 27: Pumping Lemma Problem: –Let B be the language of all palindromes over {0,1} containing an equal number of 0s and 1s. Show that B is not CFL Solution:](https://reader036.vdocuments.site/reader036/viewer/2022062320/56649c735503460f94926242/html5/thumbnails/27.jpg)
q0 q1 q2 q3
a a a b b b$
aa
$$, ,
aa ,
ab,
ab,
Input
Stack
![Page 28: Pumping Lemma Problem: –Let B be the language of all palindromes over {0,1} containing an equal number of 0s and 1s. Show that B is not CFL Solution:](https://reader036.vdocuments.site/reader036/viewer/2022062320/56649c735503460f94926242/html5/thumbnails/28.jpg)
q0 q1 q2 q3
a a a b b b$
a
$$, ,
aa ,
ab,
ab,
Input
Stack
![Page 29: Pumping Lemma Problem: –Let B be the language of all palindromes over {0,1} containing an equal number of 0s and 1s. Show that B is not CFL Solution:](https://reader036.vdocuments.site/reader036/viewer/2022062320/56649c735503460f94926242/html5/thumbnails/29.jpg)
q0 q1 q2 q3
a a a b b b
accept
$
$$, ,
aa ,
ab,
ab,
Input
Stack
![Page 30: Pumping Lemma Problem: –Let B be the language of all palindromes over {0,1} containing an equal number of 0s and 1s. Show that B is not CFL Solution:](https://reader036.vdocuments.site/reader036/viewer/2022062320/56649c735503460f94926242/html5/thumbnails/30.jpg)
• A string is accepted if there is a computation such that:– All the input is consumed– The last state is an accepting state
– At the end of the computation,– we do not care about the stack contents– (the stack can be empty at the last state)
![Page 31: Pumping Lemma Problem: –Let B be the language of all palindromes over {0,1} containing an equal number of 0s and 1s. Show that B is not CFL Solution:](https://reader036.vdocuments.site/reader036/viewer/2022062320/56649c735503460f94926242/html5/thumbnails/31.jpg)
}0:{ nbaL nn
is the language accepted by the PDA:
q0 q1 q2 q3 ,
aa ,
ab,
ab,
$$,
![Page 32: Pumping Lemma Problem: –Let B be the language of all palindromes over {0,1} containing an equal number of 0s and 1s. Show that B is not CFL Solution:](https://reader036.vdocuments.site/reader036/viewer/2022062320/56649c735503460f94926242/html5/thumbnails/32.jpg)
aab
,
aa ,
ab,q0 q1 q2 q3
ab,
$$,
![Page 33: Pumping Lemma Problem: –Let B be the language of all palindromes over {0,1} containing an equal number of 0s and 1s. Show that B is not CFL Solution:](https://reader036.vdocuments.site/reader036/viewer/2022062320/56649c735503460f94926242/html5/thumbnails/33.jpg)
0q q1 q2 q3
a a b$
,
aa ,
ab,
ab,
$$,
![Page 34: Pumping Lemma Problem: –Let B be the language of all palindromes over {0,1} containing an equal number of 0s and 1s. Show that B is not CFL Solution:](https://reader036.vdocuments.site/reader036/viewer/2022062320/56649c735503460f94926242/html5/thumbnails/34.jpg)
0q q1 q2 q3
a a b$
,
aa ,
ab,
ab,
$$,
![Page 35: Pumping Lemma Problem: –Let B be the language of all palindromes over {0,1} containing an equal number of 0s and 1s. Show that B is not CFL Solution:](https://reader036.vdocuments.site/reader036/viewer/2022062320/56649c735503460f94926242/html5/thumbnails/35.jpg)
0q q1 q2 q3
a a b$
a
,
aa ,
ab,
ab,
$$,
![Page 36: Pumping Lemma Problem: –Let B be the language of all palindromes over {0,1} containing an equal number of 0s and 1s. Show that B is not CFL Solution:](https://reader036.vdocuments.site/reader036/viewer/2022062320/56649c735503460f94926242/html5/thumbnails/36.jpg)
0q q1 q2 q3
a a b$
aa
,
aa ,
ab,
ab,
$$,
![Page 37: Pumping Lemma Problem: –Let B be the language of all palindromes over {0,1} containing an equal number of 0s and 1s. Show that B is not CFL Solution:](https://reader036.vdocuments.site/reader036/viewer/2022062320/56649c735503460f94926242/html5/thumbnails/37.jpg)
0q q1 q2 q3
a a b$
aa
,
aa ,
ab,
ab,
$$,
![Page 38: Pumping Lemma Problem: –Let B be the language of all palindromes over {0,1} containing an equal number of 0s and 1s. Show that B is not CFL Solution:](https://reader036.vdocuments.site/reader036/viewer/2022062320/56649c735503460f94926242/html5/thumbnails/38.jpg)
0q q1 q2 q3
a a b$
aa
reject
,
aa ,
ab,
ab,
$$,
![Page 39: Pumping Lemma Problem: –Let B be the language of all palindromes over {0,1} containing an equal number of 0s and 1s. Show that B is not CFL Solution:](https://reader036.vdocuments.site/reader036/viewer/2022062320/56649c735503460f94926242/html5/thumbnails/39.jpg)
Another PDA example
$$, q1 q2
bb
aa
,
,
,q0
bb
aa
,
,
}},{:{)( bavvvML R
![Page 40: Pumping Lemma Problem: –Let B be the language of all palindromes over {0,1} containing an equal number of 0s and 1s. Show that B is not CFL Solution:](https://reader036.vdocuments.site/reader036/viewer/2022062320/56649c735503460f94926242/html5/thumbnails/40.jpg)
}},{:{)( bavvvML R
1. Push v
on stack2. Guess
middle
of input
3. Match vR on input
with v on stack
4. Match
found
$$, q1 q2
,q0
bb
aa
,
,
bb
aa
,
,
![Page 41: Pumping Lemma Problem: –Let B be the language of all palindromes over {0,1} containing an equal number of 0s and 1s. Show that B is not CFL Solution:](https://reader036.vdocuments.site/reader036/viewer/2022062320/56649c735503460f94926242/html5/thumbnails/41.jpg)
$
q1 q2q0
a ab b
bb
aa
,
,
,
bb
aa
,
,
$$,
![Page 42: Pumping Lemma Problem: –Let B be the language of all palindromes over {0,1} containing an equal number of 0s and 1s. Show that B is not CFL Solution:](https://reader036.vdocuments.site/reader036/viewer/2022062320/56649c735503460f94926242/html5/thumbnails/42.jpg)
a ab
$
q1 q2q0
ab
bb
aa
,
,
,
bb
aa
,
,
$$,
![Page 43: Pumping Lemma Problem: –Let B be the language of all palindromes over {0,1} containing an equal number of 0s and 1s. Show that B is not CFL Solution:](https://reader036.vdocuments.site/reader036/viewer/2022062320/56649c735503460f94926242/html5/thumbnails/43.jpg)
$
q1 q2q0
aa ab b
b
bb
aa
,
,
,
bb
aa
,
,
$$,
![Page 44: Pumping Lemma Problem: –Let B be the language of all palindromes over {0,1} containing an equal number of 0s and 1s. Show that B is not CFL Solution:](https://reader036.vdocuments.site/reader036/viewer/2022062320/56649c735503460f94926242/html5/thumbnails/44.jpg)
$
q1 q2q0
aa ab b
b
Guess the middle
of string
bb
aa
,
,
,
bb
aa
,
,
$$,
![Page 45: Pumping Lemma Problem: –Let B be the language of all palindromes over {0,1} containing an equal number of 0s and 1s. Show that B is not CFL Solution:](https://reader036.vdocuments.site/reader036/viewer/2022062320/56649c735503460f94926242/html5/thumbnails/45.jpg)
$
q1 q2q0
aa ab b
b
bb
aa
,
,
,
bb
aa
,
,
$$,
![Page 46: Pumping Lemma Problem: –Let B be the language of all palindromes over {0,1} containing an equal number of 0s and 1s. Show that B is not CFL Solution:](https://reader036.vdocuments.site/reader036/viewer/2022062320/56649c735503460f94926242/html5/thumbnails/46.jpg)
$
1q q2q0
a ab b a
bb
aa
,
,
,
bb
aa
,
,
$$,
![Page 47: Pumping Lemma Problem: –Let B be the language of all palindromes over {0,1} containing an equal number of 0s and 1s. Show that B is not CFL Solution:](https://reader036.vdocuments.site/reader036/viewer/2022062320/56649c735503460f94926242/html5/thumbnails/47.jpg)
$
q1q0
a ab b
accept
q2
bb
aa
,
,
,
bb
aa
,
,
$$,
![Page 48: Pumping Lemma Problem: –Let B be the language of all palindromes over {0,1} containing an equal number of 0s and 1s. Show that B is not CFL Solution:](https://reader036.vdocuments.site/reader036/viewer/2022062320/56649c735503460f94926242/html5/thumbnails/48.jpg)
Rejection Example:
Input
$
q1 q2q0
a b b b
bb
aa
,
,
,
bb
aa
,
,
$$,
![Page 49: Pumping Lemma Problem: –Let B be the language of all palindromes over {0,1} containing an equal number of 0s and 1s. Show that B is not CFL Solution:](https://reader036.vdocuments.site/reader036/viewer/2022062320/56649c735503460f94926242/html5/thumbnails/49.jpg)
$
q1 q2q0
aa b b b
bb
aa
,
,
,
bb
aa
,
,
$$,
![Page 50: Pumping Lemma Problem: –Let B be the language of all palindromes over {0,1} containing an equal number of 0s and 1s. Show that B is not CFL Solution:](https://reader036.vdocuments.site/reader036/viewer/2022062320/56649c735503460f94926242/html5/thumbnails/50.jpg)
$
q1 q2q0
ab
a b b b
bb
aa
,
,
,
bb
aa
,
,
$$,
![Page 51: Pumping Lemma Problem: –Let B be the language of all palindromes over {0,1} containing an equal number of 0s and 1s. Show that B is not CFL Solution:](https://reader036.vdocuments.site/reader036/viewer/2022062320/56649c735503460f94926242/html5/thumbnails/51.jpg)
$
q1 q2q0
ab
Guess the middle
of string
a b b b
bb
aa
,
,
,
bb
aa
,
,
$$,
![Page 52: Pumping Lemma Problem: –Let B be the language of all palindromes over {0,1} containing an equal number of 0s and 1s. Show that B is not CFL Solution:](https://reader036.vdocuments.site/reader036/viewer/2022062320/56649c735503460f94926242/html5/thumbnails/52.jpg)
$
q1 q2q0
ab
a b b b
bb
aa
,
,
,
bb
aa
,
,
$$,
![Page 53: Pumping Lemma Problem: –Let B be the language of all palindromes over {0,1} containing an equal number of 0s and 1s. Show that B is not CFL Solution:](https://reader036.vdocuments.site/reader036/viewer/2022062320/56649c735503460f94926242/html5/thumbnails/53.jpg)
$
1q q2q0
aa b b b
There is no possible transition.
Input is not consumed
bb
aa
,
,
,
bb
aa
,
,
$$,
![Page 54: Pumping Lemma Problem: –Let B be the language of all palindromes over {0,1} containing an equal number of 0s and 1s. Show that B is not CFL Solution:](https://reader036.vdocuments.site/reader036/viewer/2022062320/56649c735503460f94926242/html5/thumbnails/54.jpg)
Another computation on same string:
$
q1 q2q0
a b b b
bb
aa
,
,
,
bb
aa
,
,
$$,
![Page 55: Pumping Lemma Problem: –Let B be the language of all palindromes over {0,1} containing an equal number of 0s and 1s. Show that B is not CFL Solution:](https://reader036.vdocuments.site/reader036/viewer/2022062320/56649c735503460f94926242/html5/thumbnails/55.jpg)
$
q1 q2q0
aa b b b
bb
aa
,
,
,
bb
aa
,
,
$$,
![Page 56: Pumping Lemma Problem: –Let B be the language of all palindromes over {0,1} containing an equal number of 0s and 1s. Show that B is not CFL Solution:](https://reader036.vdocuments.site/reader036/viewer/2022062320/56649c735503460f94926242/html5/thumbnails/56.jpg)
$
q1 q2q0
ab
a b b b
bb
aa
,
,
,
bb
aa
,
,
$$,
![Page 57: Pumping Lemma Problem: –Let B be the language of all palindromes over {0,1} containing an equal number of 0s and 1s. Show that B is not CFL Solution:](https://reader036.vdocuments.site/reader036/viewer/2022062320/56649c735503460f94926242/html5/thumbnails/57.jpg)
$ab
a b b b
q1 q2q0
b
bb
aa
,
,
,
bb
aa
,
,
$$,
![Page 58: Pumping Lemma Problem: –Let B be the language of all palindromes over {0,1} containing an equal number of 0s and 1s. Show that B is not CFL Solution:](https://reader036.vdocuments.site/reader036/viewer/2022062320/56649c735503460f94926242/html5/thumbnails/58.jpg)
a b b b
q1 q2q0
$abbb
bb
aa
,
,
,
bb
aa
,
,
$$,
![Page 59: Pumping Lemma Problem: –Let B be the language of all palindromes over {0,1} containing an equal number of 0s and 1s. Show that B is not CFL Solution:](https://reader036.vdocuments.site/reader036/viewer/2022062320/56649c735503460f94926242/html5/thumbnails/59.jpg)
a b b b
$abbb
q1 q2q0
final state
is not reached
bb
aa
,
,
,
bb
aa
,
,
$$,
![Page 60: Pumping Lemma Problem: –Let B be the language of all palindromes over {0,1} containing an equal number of 0s and 1s. Show that B is not CFL Solution:](https://reader036.vdocuments.site/reader036/viewer/2022062320/56649c735503460f94926242/html5/thumbnails/60.jpg)
q1 q2q0
)(MLabbb
bb
aa
,
,
,
bb
aa
,
,
$$,
![Page 61: Pumping Lemma Problem: –Let B be the language of all palindromes over {0,1} containing an equal number of 0s and 1s. Show that B is not CFL Solution:](https://reader036.vdocuments.site/reader036/viewer/2022062320/56649c735503460f94926242/html5/thumbnails/61.jpg)
Another PDA example
ab
aa
,
,
q0
M
)}()( ,prefix every in
:},{{)( *
vnvnv
bawML
ba
aababbb (r)
![Page 62: Pumping Lemma Problem: –Let B be the language of all palindromes over {0,1} containing an equal number of 0s and 1s. Show that B is not CFL Solution:](https://reader036.vdocuments.site/reader036/viewer/2022062320/56649c735503460f94926242/html5/thumbnails/62.jpg)
Pushing Strings
q1 q2a, b w
Input
symbol
Pop
symbolPush
string
![Page 63: Pumping Lemma Problem: –Let B be the language of all palindromes over {0,1} containing an equal number of 0s and 1s. Show that B is not CFL Solution:](https://reader036.vdocuments.site/reader036/viewer/2022062320/56649c735503460f94926242/html5/thumbnails/63.jpg)
q1 q2a, b cdf
a
b top
a
Push
eh h
e
cdf
pushed
string
Example:
$ $
![Page 64: Pumping Lemma Problem: –Let B be the language of all palindromes over {0,1} containing an equal number of 0s and 1s. Show that B is not CFL Solution:](https://reader036.vdocuments.site/reader036/viewer/2022062320/56649c735503460f94926242/html5/thumbnails/64.jpg)
Another PDA example
$$, q1 q2
a, $ 0$a, 0 00
1,a
b, $ 1$b, 1 11
0,b
)}()(:},{{)( * wnwnbawML ba
abbbaa