In logic and mathematics – especially metalogic and computability theory – an effective method (also called an effective procedure) is a procedure which takes some class of problems and reduces the solution to a set of steps which:
An effective method for calculating the values of a function is an algorithm; functions with an effective method are sometimes called effectively calculable.
Several independent efforts to give a formal characterization of effective calculability led to a variety of proposed definitions (general recursion, Turing machines, λ-calculus) that later were shown to be equivalent; the notion captured by these definitions is known as (recursive) computability.
The Church–Turing thesis states that the two notions coincide: any number-theoretic function that is effectively calculable is recursively computable. This is not a mathematical statement and cannot be proven by a mathematical proof.
Optionally, one may require that when an effective method is applied to a problem from outside the class for which it is effective, it may halt without result or continue forever without halting, but must not return a result as if it were the answer to the problem (c.f. divergence).
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