Päätösongelman

Päätösongelman, also known as the halting problem, is a fundamental concept in computability theory and the theory of computation. It asks whether it is possible to determine, for an arbitrary computer program and an arbitrary input, whether the program will eventually stop 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.

The halting problem is undecidable, meaning there is no algorithm that can correctly solve it in all cases. If such an algorithm existed, it would lead to a contradiction. Imagine an algorithm called Halt that takes a program P and an input I and returns true if P halts on I, and false otherwise. Now, consider a new program called Paradox that takes a program P as input. Paradox runs Halt on P with P itself as the input. If Halt returns true (meaning P halts on P), Paradox enters an infinite loop. If Halt returns false (meaning P does not halt on P), Paradox halts immediately. Therefore, if Paradox is run on itself, it will halt if and only if it does not halt, which is a logical contradiction.

The undecidability of the halting problem has profound implications. It demonstrates that there are inherent limitations to what can be computed. Many other problems in computer science have been proven to be undecidable by showing that they can be used to solve the halting problem, a technique known as reduction. These limitations are not due to a lack of computing power or time, but rather are fundamental mathematical constraints. The halting problem is a cornerstone in understanding the boundaries of algorithmic computation.