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2012-08-30 Linköpings Tekniska Högskola Institutionen för Datavetenskap Gustav Nordh TDDD65 Problem Session 2 1. Show that the class of context-free languages is closed under union, concatenation, and * (Kleene star). 2. Consider the language L = {aq | q is a prime number} and the following proposed (incorrect) proof trying to show that L is not regular. Assume that L is regular. Then the pumping lemma must hold and there exists a constant p as stipulated by the pumping lemma. Consider the string s = aq where q is the first prime number larger than p + 2. According to the pumping lemma s can be split into xyz such that for each i ≥ 0, xy i z ∈ L, |y| > 0, and |xy| ≤ p. Choose x = ε, y = a, and z = aq−1 . Now consider the string xy 2 z which according to the pumping lemma must be in L. By our choice of xyz we have that xy 2 z = aq+1 which is not in L since q + 1 is an even number larger than 2 (and thus not a prime). Hence our assumption that L is regular was wrong and we conclude that L is not regular. (a) Point out the error(s) in the proposed (incorrect) proof above. (b) Correct the proposed proof and show that L is not regular. 3. For each of the languages described by the following regular expressions over Σ = {0, 1} give a CFG (context-free grammar) generating the language. (a) 0*1* (b) 0(10)*1 (c) 0*+1* (d) (000)* (e) (0+1)*0(0+1)*1(0+1)*0(0+1)* (f) ((11)*0)* 4. (a) Give a CFG G that generates the language L = {ai bj ck | i = j or j = k where i, j, k ≥ 0} (b) Show a left-most derivation of the string aabbcc. 1 (c) Is G ambiguous? (Try to justify your answer.) 5. A palindrome is a string that reads the same forward and backward, e.g., the string abba. Describe a PDA recognizing the language of all palindromes over the alphabet {a, b}. 6. Show that the class of decidable languages is closed under complement. 7. Prove that the following languages are not regular. (a) {0n 1m 0n | m, n ≥ 0} (b) {0m 1n | m 6= n} (c) {wtw | w, t ∈ {0, 1}∗ , w 6= ε, t 6= ε} 2