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Proving the language $L=\{ a^i b a^j |\ i<j,\text{ or }i>j;\ i,j>0 \} $ is not regular, I am trying to use the pumping lemma for regular languages:

pumping length $:= m$

case 1: $i<j; j=m$

$w_1 = a^i b a^m = xyz$
$|xy| < m$
$|y| > 0$

$\Rightarrow$
$y$ can be:

  • $a$s only
  • $a$s + $b$
  • just $b$

However, this leads to $xy^hz\ \not \in L\ \forall h,\ h>0$.

case 2: $i>j; i=m$

$w_2 = a^m b a^j = xyz$
$|xy| < m$
$|y| > 0$

$\Rightarrow$
$y$ can be:

  • $a$s only

Pumping with this $y$:
$xy^hz\ \in L\ \forall h$ holds true.

As this answer does not seem to even consider the second case, is the first case sufficient to proof the language is not regular?
However, taking a look at this answer I think I'd have to considere the second case as well, in order to cover all possible $xyz$ combinations.

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cocoseis
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2 Answers2

1

Here is the contrapositive form of the pumping lemma, which is what we use to prove that a language is not regular:

Let $L$ be a language. Suppose that for all $p \geq 1$ there exists a word $w \in L$ of length at least $p$ such that for every decomposition $w=xyz$ in which $|xy| \leq p$ and $y \neq \epsilon$ there exists an integer $i \geq 0$ such that $xy^iz \notin L$. Then $L$ is not regular.

Using the pumping lemma directly for the language $L$ in the question is a bit tricky. Given $p \geq 1$, let $w = a^p b a^{p+p!} \in L$; note that $|w| \geq p$. We have to show that for every decomposition $w = xyz$ in which $|xy| \leq p$ and $y \neq \epsilon$ there exists an integer $i \geq 0$ such that $xy^iz \notin L$. Consider any such decomposition. It is easy to check that $y = a^\ell$ for some $1 \leq \ell \leq p$, and that $xy^ij = a^{p+(i-1)\ell}ba^{p+p!}$. For $i = p!/\ell+1$ (which is an integer!), $xy^ij = a^{p+p!}ba^{p+p!} \notin L$. Thus the condition in the pumping lemma is satisfied, and we conclude that $L$ is not regular.

Yuval Filmus
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Let $$L_1 = \{a^iba^j | i\lt j\text{ and } i,j\gt 0\}$$ $$L_2 = \{a^iba^j | i\gt j\text{ and } i,j\gt 0\}$$

Here, $$L = L_1\cup L_2$$

From Case 1, you have shown that $L_1$ is not regular.

You are trying to conclude that if $L_1$ is not regular, $L$ is not regular. It is not sufficient that $L_1$ is not regular to show that $L$ is not regular. A simple counter example will be $L_1 = \{a^nb^n|n\gt 0\}$ and $L_2=\{a^m b^n |m\ne n\}$. Neither $L_1$ nor $L_2$ are regular, yet their union is regular.

In the first answer you have linked, it is not true that only one case has been considered. You need to find any one string $s$ whose length is at least than every possible pumping length $p$ such that no matter how you partition it into $s=xyz$, the resulting string goes out of the language.

The key difference between your case 1 and the answer you linked is that you have taken a string in $L_1\subset L$, pumped it and shown that it goes out of $L_1$ only. However, what you need to show is that it cannot lie in $L$ at all, no matter how you split it and pump it, which is what the linked answer shows.

GoodDeeds
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