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Could anyone explain me how to prove that this language is not regular using pumping lemma? I can prove easier examples but with this one I do not even know with which word i should start proving it.

$$ L = \left\{ aavau \mid u,v \in \{b,c\}^* \wedge 3\lvert u \rvert_b = 2 \lvert v \rvert_b \wedge \lvert v \rvert > 1 \right\} $$

kvway
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2 Answers2

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We assume $L$ is regular. Then by the pumping lemma we have a constant pumping length $p \ge 1$, on which the granted split $$ aavau = x y z $$ for all word $aavau \in L$ with $\lvert aavau \rvert \ge p$ will feature repetition within the first $p$ symbols in the sense of $$ \forall i \in \mathbb{N}_0: \, x y^i z \in L \quad (*) $$ where $\lvert x y \rvert \le p$ and $\lvert y \rvert \ge 1$.

$u$ and $v$ will feature only non-$a$ symbols, $v$ at least one, $u$ might be empty.

Let us examine the word $$ w = aa\underbrace{b^{3p}}_va\underbrace{b^{2p}}_u\in L $$ We have $\lvert w \rvert = 3 + 5p \ge p$, so the pumping lemma will grant the split $$ w = x y z $$

If $y$ contains an $a$, then $y^4$ will contain at least four of them breaking the $aavau$ pattern with exactly three $a$ symbols.

If $y$ contains a $b$ then $y$ will contain

  1. just part of the first group $b^{3p}$ or
  2. just part of the second group $b^{2p}$ or
  3. parts of both groups.

In the first or second case, repetition $y^i$ will affect only one group ($u$ or $v$) and the condition $3\lvert u \rvert_b = 2 \lvert v \rvert_b$, which links the number of $b$ symbols for both $u$ and $v$ will get violated for some $i$.

In the third case, the $a$ between the two groups will be contained in $y$ and thus again the pattern $aauav$ with excatly three $a$ symbols will be violated by $y^i$ for some $i$.

Thus $w \in L$ and $L$ assumed to be regular will imply (via the pumping lemma) that words outside the intended set of $L$ have to be elements of $L$, thus $L$ can not be regular.

mvw
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  • Thanks, I forgot that requirement. – mvw Dec 23 '16 at 12:29
  • And now we have the same answer… note that the violation always occurs with $i=0$. – gniourf_gniourf Dec 23 '16 at 12:47
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    That is a good sign. – mvw Dec 23 '16 at 13:58
  • @mvw When i look at it again i'm kind of confused now. Should not be exponents of those u, v be different? because when u put p = 1 then u have aabbbabb which is not true for the given condition since 3 != 2. I think the condition says multiply v by 3 and multiplay u by 2 and get the same amount of b. So you choosed the word which is not from the language ? – kvway Dec 23 '16 at 14:10
  • $u = bb$, $v=bbb$ then we have $3\cdot 2 = 2 \cdot 3$ which is true statement. – mvw Dec 24 '16 at 11:09
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Recall the pumping lemma:

If $L$ is a regular language then there exists $p\geq1$ such that for all $w\in L$ such that $\lvert w\rvert\geq p$, $w$ can be written as $w=xyz$ where $\lvert xy\rvert\leq p$, $\lvert y\rvert\geq1$ and for all $n\geq0$, $xy^nz\in L$.

Assume that your language $L$ is regular and let $p\geq1$ be as in the pumping lemma.

  • Explain why $w=a^2b^{3p}ab^{2p}\in L$ and why $\lvert w\rvert\geq p$;
  • Decompose $w$ as $w=xyz$ with $\lvert y\rvert\geq1$ and $\lvert xy\rvert\leq p$.
    • If $y$ contains an $a$, $xz\not\in L$ since $xz$ will contain less than three $a$'s, yet the words of $L$ contain exactly three $a$'s;
    • Otherwise, $y$ contains no $a$'s and $k$ letters $b$ (with $1\leq k\leq p$, and these letters must hence be consecutive since $y$ contains no $a$'s), and there are two cases: either the $b$'s are within the first group or they are within the second group. In the former case, $xz=a^2b^{3p-k}ab^{2p}\not\in L$ and in the latter case, $xz=a^2b^{3p}ab^{2p-k}\not\in L$.
  • Conclusion: it's impossible to write $w=xyz$ with $\lvert y\rvert\geq1$ and $\lvert xy\rvert\leq p$ such that $xz\in L$.

Hence $L$ doesn't fulfill the requirements of the pumping lemma, hence $L$ is not regular.

gniourf_gniourf
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