5. (20 marks maximum) Prove that the following languages are context free: (a) (10 marks)(an+lyn+2Cn+4 1...
4. (Non-CFLs) Prove that the following languages are not context-free. (b) The following language over the alphabet {a, b, c}: B = {aix | i ≥ 0, x ∈ {b, c}* , and if i = 1 then x = ww for some string w}. (Careful: B satisfies the pumping lemma for CFLs! Make sure you understand why, but you don’t need to write it down.)
1. Show that the following languages are context-free. You can do this by writing a context free grammar or a PDA, or you can use the closure theorems for context-free languages. For example, you could show that L is the union of two simpler context-free languages. (b) L {0, 1}* - {0"1" :n z 0}
6.) Is the languages Context Free or not? (prove / disprove using pumping lemma for CFL ) L = {0n 1 0n 10n | n >= 1}
1. Show that the following languages are context-free. You can do this by writing a context free grammar or a PDA, or you can use the closure theorems for context-free languages. For example, you could show that L is the union of two simpler context-free languages. (d) L = {0, 1}* - L1, where L1 is the language {1010010001…10n-110n1 : n n ≥ 1}.
Formal Languages & Automata Theory 1411372
Pages 133,134
Problems: 7(a,b), 8 (b,c)
5.1 CoNTEXT-FREE GRAMMARS 133 EXERGISES 7. Find context-free grammars for the following languages (with n 2 0, m 0) (a) L = {a"b"": n < m + 3).
Prove the following languages are not context-free by using the
pumping
lemma.
{b(n) #6(n + 1) | n є N, n-1} where b(n) is binary representation of n with no leading 0
{b(n) #6(n + 1) | n є N, n-1} where b(n) is binary representation of n with no leading 0
Question 3 Give a context-free grammar for each of the following languages over = {a,b,c}. 1. {a^jen: n >0}. 2. {animck : k=n+m} 3. Strings of a's and b's that contain twice as many a's as b's (for example, aba).
Give context-free grammars that generate the following languages (E = {a,b}). (a) (1 point) L1 = {w | W contains at least two b's} (b) (1 point) L2 = {w/w = wf, w is a palindrome} (c) (1 point) L3 = {w w contains less a's than b's}. (d) (1 point) LA = {w w = ayn+1, n > 2} (e) (1 points) Ls = {w w = a";2(m+n)cm, m, n >0}; (S = {a,b,c}).
Question 3 (5 Points) Find context free grammars L = a"b",n is a multiple of three Find context-free grammars for the following languages (with n 2 0, m 2 0) ** L = {a"bm : n < m+3}. (a) (b) L= {a"bm : n = m - 1}. L = {a"bm 2m}. L {a"b" 2n < m < 3n}. (c) (d)