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It is known that the halting problem is undecidable even when we fix either the Turing machine $M$ or the input $w$. You have to be more careful about this statement. It's not true for any fixed Turing machine $M$ that the halting problem $\text{HALT}_M$ (deciding on input $w$ if $M$ halts on $w$) is undecidable. For example, if $M$ is a machine that always ...


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