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  1. How can I prove that the output sequence of Mealy Moore Machine is a Regular Language ?
  2. And what if transition outputs more than one character ? How can we prove for this case that the language in (1) and (2) are same ?

Can I do something like this -

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The right side one is Mealy Machine. So can I prove like making a machine in left with transitions using output sequence of Mealy Machine ?

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By discarding the input-letters you only consider the output language of the respective Mealy automaton. Effectively you obtain a finite state automaton that recognizes the output language and because it is a finite state automaton, which may be non-deterministic, the language it accepts is regular.

The length of the output of a Mealy-transition is irrelevant as you may have for example$aa\in\Gamma$, where $\Gamma$ denotes the output alphabet of a Mealy automaton.

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