The pumping lemma states that for any regular language L, there exists an integer n such that any string x in L with length at least n can be divided into uvw such that: (1) x=uvw, (2) the length of uv is at most n, (3) v is not empty, and (4) for all i>=0, uviw is in L. This allows proving that a language is non-regular by finding a string that cannot be pumped.

1. THE PUMPING LEMMA

2. THE PUMPING LEMMA
x
Theorem. For any regular language L there exists an integer
n, such that for all x ∈ L with |x| ≥ n, there exist u, v, w ∈ Σ∗
,
such that
(1) x = uvw
(2) |uv| ≤ n
(3) |v| ≥ 1
(4) for all i ≥ 0: uvi
w ∈ L.
x
L

3. THE PUMPING LEMMA
n
x
Theorem. For any regular language L there exists an integer
n, such that for all x ∈ L with |x| ≥ n, there exist u, v, w ∈ Σ∗
,
such that
(1) x = uvw
(2) |uv| ≤ n
(3) |v| ≥ 1
(4) for all i ≥ 0: uvi
w ∈ L.
x
L

4. THE PUMPING LEMMA
n
x
Theorem. For any regular language L there exists an integer
n, such that for all x ∈ L with |x| ≥ n, there exist u, v, w ∈ Σ∗
,
such that
(1) x = uvw
(2) |uv| ≤ n
(3) |v| ≥ 1
(4) for all i ≥ 0: uvi
w ∈ L.
x
L

5. THE PUMPING LEMMA
n
x
Theorem. For any regular language L there exists an integer
n, such that for all x ∈ L with |x| ≥ n, there exist u, v, w ∈ Σ∗
,
such that
(1) x = uvw
(2) |uv| ≤ n
(3) |v| ≥ 1
(4) for all i ≥ 0: uvi
w ∈ L.
x
L

6. THE PUMPING LEMMA
n
x
Theorem. For any regular language L there exists an integer
n, such that for all x ∈ L with |x| ≥ n, there exist u, v, w ∈ Σ∗
,
such that
(1) x = uvw
(2) |uv| ≤ n
(3) |v| ≥ 1
(4) for all i ≥ 0: uvi
w ∈ L.
x
u v w
L

7. THE PUMPING LEMMA
n
x
Theorem. For any regular language L there exists an integer
n, such that for all x ∈ L with |x| ≥ n, there exist u, v, w ∈ Σ∗
,
such that
(1) x = uvw
(2) |uv| ≤ n
(3) |v| ≥ 1
(4) for all i ≥ 0: uvi
w ∈ L.
x
u w
u v w
L

8. THE PUMPING LEMMA
n
x
Theorem. For any regular language L there exists an integer
n, such that for all x ∈ L with |x| ≥ n, there exist u, v, w ∈ Σ∗
,
such that
(1) x = uvw
(2) |uv| ≤ n
(3) |v| ≥ 1
(4) for all i ≥ 0: uvi
w ∈ L.
x
u v wv
u w
u v w
L

9. THE PUMPING LEMMA
n
x
Theorem. For any regular language L there exists an integer
n, such that for all x ∈ L with |x| ≥ n, there exist u, v, w ∈ Σ∗
,
such that
(1) x = uvw
(2) |uv| ≤ n
(3) |v| ≥ 1
(4) for all i ≥ 0: uvi
w ∈ L.
x
u v wv
u w
u v w
u v wv v
L

10. PROOF OF P.L. (SKETCH)

11. PROOF OF P.L. (SKETCH)
Let M be a DFA for L. Take n be the number of states of M
plus 1.

12. PROOF OF P.L. (SKETCH)
Let M be a DFA for L. Take n be the number of states of M
plus 1.
Take any x ∈ L with |x| ≥ n. Consider the path (from start
state to an accepting state) in M that corresponds to x. The
length of this path is |x| ≥ n.

13. PROOF OF P.L. (SKETCH)
Let M be a DFA for L. Take n be the number of states of M
plus 1.
Since M has at most n − 1 states, some state must be visited
twice or more in the first n steps of the path.
Take any x ∈ L with |x| ≥ n. Consider the path (from start
state to an accepting state) in M that corresponds to x. The
length of this path is |x| ≥ n.

14. PROOF OF P.L. (SKETCH)
Let M be a DFA for L. Take n be the number of states of M
plus 1.
Since M has at most n − 1 states, some state must be visited
twice or more in the first n steps of the path.
Take any x ∈ L with |x| ≥ n. Consider the path (from start
state to an accepting state) in M that corresponds to x. The
length of this path is |x| ≥ n.
u
v
w

15. PROOF OF P.L. (SKETCH)
Let M be a DFA for L. Take n be the number of states of M
plus 1.
Since M has at most n − 1 states, some state must be visited
twice or more in the first n steps of the path.
Take any x ∈ L with |x| ≥ n. Consider the path (from start
state to an accepting state) in M that corresponds to x. The
length of this path is |x| ≥ n.
u
v
w
x = uvw ∈ L
uw ∈ L
uvvw ∈ L
uvvvw ∈ L
. . .

16. L regular =⇒ L satisfies P.L.
L non-regular =⇒ ?
L non-regular ⇐= L doesn’t satisfy P.L.
Negation:
∃ n ∈ N ∀ x ∈ L with |x| ≥ n
∃ u, v, w ∈ Σ∗
all of these hold:
(1) x = uvw
(2) |uv| ≤ n
(3) |v| ≥ 1
(4) ∀ i ≥ 0: uvi
w ∈ L.
USING PUMPING LEMMA TO PROVE NON-REGULARITY

17. L regular =⇒ L satisfies P.L.
L non-regular =⇒ ?
L non-regular ⇐= L doesn’t satisfy P.L.
Negation:
∃ n ∈ N ∀ x ∈ L with |x| ≥ n
∃ u, v, w ∈ Σ∗
all of these hold:
(1) x = uvw
(2) |uv| ≤ n
(3) |v| ≥ 1
(4) ∀ i ≥ 0: uvi
w ∈ L.
USING PUMPING LEMMA TO PROVE NON-REGULARITY

18. L regular =⇒ L satisfies P.L.
L non-regular =⇒ ?
L non-regular ⇐= L doesn’t satisfy P.L.
Negation:
∃ n ∈ N ∀ x ∈ L with |x| ≥ n
∃ u, v, w ∈ Σ∗
all of these hold:
(1) x = uvw
(2) |uv| ≤ n
(3) |v| ≥ 1
(4) ∀ i ≥ 0: uvi
w ∈ L.
∀ n ∈ N ∃ x ∈ L with |x| ≥ n
∀ u, v, w ∈ Σ∗
not all of these hold:
(1) x = uvw
(2) |uv| ≤ n
(3) |v| ≥ 1
(4) ∀ i ≥ 0: uvi
w ∈ L.
USING PUMPING LEMMA TO PROVE NON-REGULARITY

19. L regular =⇒ L satisfies P.L.
L non-regular =⇒ ?
L non-regular ⇐= L doesn’t satisfy P.L.
Negation:
∃ n ∈ N ∀ x ∈ L with |x| ≥ n
∃ u, v, w ∈ Σ∗
all of these hold:
(1) x = uvw
(2) |uv| ≤ n
(3) |v| ≥ 1
(4) ∀ i ≥ 0: uvi
w ∈ L.
∀ n ∈ N ∃ x ∈ L with |x| ≥ n
∀ u, v, w ∈ Σ∗
not all of these hold:
(1) x = uvw
(2) |uv| ≤ n
(3) |v| ≥ 1
(4) ∀ i ≥ 0: uvi
w ∈ L.
Equivalently:
(1) ∧ (2) ∧ (3) ⇒ not(4)
where not(4) is:
∃ i : uvi
w ∈ L
USING PUMPING LEMMA TO PROVE NON-REGULARITY

20. EXAMPLE 1
Prove that L = {0i
1i
: i ≥ 0} is NOT regular.

21. EXAMPLE 1
Prove that L = {0i
1i
: i ≥ 0} is NOT regular.
Proof. Show that P.L. doesn’t hold (note: showing P.L. holds
doesn’t mean regularity).

22. EXAMPLE 1
Prove that L = {0i
1i
: i ≥ 0} is NOT regular.
Proof. Show that P.L. doesn’t hold (note: showing P.L. holds
doesn’t mean regularity).
If L is regular, then by P.L. ∃ n such that . . .

23. EXAMPLE 1
Prove that L = {0i
1i
: i ≥ 0} is NOT regular.
Proof. Show that P.L. doesn’t hold (note: showing P.L. holds
doesn’t mean regularity).
If L is regular, then by P.L. ∃ n such that . . .
Now let x = 0n
1n

24. EXAMPLE 1
Prove that L = {0i
1i
: i ≥ 0} is NOT regular.
Proof. Show that P.L. doesn’t hold (note: showing P.L. holds
doesn’t mean regularity).
If L is regular, then by P.L. ∃ n such that . . .
Now let x = 0n
1n
x ∈ L and |x| ≥ n, so by P.L. ∃u, v, w such that (1)–(4) hold.

25. EXAMPLE 1
Prove that L = {0i
1i
: i ≥ 0} is NOT regular.
Proof. Show that P.L. doesn’t hold (note: showing P.L. holds
doesn’t mean regularity).
If L is regular, then by P.L. ∃ n such that . . .
Now let x = 0n
1n
x ∈ L and |x| ≥ n, so by P.L. ∃u, v, w such that (1)–(4) hold.
We show that ∀ u, v, w (1)–(4) don’t all hold.

26. EXAMPLE 1
Prove that L = {0i
1i
: i ≥ 0} is NOT regular.
Proof. Show that P.L. doesn’t hold (note: showing P.L. holds
doesn’t mean regularity).
If L is regular, then by P.L. ∃ n such that . . .
Now let x = 0n
1n
x ∈ L and |x| ≥ n, so by P.L. ∃u, v, w such that (1)–(4) hold.
We show that ∀ u, v, w (1)–(4) don’t all hold.
If (1), (2), (3) hold then x = 0n
1n
= uvw with |uv| ≤ n and
|v| ≥ 1.

27. EXAMPLE 1
Prove that L = {0i
1i
: i ≥ 0} is NOT regular.
Proof. Show that P.L. doesn’t hold (note: showing P.L. holds
doesn’t mean regularity).
If L is regular, then by P.L. ∃ n such that . . .
Now let x = 0n
1n
x ∈ L and |x| ≥ n, so by P.L. ∃u, v, w such that (1)–(4) hold.
We show that ∀ u, v, w (1)–(4) don’t all hold.
If (1), (2), (3) hold then x = 0n
1n
= uvw with |uv| ≤ n and
|v| ≥ 1.
So, u = 0s
, v = 0t
, w = 0p
1n
with

28. EXAMPLE 1
Prove that L = {0i
1i
: i ≥ 0} is NOT regular.
Proof. Show that P.L. doesn’t hold (note: showing P.L. holds
doesn’t mean regularity).
If L is regular, then by P.L. ∃ n such that . . .
Now let x = 0n
1n
x ∈ L and |x| ≥ n, so by P.L. ∃u, v, w such that (1)–(4) hold.
We show that ∀ u, v, w (1)–(4) don’t all hold.
If (1), (2), (3) hold then x = 0n
1n
= uvw with |uv| ≤ n and
|v| ≥ 1.
So, u = 0s
, v = 0t
, w = 0p
1n
with
s + t ≤ n, t ≥ 1, p ≥ 0, s + t + p = n.

29. EXAMPLE 1
Prove that L = {0i
1i
: i ≥ 0} is NOT regular.
Proof. Show that P.L. doesn’t hold (note: showing P.L. holds
doesn’t mean regularity).
If L is regular, then by P.L. ∃ n such that . . .
Now let x = 0n
1n
x ∈ L and |x| ≥ n, so by P.L. ∃u, v, w such that (1)–(4) hold.
We show that ∀ u, v, w (1)–(4) don’t all hold.
If (1), (2), (3) hold then x = 0n
1n
= uvw with |uv| ≤ n and
|v| ≥ 1.
But then (4) fails for i = 0:
So, u = 0s
, v = 0t
, w = 0p
1n
with
s + t ≤ n, t ≥ 1, p ≥ 0, s + t + p = n.

30. EXAMPLE 1
Prove that L = {0i
1i
: i ≥ 0} is NOT regular.
Proof. Show that P.L. doesn’t hold (note: showing P.L. holds
doesn’t mean regularity).
If L is regular, then by P.L. ∃ n such that . . .
Now let x = 0n
1n
x ∈ L and |x| ≥ n, so by P.L. ∃u, v, w such that (1)–(4) hold.
We show that ∀ u, v, w (1)–(4) don’t all hold.
If (1), (2), (3) hold then x = 0n
1n
= uvw with |uv| ≤ n and
|v| ≥ 1.
But then (4) fails for i = 0:
uv0
w = uw = 0s
0p
1n
= 0s+p
1n
∈ L, since s + p = n
So, u = 0s
, v = 0t
, w = 0p
1n
with
s + t ≤ n, t ≥ 1, p ≥ 0, s + t + p = n.

31. IN PICTURE
∃ u, v, w such that
(1) x = uvw
(2) |uv| ≤ n
(3) |v| ≥ 1
(4) ∀ i ∈ N : uvi
w ∈ L.

32. IN PICTURE
∃ u, v, w such that
(1) x = uvw
(2) |uv| ≤ n
(3) |v| ≥ 1
(4) ∀ i ∈ N : uvi
w ∈ L.
u
00000
v
0 . . .
w
01111 . . . 1 ∈ L

33. IN PICTURE
∃ u, v, w such that
(1) x = uvw
(2) |uv| ≤ n
(3) |v| ≥ 1
(4) ∀ i ∈ N : uvi
w ∈ L.
u
00000
v
0 . . .
w
01111 . . . 1 ∈ L
non-empty

34. IN PICTURE
∃ u, v, w such that
(1) x = uvw
(2) |uv| ≤ n
(3) |v| ≥ 1
(4) ∀ i ∈ N : uvi
w ∈ L.
u
00000
v
0 . . .
w
01111 . . . 1 ∈ L
non-empty
If (1), (2), (3) hold then (4) fails: it is not the case that for all
i, uvi
w is in L.
In particular, let i = 0. uw ∈ L.

35. EXAMPLE 2
Prove that L = {0i
: i is a prime} is NOT regular.

36. EXAMPLE 2
Prove that L = {0i
: i is a prime} is NOT regular.
Proof. We show that P.L. doesn’t hold.

37. EXAMPLE 2
If L is regular, then by P.L. ∃ n such that . . .
Prove that L = {0i
: i is a prime} is NOT regular.
Proof. We show that P.L. doesn’t hold.

38. EXAMPLE 2
If L is regular, then by P.L. ∃ n such that . . .
Prove that L = {0i
: i is a prime} is NOT regular.
Now let x = 0m
where m ≥ n + 2 is prime.
Proof. We show that P.L. doesn’t hold.

39. EXAMPLE 2
If L is regular, then by P.L. ∃ n such that . . .
Prove that L = {0i
: i is a prime} is NOT regular.
Now let x = 0m
where m ≥ n + 2 is prime.
x ∈ L and |x| ≥ n, so by P.L. ∃u, v, w such that (1)–(4) hold.
Proof. We show that P.L. doesn’t hold.

40. EXAMPLE 2
If L is regular, then by P.L. ∃ n such that . . .
Prove that L = {0i
: i is a prime} is NOT regular.
Now let x = 0m
where m ≥ n + 2 is prime.
x ∈ L and |x| ≥ n, so by P.L. ∃u, v, w such that (1)–(4) hold.
We show that ∀ u, v, w (1)–(4) don’t all hold.
Proof. We show that P.L. doesn’t hold.

41. EXAMPLE 2
If L is regular, then by P.L. ∃ n such that . . .
Prove that L = {0i
: i is a prime} is NOT regular.
Now let x = 0m
where m ≥ n + 2 is prime.
x ∈ L and |x| ≥ n, so by P.L. ∃u, v, w such that (1)–(4) hold.
We show that ∀ u, v, w (1)–(4) don’t all hold.
If 0m
is written as 0m
= uvw, then 0m
= 0|u|
0|v|
0|w|
.
Proof. We show that P.L. doesn’t hold.

42. EXAMPLE 2
If L is regular, then by P.L. ∃ n such that . . .
Prove that L = {0i
: i is a prime} is NOT regular.
Now let x = 0m
where m ≥ n + 2 is prime.
x ∈ L and |x| ≥ n, so by P.L. ∃u, v, w such that (1)–(4) hold.
We show that ∀ u, v, w (1)–(4) don’t all hold.
If 0m
is written as 0m
= uvw, then 0m
= 0|u|
0|v|
0|w|
.
If |uv| ≤ n and |v| ≥ 1, then consider i = |v| + |w|:
Proof. We show that P.L. doesn’t hold.

43. EXAMPLE 2
If L is regular, then by P.L. ∃ n such that . . .
Prove that L = {0i
: i is a prime} is NOT regular.
Now let x = 0m
where m ≥ n + 2 is prime.
x ∈ L and |x| ≥ n, so by P.L. ∃u, v, w such that (1)–(4) hold.
We show that ∀ u, v, w (1)–(4) don’t all hold.
If 0m
is written as 0m
= uvw, then 0m
= 0|u|
0|v|
0|w|
.
If |uv| ≤ n and |v| ≥ 1, then consider i = |v| + |w|:
uvi
w = 0|v|
0|v|(|v|+|w|)
0|w|
= 0(|v|+1)(|v|+|w|)
∈ L
Proof. We show that P.L. doesn’t hold.

44. EXAMPLE 2
If L is regular, then by P.L. ∃ n such that . . .
Prove that L = {0i
: i is a prime} is NOT regular.
Now let x = 0m
where m ≥ n + 2 is prime.
x ∈ L and |x| ≥ n, so by P.L. ∃u, v, w such that (1)–(4) hold.
We show that ∀ u, v, w (1)–(4) don’t all hold.
If 0m
is written as 0m
= uvw, then 0m
= 0|u|
0|v|
0|w|
.
If |uv| ≤ n and |v| ≥ 1, then consider i = |v| + |w|:
Both factors ≥ 2
uvi
w = 0|v|
0|v|(|v|+|w|)
0|w|
= 0(|v|+1)(|v|+|w|)
∈ L
Proof. We show that P.L. doesn’t hold.

45. EXAMPLE 3
Prove that L = {yy : y ∈ {0, 1}∗
} is NOT regular.

46. EXAMPLE 3
Again we try to show that P.L. doesn’t hold.
Prove that L = {yy : y ∈ {0, 1}∗
} is NOT regular.

47. EXAMPLE 3
If L is regular, then by P.L. ∃ n such that . . .
Again we try to show that P.L. doesn’t hold.
Prove that L = {yy : y ∈ {0, 1}∗
} is NOT regular.

48. EXAMPLE 3
If L is regular, then by P.L. ∃ n such that . . .
Let us consider x = 0n
0n
∈ L. Obviously |x| ≥ n.
Again we try to show that P.L. doesn’t hold.
Prove that L = {yy : y ∈ {0, 1}∗
} is NOT regular.

49. EXAMPLE 3
If L is regular, then by P.L. ∃ n such that . . .
Let us consider x = 0n
0n
∈ L. Obviously |x| ≥ n.
Can 0n
0n
be written as 0n
0n
= uvw such that |uv| ≤ n |v| ≥ 1
and that for all i: uvi
w ∈ L?
Again we try to show that P.L. doesn’t hold.
Prove that L = {yy : y ∈ {0, 1}∗
} is NOT regular.

50. EXAMPLE 3
If L is regular, then by P.L. ∃ n such that . . .
Let us consider x = 0n
0n
∈ L. Obviously |x| ≥ n.
YES! Let u = , v = 00, and w = 02n−2
.
Can 0n
0n
be written as 0n
0n
= uvw such that |uv| ≤ n |v| ≥ 1
and that for all i: uvi
w ∈ L?
Again we try to show that P.L. doesn’t hold.
Prove that L = {yy : y ∈ {0, 1}∗
} is NOT regular.

51. EXAMPLE 3
If L is regular, then by P.L. ∃ n such that . . .
Let us consider x = 0n
0n
∈ L. Obviously |x| ≥ n.
YES! Let u = , v = 00, and w = 02n−2
.
Then ∀i , uvi
w is of the form 02k
= 0k
0k
.
Can 0n
0n
be written as 0n
0n
= uvw such that |uv| ≤ n |v| ≥ 1
and that for all i: uvi
w ∈ L?
Again we try to show that P.L. doesn’t hold.
Prove that L = {yy : y ∈ {0, 1}∗
} is NOT regular.

52. EXAMPLE 3
If L is regular, then by P.L. ∃ n such that . . .
Let us consider x = 0n
0n
∈ L. Obviously |x| ≥ n.
YES! Let u = , v = 00, and w = 02n−2
.
Then ∀i , uvi
w is of the form 02k
= 0k
0k
.
Does this mean that L is regular?
Can 0n
0n
be written as 0n
0n
= uvw such that |uv| ≤ n |v| ≥ 1
and that for all i: uvi
w ∈ L?
Again we try to show that P.L. doesn’t hold.
Prove that L = {yy : y ∈ {0, 1}∗
} is NOT regular.

53. EXAMPLE 3
If L is regular, then by P.L. ∃ n such that . . .
Let us consider x = 0n
0n
∈ L. Obviously |x| ≥ n.
YES! Let u = , v = 00, and w = 02n−2
.
Then ∀i , uvi
w is of the form 02k
= 0k
0k
.
Does this mean that L is regular?
NO.We have chosen a bad string x. To show that L fails the
P.L., we only need to exhibit some x that cannot be “pumped”
(and |x| ≥ n).
Can 0n
0n
be written as 0n
0n
= uvw such that |uv| ≤ n |v| ≥ 1
and that for all i: uvi
w ∈ L?
Again we try to show that P.L. doesn’t hold.
Prove that L = {yy : y ∈ {0, 1}∗
} is NOT regular.

54. EXAMPLE 3, 2ND ATTEMPT
Prove that L = {yy : y ∈ {0, 1}∗
} is NOT regular.

55. EXAMPLE 3, 2ND ATTEMPT
Given n from the P.L., let x = (01)n
(01)n
. Obviously x ∈ L
and |x| ≥ n.
Prove that L = {yy : y ∈ {0, 1}∗
} is NOT regular.

56. EXAMPLE 3, 2ND ATTEMPT
Q: Can x be “pumped” for some choice of u, v, w with |uv| ≤ n
and |v| ≥ 1?
Given n from the P.L., let x = (01)n
(01)n
. Obviously x ∈ L
and |x| ≥ n.
Prove that L = {yy : y ∈ {0, 1}∗
} is NOT regular.

57. EXAMPLE 3, 2ND ATTEMPT
Q: Can x be “pumped” for some choice of u, v, w with |uv| ≤ n
and |v| ≥ 1?
Given n from the P.L., let x = (01)n
(01)n
. Obviously x ∈ L
and |x| ≥ n.
A: Yes! Take u = , v = 0101, w = (01)2n−2
.
Prove that L = {yy : y ∈ {0, 1}∗
} is NOT regular.

58. EXAMPLE 3, 2ND ATTEMPT
Q: Can x be “pumped” for some choice of u, v, w with |uv| ≤ n
and |v| ≥ 1?
Given n from the P.L., let x = (01)n
(01)n
. Obviously x ∈ L
and |x| ≥ n.
A: Yes! Take u = , v = 0101, w = (01)2n−2
.
Another bad choice of x!
Prove that L = {yy : y ∈ {0, 1}∗
} is NOT regular.

59. EXAMPLE 3, 3RD ATTEMPT
Prove that L = {yy : y ∈ {0, 1}∗
} is NOT regular.

60. EXAMPLE 3, 3RD ATTEMPT
Given n from the P.L., let x = 0n
10n
1. Again x ∈ L and
|x| ≥ n.
Prove that L = {yy : y ∈ {0, 1}∗
} is NOT regular.

61. EXAMPLE 3, 3RD ATTEMPT
Given n from the P.L., let x = 0n
10n
1. Again x ∈ L and
|x| ≥ n.
∀u, v, w such that 0n
10n
1 = uvw and |uv| ≤ n and |v| ≥ 1:
Prove that L = {yy : y ∈ {0, 1}∗
} is NOT regular.

62. EXAMPLE 3, 3RD ATTEMPT
Given n from the P.L., let x = 0n
10n
1. Again x ∈ L and
|x| ≥ n.
∀u, v, w such that 0n
10n
1 = uvw and |uv| ≤ n and |v| ≥ 1:
must have uv contained in the first group of 0n
. Thus consider
Prove that L = {yy : y ∈ {0, 1}∗
} is NOT regular.

63. EXAMPLE 3, 3RD ATTEMPT
Given n from the P.L., let x = 0n
10n
1. Again x ∈ L and
|x| ≥ n.
∀u, v, w such that 0n
10n
1 = uvw and |uv| ≤ n and |v| ≥ 1:
must have uv contained in the first group of 0n
. Thus consider
uv0
w = 0n−|v|
10n
1.
Prove that L = {yy : y ∈ {0, 1}∗
} is NOT regular.

64. EXAMPLE 3, 3RD ATTEMPT
Given n from the P.L., let x = 0n
10n
1. Again x ∈ L and
|x| ≥ n.
∀u, v, w such that 0n
10n
1 = uvw and |uv| ≤ n and |v| ≥ 1:
must have uv contained in the first group of 0n
. Thus consider
uv0
w = 0n−|v|
10n
1.
Since |v| is at least 1, this is clearly not of the form yy.
Prove that L = {yy : y ∈ {0, 1}∗
} is NOT regular.