Halting Problem

The Halting Problem is an important problem in the theory of computability in logic and mathematics, proposed by British mathematician Alan Turing in 1936. The relevant paper is Turing's famous paper "On Computable Numbers, with an Application to the Entscheidungsproblem", in which the halting problem and the concept of a Turing machine were first proposed. This paper, published in the Proceedings of the London Mathematical Society, laid the foundation for modern computer science. Although Turing's original paper did not directly discuss the problem that later became known as the halting problem, his work laid the foundation for understanding the boundaries of computing. The unsolvability of the halting problem is a core concept in computational theory, which shows that some problems are inherently unsolvable by algorithms.

The halting problem is about determining whether any program can finish running within a finite time. In other words, given a program and input, is it possible to determine whether the program will eventually stop running instead of running forever? Turing proved through his diagonal argument that the halting problem is unsolvable, that is, there is no universal algorithm that can determine whether a program will stop running or loop infinitely for all programs and inputs. The proof of this problem is one of the cornerstones of computer science theory, which reveals the limitations of computing.