A is regular , then FirstHalves or Half(A) is also regular.
1. Design DFA, M of language A
2. Find the reversal of DFA M , say N.
3. Traverse M for one transition (for given Σ ) and N for one transition (for given Σ ) , and repeat the process.
4. if We reach at common states, Mark them as New finals in DFA M (remove old final and do minimization )
5. Result is DFA of Half(L) or FirstHalve(L)
[Note: in Short, design DFA for A , traverse from both directions (start and final),if reach at common states with same no of transitions, make them as new finals, result will be DFA for Half(L) ]