Question

in the general halting problem, we ask for an algorithm that gives the correct answer for any m and w. we can relax this generality, for example, by looking for an algorithm that works for all m but only a single w. we say that such a problem is decidable if for every w there exists a (possibly different) algorithm that determines whether or not (m, w) halts. show that even in this restricted setting the problem is undecidable.

Answer 1

In the restricted setting of the general halting problem, the original halting problem is undecidable, this means that the restricted halting problem is also undecidable.

Step-by-step explanation:

In the restricted setting of the general halting problem, we are asked to find an algorithm that works for all possible inputs m, but only a single fixed input w.

This problem is still undecidable, meaning that no such algorithm exists. This can be proven using a reduction from the original halting problem.

To show that the restricted halting problem is undecidable, we can assume that there exists an algorithm A that solves it. We can then construct a new algorithm B that takes inputs (m, w), where m is any program and w is an input for that program.

Algorithm B simulates A on input (m, m) and outputs the opposite of what A outputs, effectively solving the original halting problem.

Since the original halting problem is undecidable, this means that the restricted halting problem is also undecidable.