Is this language regular? ${w ∈ (a + b)∗ : |u_{a}|>= 2009 · |u_{b}|}$ for every non empty prefix $u$ of string $w$} I think it's non-regular. I tried concatenation of $L_{prefix} $={${ u : |u_{a}|> 2009 · |u_{b}|}$} and $L_{2}= a^*b^*$. $L_{2}$ is regular. I tried to show that $L_{prefix}$ is non-regular using pumping lemma. for string z= $a^{2009n} b^n$ pumping 'a' for i=0 : $$ z=uv^iw $$ $$z= uv^0w $$ $$z=a^{2009n-p} b^n$$ so there is less 'a' than 2009'b'
pumping 'b' for i=2 $$z=uv^2w $$ $$z=a^{2009n} b^{n+p} $$ again to much 'b' and less 'a' than should be. So this is non-regular language and concatenation of this two also is non-regular . Is it correct? Or maybe it's somehow regular?
pimping
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