Computability

Computability concerns the existence of an effective procedure, not the speed of the current machine. A decision procedure must handle every permitted instance and provide the required answer. Solving many examples is not enough to establish that one general program exists for all possible inputs within the specified problem.

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Turing machines, lambda calculus, and recursive functions formalize stepwise calculation. Equivalence results among classical models make the analysis independent of one physical machine. The Church–Turing thesis connects those formal capabilities with the informal notion of effective methods; it should not be presented as a theorem about every imaginable physical process.

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A procedure may halt when a positive answer has evidence but search forever when the answer is negative. Such recognition differs from deciding both outcomes in finite time. The distinction prevents an open-ended search for a witness from being mistaken for a complete decision method covering all allowed cases.

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A computable task can still require impractical time or memory as its input grows. That is a question of complexity and implementation. Uncomputability instead rules out the requested general method within the stated model and domain. Faster hardware does not remove such a logical limitation merely by reducing execution time.

References: [2]