Décidabilité

Décidabilité refers to a fundamental concept in theoretical computer science and mathematical logic concerning whether a problem can be solved by an algorithm. A problem is decidable if there exists a Turing machine, or an equivalent computational model, that can always halt and provide a correct yes or no answer for any given input. The field of decidability explores the boundaries of what can be computed.

Problems that are not decidable are called undecidable. The most famous example of an undecidable problem is the Halting Problem, which asks whether an arbitrary program will finish running or continue to run forever on a given input. Alan Turing proved that no general algorithm can solve the Halting Problem for all possible program-input pairs.

The study of decidability is crucial for understanding the limits of computation and the fundamental capabilities of algorithms. It has implications in areas such as artificial intelligence, program verification, and the design of programming languages. The set of decidable problems forms a class of computable functions, and understanding which problems fall into this class is a core objective of computability theory. The existence of undecidable problems highlights that not all mathematical or computational questions can be answered algorithmically.