Prove that every nfa can be converted to an equivalent one that has a single accept state.
Answers
Every NFA can be converted into an equivalent NFA that has a single accept state. Simply add a new final state to the original automaton, add epsilon transitions from every old final state to the new final state, and change every old final state into a regular state. This new NFA accepts exactly the same language as the original NFA.
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Create a single new final state. Create λ-edges from the original final states to this new final state. Make the original final states non-final. If the processing of an input word completes in one of the original final states, the new final state is immediately accessible without any additional symbols. Furthermore, the only way to access the new final state is if the processing completes in one of the original final states. Hence, all words previously accepted will remain accepted and no additional words will be accepted, meaning the two machines accept the same language and are therefore equivalent.