1. Limitations of pushdown automata
The Chinese University of Hong Kong
Fall 2010
CSCI 3130: Automata theory and formal languages
Limitations of pushdown automata
Andrej Bogdanov

2. Non context-free languages
L1 = {anbn: n ≥ 0}
L2 = {s: s has same number of as and bs}
L3 = {anbncn: n ≥ 0}
L4 = {ssR: s ∈ {a, b}*}
L5 = {ss: s ∈ {a, b}*} ✔ ✔ ?
These are not regular
Are they context-free?

3. An attempt L3 = {anbncn: n ≥ 0} Let’s try to design a CFG or PDA
read a / push x
S → aBc | e
???
B → ??
read c / pop x

4. What would happen if...
Suppose we could construct some CFG for L3, e.g.
Let’s do some long derivations
S  BC  CSC  aSC  aBCC  abCC  abaC  abaSB
 abaBCB  ababCB  ababaB
 ababab
S  BC
B  CS | b
C  SB | a
. . .

5. Repetition in long derivations
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. Pumping example Then we can “cut and paste” part of parse tree SB C S a b S
ababab ✗
ababbabb B C C S S B B S C a b B C B C a b a b a b

7. Pumping example We can repeat this many times
Every sufficiently large derivation will have a middle part that can be repeated indefinitely
ababab
ababbabb
ababbbabbb ✗ ✗
ababnabnbb

8. Pumping in general uvwxy uv2wx2y uv3wx3y uwy u v w x y A u v w x y A u

9. Example L3 = {anbncn: n ≥ 0} If L3 has a context-free grammar G, then
What happens for anbncn?
No matter how it is split, uv2wx2y ∉ L4!
If uvwxy is in G, so are uv2wx2y, uv3wx3y, uwy, ...
a a a ... a a b b b ... b b c c c ... c c w u y x v

10. Pumping lemma for context-free languages
Pumping lemma: 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
Like for regular languages, you need a 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
At least one is not empty u v w x y
≤ n

13. Example L3 = {anbncn: n ≥ 0} adversary
choose n write z = uvwxy (|vwx| ≤ n,|vx| ≥ 1)
you
choose z  L choose i you win if uviwxiy  L 1 2
L3 = {anbncn: n ≥ 0}
choose n
z = anbncn 1 2
write z = uvwxy
i = ? 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 L3 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 x v
a a a ... a a b b b ... b b c c c ... c c x v
a a a ... a a b b b ... b b c c c ... c c

15. More examples ✔ ✔ ✔ Which is context-free? L1 = {anbn: n ≥ 0}
L2 = {s: s has same number of as and bs}
L3 = {anbncn: n ≥ 0}
L4 = {ssR: s ∈ {a, b}*}
L5 = {ss: s ∈ {a, b}*} ✔ ✔ ✘ ✔
Which is context-free?

16. Example L5 = {ss: s ∈ {a, b}*} a a a a a a a a a b a a a a a a a a a b
choose n
z = anbanb 1 2
write z = uvwxy
i = ? w u y x v
a a a a a a a a a b a a a a a a a a a b w x v u y
a a a a a a a a a b a a a a a a a a a b
What if:

17. Example L5 = {ss: s ∈ {a, b}*}
choose n
z = anbnanbn 1 2
write z = uvwxy
i = ? w u y x v
a a a a a a b b b b b b a a a a a a b b b b b b
Recall that |vwx| ≤ n

18. Example Three cases a a a a a a b b b b b b a a a a a a b b b b b bv w x
vwx is in the first half of anbnanbn
a a a a a a b b b b b b a a a a a a b b b b b b
Case 2: v w x
vwx is in the middle part of anbnanbn
a a a a a a b b b b b b a a a a a a b b b b b b
Case 3: v w x
vwx is in the second half of anbnanbn

19. Example Apply pumping with i = 0
a a a a a a b b b b b b a a a a a a b b b b b b
Case 1: v w x
uwy looks like aibjanbn, where i < n or j < n
a a a a a a b b b b b b a a a a a a b b b b b b
Case 2: v w x
uwy looks like anbiajbn, where i < n or j < n
a a a a a a b b b b b b a a a a a a b b b b b b
Case 3: v w x
uwy looks like anbnaibj, where i < n or j < n

20. Example Apply pumping with i = 0 L5 = {ss: s ∈ {a, b}*}
Case 1:
uv0wx0y looks like aibjanbn, where i < n or j < n
Not of the form ss
Case 2:
uv0wx0y looks like anbiajbn, where i < n or j < n
Not of the form ss
Case 3:
uv0wx0y looks like anbnaibj, where i < n or j < n
Not of the form ss
This covers all the cases, so L5 is not context-free