CSC 3130: Automata theory and formal languages

Fall 2009 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

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 = {zzR: z∈ {0, 1}*} L6 = {zz: z∈ {0, 1}*}

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

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 . . .

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

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

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

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 uvvwxxy uv3wx3y w

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

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

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

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

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} choose n z= anbncn 1 write z = uvwxy i = ? 2 a a a ... a a b b b ... b b c c c ... c c x v w y u

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

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 = {zzR: z∈ {a, b}*} L6 = {zz: z∈ {a, b}*}

Example L6 = {zz: z∈ {a, b}*} choose n z= anb 1 write z = uvwxy i = ? 2 a a a a a a a a a b a a a a a a a a a b What if: a a a a a a a a a b a a a a a a a a a b x v w y u x v w y u

Example L6 = {zz: z∈ {a, b}*} choose n z= anbn 1 write z = uvwxy i = ? 2 Recall that |vwx| ≤ 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 x v w y u

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 b Case 1: x v w 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: x v w 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: x v w vwx is in the second half of anbnanbn

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: x v w uv0wx0y 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: x v w uv0wx0y 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: x v w uv0wx0y looks like anbnaibj, where i < n or j < n

Example Apply pumping with i = 0 Case 1: uv0wx0y looks like aibjanbn, where i < n or j < n Not of the form zz Case 2: uv0wx0y looks like anbiajbn, where i < n or j < n Not of the form zz uv0wx0y looks like anbnaibj, where i < n or j < n Case 3: Not of the form zz This covers all the cases, so L6 is not context-free

CSC 3130: Automata theory and formal languages