Let $L$ be a regular language. Prove that:
$L_{+--}=\left\{w: \exists_u |u|=2|w| \wedge wu\in L\right\}$
$L_{++-}=\left\{w: \exists_u 2|u|=|w| \wedge wu\in L \right\}$
$L_{-+-}=\left\{w:\exists_{u,v} |u|=|w|=|v| \wedge uwv\in L\right\}$
are regular and:
- $L_{+-+}=\left\{ uv:\exists_w |u|=|w|=|v| \wedge uwv\in L \right\}$
is not regular.
Seems very hard to me. I suppose 1-3 are similar (but I may be wrong), but I don't know how to approach. General idea is usually to modify finite state machine for $L$ to accept other language. But those constructions are often very sophisticated and I still can't come up with it alone.
