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    <p>The definition of <code>T</code> in that grammer does indeed appear to be unnecessary complication. <code>T</code> can generate any string of <code>a</code>s and <code>b</code>s, so the simpler definition would have been just as good.</p> <p>I can only guess that the productions are given as they are because of the sausage-factory nature of writing a book.</p> <p><strong>ORIGINAL WRONG ANSWER:</strong></p> <p>They are not equivalent, because <code>X</code> itself cannot be <code>&lt;epsilon&gt;</code>, and <code>T</code> is not any combination of <code>a</code> and <code>b</code>. <code>T</code> can only expand to a palindrome (including the empty palindrome, a single character, or a palindrome with an unpaired central character).</p> <p>If <code>X</code> could be empty, then <code>T</code> could expand to anything, but it can't.</p> <p><em>NOTE</em></p> <p>This answer is based on the supposition that the author’s intention for the production <code>T -&gt; XTX</code> is that the two identical non-terminals in the substitution must represent identical strings of characters. Since I don't have the text to look at, I don't know if this assumption is well-founded except that it is motivated by the question itself. This supposition could be a mistake by the author if that is not the case elsewhere. I think that, in general, this requirement is not true of context-free grammers.</p> <p>The correct productions would be:</p> <pre><code>R -&gt; aRa | bRb | S S -&gt; aTb | bTa T -&gt; aTa | bTb | a | b | &lt;epsilon&gt; </code></pre>
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    1. POContext free grammar for non-palindrome
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    1. USJeffrey L Whitledge
    1. USJeffrey L Whitledge
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    1. COI guess what I'm saying is that, yes, you're right, this grammer is messed up. I couldn't find anything about it on the errata page for that book, so you might want to tell the author about it.
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      1. USJeffrey L Whitledge
    2. COMy mistake. I did not mean to show that X could become <epsilon>. X remains unchanged. It can become only 'a' or 'b'. I have corrected the question now. Further I don't understand how T can expand only to become a Palindrome as you say. Consider aabb, a non-palindrome. It can be generated as T->XTX->XXTXX->aa<epsilon>bb = aabb. Use of same variables does not mean that they be substituted the same non-terminal.
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      1. USAbhijith Madhav
    3. CO@Abhijith Madhav - I updated my answer. I agree with you.
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      1. USJeffrey L Whitledge
 

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