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copied!<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><epsilon></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 -> 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 -> aRa | bRb | S S -> aTb | bTa T -> aTa | bTb | a | b | <epsilon> </code></pre>

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