# Effective method

(Redirected from Effective procedure)

In logic, mathematics and computer science, especially metalogic and computability theory, an effective method[1] or effective procedure is a procedure for solving a problem from a specific class. An effective method is sometimes also called mechanical method or procedure.[2]

## Definition

A method is called effective for a class of problems iff

• it consists of a finite number of exact, finite instructions
• when applied to a problem from its class, it always finishes (terminates) after a finite number of steps
• when applied to a problem from its class, 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 be followed rigorously to succeed; in particular, it requires no ingenuity to do so.[3]

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 diverge, but must not return a result as if it were the answer to the problem. 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 (general recursion, Turing machines, λ-calculus) 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.