1. Limitations of context-free languages
The Chinese University of Hong Kong
Fall 2008
CSC 3130: Automata theory and formal languages
Limitations of context-free languages
Andrej Bogdanov

2. Non context-free languages
Recall the pumping lemma for regular languages allows us to show some languages are not regular Are these languages context-free?
L1 = {anbn: n ≥ 0}
L2 = {x: x has same number of as and bs}
L3 = {1n: n is prime}
L4 = {anbncn: n ≥ 0}
L5 = {x#xR: x ∈ {0, 1}*}
L6 = {x#x: x ∈ {0, 1}*}

3. Some intuition L4 = {anbncn: n ≥ 0}
Let’s try to show this is context-free
read a / push 1
S → aBc
???
B → ??
read c / pop 1
context-free grammar
pushdown automaton

4. More intuition Suppose we could construct some CFG for L4, e.g.
We do some derivations of “long” strings
S  BC  CSC  aSC  aBCC  abCC  abaC  abaSB
 abaBCB  ababCB  ababaB
 ababab
S  BC
B  CS | b
C  SB | a
. . .

5. More intuition
If derivation is long enough, some variable must appear twice on same path in parse tree S
S  BC  CSC  aSC  aBCC  abCC  abaC  abaSB
 abaBCB  ababCB  ababaB
 ababab B C C S S B B C B C a b a b a b

6. More intuition Then we can “cut and paste” part of parse tree ababbabb
ababab ✗ B C C S S B B S C a b B C B C a b a b a b

7. More intuition We can repeat this many times
Every sufficiently large derivation will have a part that can be repeated indefinitely
This is caused by cycles in the grammar
ababab
ababbabb
ababbbabbb ✗ ✗
ababnabnbb

8. General picture u u y y v v x x w v x uvwxy xvvwxxy uv3wx3y w A A A A

9. Example L4 = {anbncn: n ≥ 0} If L4 has a context-free grammar G, then
What happens for anbncn?
No matter how it is split, uv2wx2y ∉ L4!
If uvwxy can be derived in G, so can uviwxiy for every i
a a a ... a a b b b ... b b c c c ... c c u v w x y

10. Pumping lemma for context-free languages
Theorem: For every context-free language L
There exists a number n such that for every string z in L, we can write z = uvwxy where  |vwx| ≤ n  |vx| ≥ 1  For every i ≥ 0, the string uviwxiy is in L. u v w x y

11. Pumping lemma for context-free languages
So to prove L is not context-free, it is enough that
For every n there exists z in L, such that for every way of writing z = uvwxy where  |vwx| ≤ n and  |vx| ≥ 1, the string uviwxiy is not in L for some i ≥ 0. u v w x y

12. Proving language is not context-free
Just like for regular languages, need strategy that, regardless of adversary, always wins you this game
adversary
choose n write z = uvwxy (|vwx| ≤ n,|vx| ≥ 1)
you
choose z  L choose i you win if uviwxiy  L 1 2

13. Example L4 = {anbncn: n ≥ 0} a a a ... a a b b b ... b b c c c ... c c
adversary
choose n write z = uvwxy (|vwx| ≤ n,|vx| ≥ 1)
you
choose z  L choose i you win if uviwxiy  L 1 2
L4 = {anbncn: n ≥ 0}
adversary
n write z = uvwxy
you
z = anbncn i = ? 1 2 w u y x v
a a a ... a a b b b ... b b c c c ... c c

14. Example
Case 1: v or x contains two kinds of symbols Then uv2wx2y not in L because pattern is wrong
Case 2: v and x both contain one kind of symbol Then uv2wx2y does not have same number of as, bs, cs
a a a ... a a b b b ... b b c c c ... c c v x
a a a ... a a b b b ... b b c c c ... c c v x

15. More examples Which of these is context-free? L1 = {anbn: n ≥ 0}
L2 = {x: x has same number of as and bs}
L3 = {1n: n is prime}
L4 = {anbncn: n ≥ 0}
L5 = {x#xR: x ∈ {0, 1}*}
L6 = {x#x: x ∈ {0, 1}*}