# Prove that half(L) is also regular if L is a regular language [duplicate]

Given a regular language $L \subseteq \sum^*$ prove that the language

$half(L) = \{w \space | \space ww \space \in \space L \}$

is also a regular language.

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While trying to make one for $half(L)$, I came up with a DFA which uses the DFA for $reverse(L)$, but that one, it turns out, accepts a different regular language called $reflect(L) = \{w \space | \space ww^R \in L\}$. The idea was to make a DFA by first making an NFA using the standard "first reverse transitions, add new start state, make previous start state the sole accept state, etc." technique. Then we make a DFA where the states are a pair of the states of the original DFA and this reverse(L) DFA's states. However, I won't digress into that because this doesn't accept $half(L)$.