Which word could I use for the pumping lemma?

Question

I have a problem to start my proof because I do not find a word $w$ where I can use the pumping lemma.

Task:

Be $\sum { =\left\{ a,b,c \right\} } $ and $S=\left\{ bx{ c }^{ m }|x\in { \left\{ a,b \right\} }^{ * }\wedge m\in N\wedge { |bx| }_{ a }-{ |bx| }_{ b }>m \right\} $

Proof with the the pumping lemma, that S is no regular language.

What I know is, that the word has to start we one $b$ following from n times $a$ or $b$ but $w$ needs to have one more $a$ than $b$ in it and $w$ also needs to have one more $b$ than $c$ in it to fulfill the condition. So it has to be something like this $W=b{a}^{2m+2}b^mc^m$? This language is very difficult for me, hope you see more than I.

P.S. Please no complete solutions of the full task.

Answer 1

So it has to be something like this $w=b{a}^{2m+2}b^mc^m$.

$b{a}^{2m+2}b^mc^m$ is not the general form for $w=bxc^m\in S$. If $x$ must be in the form of $a^*b^*$, $w$ will be something like $b{a}^{m+n+k+2}b^nc^m$ for some $n,k\ge0$. Just for completeness, $w$ could be like $baba^{m+1}c^m$ or $baba^{m+1}a^kb^kc^m$ or many other forms. Of course, it is totally fine if you just meant to choose one word so as to apply the pumping lemma.

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Can you check word $ba^{p+2}c^{p}$, assuming the pumping length is $p$? This word is about as simple as you could get.

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Exercise 1. Assuming the pumping length is $p$, can we pump $ba^{p+3}bc^p\in S$?

Exercise 2. Assuming the pumping length is $p$, can we pump $ba^{2p+2}b^pc^p\in S$?