Halting problem

In computability theory, the halting problem is the decision problem of, given an arbitrary computer program and an input, determining whether said program will eventually finish running and halt, or will continue to run forever. Alan Turing proved in 1937 that the halting problem is undecidable, meaning that no general algorithm exists that can correctly solve the problem for all possible program–input pairs. The problem comes up often in discussions of computability since it demonstrates that some functions are mathematically definable but not computable. A key part of the formal statement of the problem is a mathematical definition of a computer and program, usually via a Turing machine. The proof then shows, for any program f that might determine whether programs halt, that a "pathological" program g exists for which f makes an incorrect determination. Specifically, g is the program that, when called with some input, passes its own source and its input to f and does the opposite of what f predicts g will do. The behavior of f on g shows undecidability as it means no program f will solve the halting problem in every possible case.