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

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

Publisher:Abraham Silberschatz Professor, Henry F. Korth, S. Sudarshan

Chapter1: Introduction

Section: Chapter Questions

Problem 1PE

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Let M be the language over {a, b, c, d, e, f} accepting all strings so that: 1. There are precisely two e's in the string. 2. Every a is immediately followed by an even number of f's. 3. Every c is immediately followed by an odd number of b's. 4. b's and f's don't occur except as provided in rules 2 and 3. 5. All a's occur after the first e. 6. All c's occur before the second e. 7. In between the two e's there are exactly half as many a's as c's. Construct a context-free grammar generating M. You do not need an inductive proof, but you should explain how your construction accounts for each rule. We could eliminate one rule from m to make it regular. Which one? Why?
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