12 Facts About Halting problem

In computability theory, the halting problem is the problem of determining, from a description of an arbitrary computer program and an input, whether the program will finish running, or continue to run forever.

Halting problem is a decision problem about properties of computer programs on a fixed Turing-complete model of computation, i e, all programs that can be written in some given programming language that is general enough to be equivalent to a Turing machine.

The Halting problem is to determine, given a program and an input to the program, whether the program will eventually halt when run with that input.

One approach to the Halting problem might be to run the program for some number of steps and check if it halts.

However, an interpreter will not halt if its input program does not halt, so this approach cannot solve the halting problem as stated; it does not successfully answer "does not halt" for programs that do not halt.

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Halting problem is theoretically decidable for linear bounded automata or deterministic machines with finite memory.

Halting problem is historically important because it was one of the first problems to be proved undecidable.

Typical method of proving a problem to be undecidable is to reduce it to the halting problem.

Universal halting problem, known as totality, is the problem of determining, whether a given computer program will halt for every input.

However the problem "given program p, is it a partial halting solver" is at least as hard as the halting problem.

The halting problem is decidable for a lossy Turing machine but non-primitive recursive.

Machine with an oracle for the halting problem can determine whether particular Turing machines will halt on particular inputs, but they cannot determine, in general, if machines equivalent to themselves will halt.