1. Non-Context-Free Languages
CSC 4170
Theory of Computation
Non-Context-Free Languages
Section 2.3

2. The pumping lemma for context-free languages
Giorgi Japaridze Theory of Computability
Theorem 2.34 (Pumping lemma for context-free languages)
If L is a context-free language, then there is a number p (the pumping
length) where, if s is any string in L of length at least p, then s may
be divided into five pieces s = uvxyz satisfying the conditions:
1. For each i0, uvixyiz L;
2. |vy| > 0;
3. |vxy| p.
uxz
uvxyz
uvvxyyz
uvvvxyyyz
uvvvvxyyyyz
uvvvvvxyyyyyz

3. The pumping lemma in work: example
2.3.b
Giorgi Japaridze Theory of Computability
S  “R” is a regular expression
R  0 | ( R )*
“0” is a regular expression
“(0)*” is a regular expression
“((0)*)*” is a regular expression
“(((0)*)*)*” is a regular expression …
u = “(
v = (
x = 0
y = )*
z = )*” is a regular expression
“((0)*)*” is a regular expression
uv0xy0z: “(0)*” is a regular expression
uv1xy1z: “((0)*)*” is a regular expression
uv2xy2z: “(((0)*)*)*” is a regular expression
uv3xy3z: “((((0)*)*)*)*” is a regular expression

4. Using the pumping lemma for proving that certain languages are not CF
Giorgi Japaridze Theory of Computability
Example 2.36: Show that the following language is not CF:
B = {anbncn | n0}
Proof by contradiction:
Assume B is CF. Let then p be its pumping length.
Select wB with |w|  p.
By the pumping lemma, w=uvxyz and v and y can be pumped.
If either v or y contain more than one type of symbols, then pumping
would intermix these symbols in a wrong way.
aaaabbbbcccc B aaaababbbbccccc B
Thus, one of the three symbols should be neither in neither v, nor in y.
aaaabbbbcccc B
But then, after pumping, the number of that symbol will not change,
while the number of the other symbols will increase.
aaaaaabbbbbbbbcccc B

5. Regular vs context-free vs computer-recognizable languages
2.3.d
Giorgi Japaridze Theory of Computability
Computer-recognizable languages
Context-free languages
Regular languages
{anbn | n0}
{anbncn | n0}