Effective method

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title: "Effective method" type: doc version: 1 created: 2026-02-28 author: "Wikipedia contributors" status: active scope: public tags: ["metalogic", "computability-theory", "theory-of-computation"] description: "Problem-solving procedures with certain characteristics" topic_path: "technology/computing" source: "https://en.wikipedia.org/wiki/Effective_method" license: "CC BY-SA 4.0" wikipedia_page_id: 0 wikipedia_revision_id: 0

::summary Problem-solving procedures with certain characteristics ::

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 (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.

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.

References

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

References

1. {{Hunter 1996
2. 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).
3. (1980). "Church's Thesis and the Principles for Mechanisms". The Kleene Symposium.
4. Copeland, B.J.. (June 2000). "The Turing-Church Thesis". Turing Archive for the History of Computing.
5. The Cambridge Dictionary of Philosophy, ''effective procedure''

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