Let Σ be a non-empty alphabet and LCE* be a context-free language. Which of the following languages is/are necessarily context-free? Note: For some string w€ Σ*, wR is the string reversal. 1. reverse_1(L) = {xy : ]x, y ≤ Σ*, x € L and yR € L} 2. reverse 2(L) = {x : 3x € Σ*, x € L and x = L} If a language is necessarily context-free, prove your claim. Any construction you make does not require a proof of correctness. If a language is not necessarily context-free, give a counterexample language L which is context- free but for which reverse_i(L) (i = 1, 2) is not context-free. To do so, you must show that your language L is context-free by giving either a PDA which accepts it or a CFG which generates it. You do not need to prove the correctness of your construction but you should explain how it works. In addition, you must give a Pumping lemma proof which shows that reverse; (L) is not

Database System Concepts
7th Edition
ISBN:9780078022159
Author:Abraham Silberschatz Professor, Henry F. Korth, S. Sudarshan
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Chapter1: Introduction
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Let Σ be a non-empty alphabet and LCE* be a context-free language.
Which of the following languages is/are necessarily context-free? Note: For some string w = Σ*,
w is the string reversal.
€
1. reverse_1(L) = {xy : ³x, y ≤ Σ*, x € L and yR € L}
R
2. reverse_2(L) = {x : 3x € Σ*, x € L and x² EL
If a language is necessarily context-free, prove your claim. Any construction you make does not
require a proof of correctness.
If a language is not necessarily context-free, give a counterexample language L which is context-
free but for which reverse_i(L) (i 1, 2) is not context-free. To do so, you must show that
your language L is context-free by giving either a PDA which accepts it or a CFG which generates
it. You do not need to prove the correctness of your construction but you should explain how it
works. In addition, you must give a Pumping lemma proof which shows that reverse; (L) is not
context-free.
=
Transcribed Image Text:Let Σ be a non-empty alphabet and LCE* be a context-free language. Which of the following languages is/are necessarily context-free? Note: For some string w = Σ*, w is the string reversal. € 1. reverse_1(L) = {xy : ³x, y ≤ Σ*, x € L and yR € L} R 2. reverse_2(L) = {x : 3x € Σ*, x € L and x² EL If a language is necessarily context-free, prove your claim. Any construction you make does not require a proof of correctness. If a language is not necessarily context-free, give a counterexample language L which is context- free but for which reverse_i(L) (i 1, 2) is not context-free. To do so, you must show that your language L is context-free by giving either a PDA which accepts it or a CFG which generates it. You do not need to prove the correctness of your construction but you should explain how it works. In addition, you must give a Pumping lemma proof which shows that reverse; (L) is not context-free. =
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