Prove that this language is undecidable

Question

Is the following language L undecidable?

L = {M | M is a Turing machine description and there exists an input x of length k such that M halts after at most k steps}

I think it is but I couldn't prove it. I tried to think of a reduction from the halting problem.

Answer 1

Review: An instance of the halting problem asks whether Turning machine N halts on input y. The problem is known to be undecidable (but semidecidable).

Your language L is indeed undecidable. This can be shown by reducing the halting problem to L:

1. For the halting problem instance (N, y), create a new machine M for the L problem.
2. On input x, M simulates (N, y) for length(x) steps.
3. If the simulation halted within that number of steps, then M halts. Otherwise, M deliberately goes into an infinite loop.

This reduction is valid because:

• If (N, y) does halt eventually in k steps, then M will halt for all inputs of length k or greater, thus M is in L.
• Otherwise (N, y) does not halt, then M will not halt for any input string no matter how long it is, thus M is not in L.

Finally, the halting problem is undecidable, therefore L is undecidable.