CSC 3130: Automata theory and formal languages

1. Fall 2008 The Chinese University of Hong Kong CSC 3130: Automata theory and formal languages Limitations of context-free languages Andrej Bogdanov http://www.cse.cuhk.edu.hk/~andrejb/csc3130

2. Non context-free languages • Recall the pumping lemma for regular languagesallows us to show some languages are not regularAre 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 • Let’s try to show this is context-free L4 = {anbncn: n ≥ 0} 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 derivationsof “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 mustappear 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 S S ababbabb ababab ✗ C B B C C S B S a b C C B B C S S B b b a S B B C B C b B C a b 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 ababbabb ababbbabbb ababab ✗ ✗ ababnabnbb

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

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 x v w y u

10. Pumping lemma for context-free languages • Theorem: For every context-free language L There exists a number nsuch 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. x v w y u

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

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 nwrite z = uvwxy(|vwx| ≤ n,|vx| ≥ 1) you choose zLchoose iyou win if uviwxiyL 1 2

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

14. Example • Case 1:v or x contains two kinds of symbolsThen uv2wx2y not in L because pattern is wrong • Case 2: v and x both contain one kind of symbolThen 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}*}