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I am not quite sure, whether automata can construct languages over their types. For example, a Push down automaton can construct a language of type 2 - does that mean that a PDA also can construct a language of type 3? The same for turing machine?

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The set of languages of type $3$ is a proper subset of the of languages of type $2$. In other words, a language of type $3$ is also a language of type $2$.

If some automaton can recognize languages of type $2$, then it can also recognize languages of type $3$.

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