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.--Please give a hint or two. Have thought about it a lot already.--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 $half(L)$.
Prove that half(L) is also regular if L is a regular language
formal languages;regular languages;finite automata