decidabilitás

Decidabilitás refers to a property of mathematical problems, particularly within the realm of theoretical computer science and mathematical logic. A problem is considered decidable if there exists an algorithm, often called a decision procedure, that can determine, for any given instance of the problem, whether the instance has a solution or not, and do so in a finite amount of time. This means the algorithm must always halt and provide a definitive yes or no answer.

The concept is closely tied to the idea of computability. If a problem is decidable, it is also computable. However, not all computable problems are decidable. For instance, a problem might be computable in the sense that there's an algorithm to find a solution, but that algorithm might not always halt, making the problem undecidable.

A famous example of an undecidable problem is the Halting Problem. This problem asks whether it's possible to create a general algorithm that can determine, for any arbitrary program and any given input, whether that program will eventually halt or run forever. Alan Turing proved that such a general algorithm cannot exist.

The study of decidability is crucial for understanding the theoretical limits of computation and for classifying problems based on their inherent difficulty. It helps in identifying problems that can be solved algorithmically and those that cannot, guiding the development of algorithms and the understanding of what is computationally feasible.