(20 pt) Show that the following languages are not regular. (a) $L = \{a^ib^j \mid i = 2j\}$. (b) $L = \{w \in \{a, b, c\}^* \mid w$ is a palindrome (i.e. $w = w^R)\}$.
Added by Victor L.
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Assume that L is regular. Then, according to the pumping lemma, there exists a pumping length p such that for any string s in L with |s| ≥ p, s can be divided into three parts, s = xyz, satisfying the following conditions: Show more…
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1. Give a CFG for each of the following languages (with n, m, k ≥ 0.) Note: You only need to give production rules for each grammar. (a) L = {a^n b^2n | n ≥ 2} (b) L = {w ∈ {a, b}* | na(w) = 3nb(w)}. (c) L = {wwR | w ∈ {a, b}*} (d) L = {w ∈ {a, b}* | w = wR, that is, w is a palindrome. } (e) L = {a^n b^m | n ≥ m + 3}. (f) L = {a^n b^m | n < m + 3}. (g) L = {a^m b^n c^k | m = n + k}. (h) L = {a^m b^n c^k | n = k + m}. (i) L = {a^m b^n c^k | n = m + 2k}. (j) L = {w ∈ {a, b, c}* | na(w) + 2nb(w) ≠ nc(w)}. (k) L = {a^m b^n c^k | m > n or n + m = k}.
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Let L be the language of all binary palindromes (i.e. w = w-reversed). (a) Construct a CFG for L and then convert it to Chomsky normal form (b) Construct a PDA for L
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Is the following language regular or not regular? Prove your answer: (1) A1 = {www | w ϵ {a, b}* } (2) A2 = {w1w2 w3 | w1, w2, w3 ϵ {a, b}* } (3) A3 = {11n 22n 33n | n > 0 } (4) A4 = {1i 2j 3k | i, j, k > 0 } Please CLEARLY explain why or why not.
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