Discussion
The decision problem "Given a Turing Machine M and a step-counting function "myfun", is the language accepted by M in Time (myfun) ?" is........... .
Answer: B
The problem is unsolvable, that is undecidable.
The input is a Turing machine \(M\), and the question asks whether the language it accepts belongs to the class \(\mathrm{TIME}(\mathit{myfun})\). That is a property of the language \(L(M)\), not of how \(M\) happens to run. The property is non-trivial: some recursively enumerable languages are decidable within the time bound and some are not. Rice's theorem says every non-trivial property of the language of a Turing machine is undecidable, so no algorithm decides it.
'Solvable' is therefore wrong. 'Stable' is not a term used for decidability. 'Partially solvable' would mean the yes-instances can at least be recognised, which Rice's theorem does not give, and the standard answer for such a question is unsolvable.
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