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    <h2>"IN CONTEXT OF PUMPING LEMMA FOR REGULAR LANGUAGES "</h2> <p><strong>Yes we agree</strong>, All finite languages are regular language means we can have Finite automata as well as regular expression for any finite language.<br> Whereas <code>a infinite language may be regular or not</code>. Venn-Diagram is shown below. So we need to only check for infinite language L where its regular of not! </p> <p>Think about FA: </p> <ul> <li><p>Any <code>automata</code> for <code>a finite language can not contains loop!</code> (also <code>regular expressions for finite language will be without * and +</code>operation). </p></li> <li><p>Any <code>automata</code> for <code>a infinite language(regular) will contain the loop</code>. <code>We can't construct an automata for infinite language without loop</code>; where loop may be a self loop of via some other state. {If its self loop then a single symbol repeats any number of time, if via other state a sequence of symbols comes in loop can be repeat any numbers of time}. </p></li> </ul> <p>Pumping means repeating. In pumping lemma <code>w</code> can be break in three parts <em>x, y, z</em>. The 'y' is in part of <code>w</code> occurs in loop (that's y>=1 ). So pumping lemma is nothing finding looping part <code>y</code> and repeating this looping part any numbers of time.<br> <img src="https://i.stack.imgur.com/oZhI8.jpg" alt="enter image description here"> You can see if you repeat loop any number of times and generates new string <code>w'</code> its still in language. </p> <p><strong>NOTE</strong>: <code>Regular Expressions for infinite language can't be without * and +</code>operation!</p> <p><strong>[answer]</strong> There is no loop in an automata for finite language so we can't pump(generate by repeating) new strings in language. And Pumping Lemma is not applicable for finite language. </p> <blockquote> <p>Although some writers also explain pumping lemma for finite language where <code>i</code> in xy<sup>i</sup>z can be repeat boundedly (say k ≤ i ≤ m )</p> </blockquote> <hr> <p><img src="https://i.stack.imgur.com/av5GV.jpg" alt="enter image description here"> </p> <p>In Venn-Diagram Every finite set is regular. Infinite set may be regular or not. <code>Regular-Languages ⊆ Non-Regular Languages</code></p> <hr>
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    1. POTo make sure: Pumping lemma for infinite regular languages only?
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    1. USGrijesh Chauhan
    1. USGrijesh Chauhan
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    1. CO`VERY OLD QUESTION`~~ `Although it was answered, I Liked to add my thoughts too!`
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      1. USGrijesh Chauhan
 

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