A term primarily used in logic, computer science, and mathematics, 'decidable' refers to a problem or statement for which an algorithm exists that can determine, in a finite amount of time, whether it is true or false, or whether a given input possesses a specific property. This algorithm must be complete, returning a definitive answer for all possible inputs. If no such algorithm exists, the problem or statement is considered undecidable. Decidability hinges on the existence of a procedure, not necessarily its practical efficiency. In contrast to this, a problem that can be solved by an algorithm in a reasonable time is called tractable.
Decidable meaning with examples
- In formal logic, determining if a proposition within a particular system is a theorem or not could be a decidable problem, as a proof could be found or refuted algorithmically. This highlights the importance of established rules within a formal system. If a proof is able to be found, then that means it is true within a specific system and that process makes it decidable.
- Computer scientists explore if specific properties are decidable in relation to formal languages, such as determining whether a given string belongs to a language's set of accepted strings. This relates to the design of compilers and programming language implementations, which are often defined by the specifications. If the language can determine those constraints, the constraints are decidable.
- In the context of a game, figuring out if a winning strategy exists for a finite, two-player game with perfect information could be a decidable problem, as an algorithm could theoretically explore all possible game states to determine if a winning path always exists for one of the players. That process must involve the totality of the system.
- A Turing machine can only solve decidable problems. Determining whether a Turing machine will eventually halt on a given input is considered an undecidable problem (the halting problem). This limitation is a core tenet of computation, and the concept of what's decidable dictates what can be processed.
- Decidability contrasts with empirical inquiries. Whether an observed pattern in nature follows a specific underlying law is not necessarily a decidable problem in a strict sense. The scientific process provides evidence, but without complete information, we cannot prove or disprove the law universally through an algorithm.