Effective method

In metalogic, mathematical logic, and computability theory, an effective method or effective procedure is a finite-time, deterministic procedure for solving a problem from a specific class. An effective method is sometimes also called a mechanical method or procedure.

Definition

Formally, a method is called effective to a specific class of problems when it satisfies the following criteria:

It consists of a finite number of exact, finite instructions.
When it is applied to a problem from its class:
* It always finishes after a finite number of steps.
* It always produces a correct answer.
In principle, it can be done by a human without any aids except writing materials.
Its instructions need only to be followed rigorously to succeed. In other words, it requires no ingenuity to succeed.

Optionally, it may also be required that the method never returns a result as if it were an answer when the method is applied to a problem from outside its class. Adding this requirement reduces the set of classes for which there is an effective method.

Algorithms

An effective method for calculating the values of a function is an algorithm. Functions for which an effective method exists are sometimes called effectively calculable.

Computable functions

Several independent efforts to give a formal characterization of effective calculability led to a variety of proposed definitions that later were shown to be equivalent. The notion captured by these definitions is known as recursive or effective computability.
The Church–Turing thesis states that the two notions coincide: any number-theoretic function that is effectively calculable is recursively computable. As this is not a mathematical statement, it cannot be proven by a mathematical proof.