# Effective method > Mediated Wiki article. Canonical URL: https://mediated.wiki/source/Effective_method > Markdown URL: https://mediated.wiki/source/Effective_method.md > Source: https://en.wikipedia.org/wiki/Effective_method > Source revision: 1339806837 > License: Creative Commons Attribution-ShareAlike 4.0 International (https://creativecommons.org/licenses/by-sa/4.0/) Problem-solving procedures with certain characteristics This article needs more citations. Please help improve this article by adding citations to reliable sources. Unsourced material may be challenged and removed. Find sources: "Effective method" – news · newspapers · books · scholar · JSTOR (May 2025) (Learn how and when to remove this message) In [metalogic](/source/Metalogic), [mathematical logic](/source/Mathematical_logic), and [computability theory](/source/Computability_theory), an **effective method**[1] or **effective procedure** is a finite-time, [deterministic procedure](/source/Deterministic_algorithm) for [solving a problem](/source/Problem_solving) from a specific class.[2][3] An effective method is sometimes also called a *mechanical* method or procedure.[4] Functions for which an effective method exists are sometimes called *[effectively calculable](/source/Computable_function)*. ## Definition Formally, a method is called *effective* to a specific class of problems when it satisfies the following criteria: - It consists of a [finite](https://en.wiktionary.org/wiki/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](/source/Rigour) to succeed. In other words, it requires no [ingenuity](/source/Creativity) to succeed.[5] 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](/source/Algorithm)". ## Computable functions Several independent efforts to give a formal characterization of effective calculability led to a variety of proposed definitions ([general recursive functions](/source/General_recursive_function), [Turing machines](/source/Turing_machine), [λ-calculus](/source/%CE%9B-calculus)) that later were shown to be equivalent. The notion captured by these definitions is known as [recursive or effective computability](/source/Computable_function). The [Church–Turing thesis](/source/Church%E2%80%93Turing_thesis) states that the two notions coincide: any [number-theoretic function](/source/Number-theoretic_function) that is effectively calculable is [recursively computable](/source/Computable_function). As this is not a mathematical statement, it cannot be proven by a [mathematical proof](/source/Mathematical_proof).[*[citation needed](https://en.wikipedia.org/wiki/Wikipedia:Citation_needed)*] ## See also - [Decidability (logic)](/source/Decidability_(logic)) - [Decision problem](/source/Decision_problem) - [Effective results in number theory](/source/Effective_results_in_number_theory) - [Function problem](/source/Function_problem) - [Model of computation](/source/Model_of_computation) - [Recursive set](/source/Recursive_set) - [Undecidable problem](/source/Undecidable_problem) ## References 1. **[^](#cite_ref-1)** [Hunter, Geoffrey](/source/Geoffrey_Hunter_(logician)) (1996) [1971]. "**1.7**: The notion of effective method in logic and mathematics". *Metalogic: An Introduction to the Metatheory of Standard First-Order Logic*. University of California Press (published 1973). [ISBN](/source/ISBN_(identifier)) [9780520023567](https://en.wikipedia.org/wiki/Special:BookSources/9780520023567). [OCLC](/source/OCLC_(identifier)) [36312727](https://search.worldcat.org/oclc/36312727). ([accessible to patrons with print disabilities](https://archive.org/details/metalogicintrodu0000hunt)) 1. **[^](#cite_ref-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). 1. **[^](#cite_ref-3)** [Gandy, Robin](/source/Robin_Gandy) (1980). ["Church's Thesis and the Principles for Mechanisms"](https://doi.org/10.1016/S0049-237X(08)71257-6). *The Kleene Symposium*. Studies in Logic and the Foundations of Mathematics. **101**: 123–148. [doi](/source/Doi_(identifier)):[10.1016/S0049-237X(08)71257-6](https://doi.org/10.1016%2FS0049-237X%2808%2971257-6). [ISBN](/source/ISBN_(identifier)) [978-0-444-85345-5](https://en.wikipedia.org/wiki/Special:BookSources/978-0-444-85345-5). Retrieved 19 April 2024. 1. **[^](#cite_ref-alanturingnet_4-0)** [Copeland, B.J.](/source/Jack_Copeland); Copeland, Jack; Proudfoot, Diane (June 2000). ["The Turing-Church Thesis"](http://www.alanturing.net/turing_archive/pages/reference%20articles/The%20Turing-Church%20Thesis.html). *AlanTuring.net*. Turing Archive for the History of Computing. Retrieved 23 March 2013. 1. **[^](#cite_ref-5)** The Cambridge Dictionary of Philosophy, *effective procedure* - [S. C. Kleene](/source/Stephen_Cole_Kleene) (1967), *Mathematical logic*. Reprinted, Dover, 2002, [ISBN](/source/ISBN_(identifier)) [0-486-42533-9](https://en.wikipedia.org/wiki/Special:BookSources/0-486-42533-9), pp. 233 ff., esp. p. 231. v t e Metalogic and metamathematics Cantor's theorem Entscheidungsproblem Church–Turing thesis Consistency Effective method Foundations of mathematics of geometry Gödel's completeness theorem Gödel's incompleteness theorems Soundness Completeness Decidability Interpretation Löwenheim–Skolem theorem Metatheorem Satisfiability Independence Type–token distinction Use–mention distinction --- Adapted from the Wikipedia article [Effective method](https://en.wikipedia.org/wiki/Effective_method) by Wikipedia contributors ([contributor history](https://en.wikipedia.org/wiki/Effective_method?action=history)). Available under [Creative Commons Attribution-ShareAlike 4.0 International](https://creativecommons.org/licenses/by-sa/4.0/). Changes may have been made.