pysähtymisongelma

The halting problem, known in Finnish as the pysähtymisongelma, is a fundamental decision problem in computability theory. It asks whether it is possible to determine, for an arbitrary program and an arbitrary input, whether the program will eventually finish running (halt) or continue to run forever. Alan Turing proved in 1936 that a general algorithm to solve the halting problem for all possible program-input pairs cannot exist. This is a landmark result in computer science, demonstrating inherent limitations of computation.

The proof of the halting problem's undecidability typically involves a self-referential argument. Imagine a hypothetical program, let's call it HaltChecker, that takes a program P and its input I and outputs "halts" if P halts on I, and "loops" if P does not halt on I. Now, construct another program, Diagonal, which takes a program P as its input. Diagonal first calls HaltChecker with P and P as arguments. If HaltChecker reports that P halts on input P, then Diagonal enters an infinite loop. If HaltChecker reports that P loops on input P, then Diagonal halts. The contradiction arises when we consider what happens when Diagonal is run with itself as input: Diagonal(Diagonal). If Diagonal(Diagonal) halts, then according to its definition, it must loop when HaltChecker reports it halts. Conversely, if Diagonal(Diagonal) loops, then HaltChecker must have reported that it loops, which means Diagonal(Diagonal) should halt. This logical inconsistency demonstrates that such a HaltChecker program cannot exist.

The undecidability of the halting problem has profound implications. It implies that there are problems that cannot be solved by any computer, no matter how powerful. This has ramifications for areas such as compiler design, program verification, and artificial intelligence, highlighting the need for alternative approaches when dealing with potentially non-terminating computations.