# Tagged Questions

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### Proof that the regular languages are closed under string homomorphism

Where can I find a proof of this? Thanks!
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### Is given language regular? [duplicate]

Let $L$ be a regular language. Is $\frac{1}{2}L := \left\{ w: \exists_u |u|=|w| \wedge wu\in L \right\}$ regular too? I think the answer is YES. But I don't know how to prove it. I was trying to ...
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### Prove that $L_1$ is regular if $L_2$, $L_1L_2$, $L_2L_1$ are regular

Prove that $L_1$ is regular if $L_2$, $L_1L_2$, $L_2L_1$ are regular. These are the things that I would use to start. As $L_1L_2$ is regular, then the homomorphism $h(L_1L_2)$ is regular. Let ...
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### Regularity of the exact middle of words from a regular language

Let $L$ be a regular language. Is the language $L_2 = \{y : \exists x,z\ \ s.t.|x|=|z|\ and\ xyz \in L \}$ regular? I know it's very similar to the question here, but the catch is that it's not a ...
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### Closure against right quotient with a fixed language

I'd really love your help with the following: For any fixed $L_2$ I need to decide whether there is closure under the following operators: $A_r(L)=\{x \mid \exists y \in L_2 : xy \in L\}$ ...
### Closure against the operator $A(L)=\{ww^Rw \mid w \in L \wedge |w| \lt 2007\}$
I would like your help with the following question: Let $L$ be a language, and operator $A(L)=\{\,ww^Rw \mid w \in L\ \wedge\ |w| \lt 2007\,\}$ where $x^R$ is the reversed string of $x$. Which of ...
L is a regular language over the alphabet $\sum = \{a,b\}$. The left quotient of L regarding $w \in \sum^*$ is the language $$w^{-1} L := \{v: wv \in L\}$$ How can I prove that $w^{-1}$ L is ...