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I'm trying to prove that the language $\mathcal L = \{w \in \{0,1\}^* ∣ w \leq w′ \text{ where }w′ \text{ is any rotation of }w\}$ is not a regular language.

Note: The inequality is with respect to lexicographic order. And if $w=xy$, then $yx$ is a rotation of $w$.

I was thinking of using the string $s = 0^{P}1$ with $P$ being the pumping length, but I can't figure out how to pump this in a way to show a contradiction. Any help will be greatly appreciated.

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HINT: If $p$ is the pumping length, start with $s=0^p10^p1$ and pump down.

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  • $\begingroup$ When you say pump down, do you mean take the divided string's pumpable portion and go from something like say 5, down to 4, 3, etc? $\endgroup$ – Vimzy Feb 9 '16 at 21:51
  • $\begingroup$ @Vimzy: The pumping lemma says that $s$ decomposes as $xyz$, where $|xy|\le p$, $|y|\ge 1$, and $xy^kz\in L$ for each $k\ge 0$. The original $s$ has $k=1$; pumping down is setting $k=0$. $\endgroup$ – Brian M. Scott Feb 9 '16 at 21:53
  • $\begingroup$ But s=0p10p1 wouldn't even be a part of the language L? On certain rotations, 0p10p1 is greater than the rotation? $\endgroup$ – Vimzy Feb 9 '16 at 22:00
  • $\begingroup$ @Vimzy: $0^p10^p1$ is in $L$: no rotation has more than $p$ leading zeroes. $\endgroup$ – Brian M. Scott Feb 9 '16 at 22:03
  • $\begingroup$ I think I may just be misunderstanding the lexicographic condition here, but isn't 0101 when rotated to 0011 come out to 0101>0011? I think I may have a misunderstanding here? $\endgroup$ – Vimzy Feb 9 '16 at 22:04

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