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Use the Pumping Lemma to show that $L$ is not regular: $$ L = \{{a^{n^3} | \ge 0}\}$$

I feel like I have a good intuition of what the Pumping Lemma states; strings that belong to a regular language have a part or parts that repeat over and over again.

Example strings:
$a^{0^3} = a^0 = $ "" ($\epsilon$)
$a^{1^3} = a^1 = $ "a"
$a^{2^3} = a^8 = $ "aaaaaaaa"
$a^{3^3} = a^{27} = $ "aaaaaaaaaaaaaaaaaaaaaaaaaaa"

I understand that this proof is done via contradiction, and looking at the examples it becomes obvious that constructing a DFA is difficult. But how do I prove that's impossible?!

Chuckles
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  • That intuition is not really helpful, since it's mostly wrong. I recommend you stick to the formal statement of the theorem for some exercises in order to build a more useful intuition. – Raphael Apr 06 '15 at 10:08
  • Considering that is was the close reading of the formal definition of the P.L. that brought me to my intuition, it doesn't seem to make sense to go back to said reading and simply cross my fingers, doesn't it? I wish I had a better way of translating my understanding (whether flawed or not) into text. Seems like my misunderstanding is only getting pumped by reading the same university PDFs and Wikipedia articles online. I'm merely asking for a little push, a little more from anyone who may be adept at using these concepts in proofs such as the problem I have described. I'm just having trouble. – Chuckles Apr 06 '15 at 15:07
  • Okay, here's a more helpful response that cleared up and confirmed some of my intution: http://stackoverflow.com/questions/461619/in-laymens-terms-what-is-the-pumping-lemma/1933405#1933405 I ran into this intuition by forcing myself to try and construct an automaton, only to realize that there is not way (an understanding I tried to illustrate with my listing of the first four strings in $L$). – Chuckles Apr 06 '15 at 15:20
  • However, for this example, I feel that I lack the mathematical sneakiness to understand how to manipulate the expression $a^{n^3}$ in order to show that $L$ is not regular. Seriously, math is not second nature to me. The reason I'm in this class a second time is that I feel I understand these concepts at a higher level, but when it comes to the proofs and tests, I might as well be standing on the double yellow lines looking straight into the headlights. – Chuckles Apr 06 '15 at 15:27
  • We already linked our reference question. What of the things written there don't you understand? Can you follow the examples there and filed unter [tag:pumping-lemma]? Have you looked at the proof of the Pumping lemma? – Raphael Apr 07 '15 at 07:03
  • I really cannot explain my confusion any further. I tried to point it out in my detailed comments above, but that simply is not enough. I'm done. – Chuckles Apr 08 '15 at 21:45

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