Answer
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The halting problem refers to the question of whether it's possible to create an algorithm that can determine, given any arbitrary program and its input, whether the program will eventually halt (terminate) or continue to run indefinitely.
A halting problem is undecidable, meaning that there does not exist a single algorithm that can correctly determine whether any arbitrary program halts or loops forever. This was proven by Alan Turing in 1936.
Turing's proof relies on a clever contradiction: Assume there exists a halting problem solver algorithm. Then, by constructing a program that takes itself as input and does the opposite of what the solver algorithm predicts, we arrive at a contradiction. If the program halts, then the solver algorithm predicts it loops forever, and vice versa, leading to an inconsistency. Thus, no algorithm can solve the halting problem for all possible programs and inputs.
