Päätettävyys

Päätettävyys, also known as decidability, is a concept in theoretical computer science and logic that refers to the ability to determine, through an algorithm, whether a given statement or problem has a solution. A problem is considered decidable if there exists an algorithm that can always produce a correct answer, either "yes" or "no," for any input within a finite amount of time. This algorithm must terminate for all possible inputs, providing a definitive answer.

In contrast, a problem is undecidable if no such algorithm exists. The concept of decidability is closely related to the halting problem, which asks whether a given program will eventually halt for a given input. Alan Turing's work in the 1930s demonstrated that the halting problem is undecidable, meaning there is no general algorithm that can determine whether any arbitrary program will halt.

Decidability is a fundamental concept in the study of computation and complexity. It helps classify problems into categories based on their solvability, which is crucial for understanding the limits of what can be computed. The study of decidable and undecidable problems has led to significant advancements in computer science, including the development of formal languages, automata theory, and the theory of computation.