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My Turing machine starts with an empty tape. It writes a random word of the set $0^n1^n$ to the tape. Hopefully i made no mistakes.

My question is about the productions coming out of state $q_0$:

$δ(q_0, blank)=(0,R)$ and $δ(q_0,blank)=(blank,L)$. (see image below).

Their purpose is to generate a random amount of $0$ before going to state $q_1$. They are nondeterministic. Is there a workaround to make them deterministic?

I heard that every Turing machine can be made deterministic. But this case is a little bit strange, because instead of testing if a word on the tape is element of a set, it writes a random word of the set to the tape.

Turing machine generating 0^n1^n

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If a language is decided by some nondeterministic Turing machine, then it is decided by some deterministic Turing machine. However, not every behavior of a nondeterministic Turing machine can be replicated deterministically, as your example demonstrates.

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