Effective method explained

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.^{[1] [2]} An effective method is sometimes also called a mechanical method or procedure.^[3] Functions for which an effective method exists are sometimes called effectively calculable.

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 (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 called "an algorithm".

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.

See also

• Decidability (logic)
• Decision problem
• Effective results in number theory
• Function problem
• Model of computation
• Recursive set
• Undecidable problem

References

• S. C. Kleene (1967), Mathematical logic. Reprinted, Dover, 2002,, pp. 233 ff., esp. p. 231.

Notes and References

1. Whether or not a process with random interior processes (not including the input) is an algorithm is debatable. Rogers opines that: "a computation is carried out in a discrete stepwise fashion, without the use of continuous methods or analog devices ... carried forward deterministically, without resort to random methods or devices, e.g., dice" (Rogers 1987:2).
2. Gandy . Robin . Robin Gandy. Church's Thesis and the Principles for Mechanisms . The Kleene Symposium . Studies in Logic and the Foundations of Mathematics . 1980 . 101 . 123–148 . 10.1016/S0049-237X(08)71257-6 . 978-0-444-85345-5 . 19 April 2024. subscription .
3. Web site: Copeland. B.J.. The Turing-Church Thesis. AlanTuring.net. Turing Archive for the History of Computing. June 2000. 23 March 2013. Jack Copeland . Copeland, Jack . Proudfoot, Diane.
4. The Cambridge Dictionary of Philosophy, effective procedure