Effective method

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 by any intuitively 'effective' means from a specific class.^[2] An effective method is sometimes also called a mechanical method or procedure.^[3]

Definition

The definition of an effective method involves more than the method itself. In order for a method to be called effective, it must be considered with respect to a class of problems. Because of this, one method may be effective with respect to one class of problems and not be effective with respect to a different class.

A method is formally called effective for a class of problems when it satisfies these criteria:

It consists of a finite number of exact, finite instructions.
When it is applied to a problem from its class:
- It always finishes (terminates) 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.^[4]

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 (general recursive functions, 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.^{[citation needed]}

References

^ Hunter, Geoffrey, Metalogic: An Introduction to the Metatheory of Standard First-Order Logic, University of California Press, 1971
^ Gandy, Robin (1980). "Church's Thesis and the Principles for Mechanisms". The Kleene Symposium. Retrieved 19 April 2024.
^ Copeland, B.J.; Copeland, Jack; Proudfoot, Diane (June 2000). "The Turing-Church Thesis". AlanTuring.net. Turing Archive for the History of Computing. Retrieved 23 March 2013.
^ The Cambridge Dictionary of Philosophy, effective procedure