{{Short description|Problem-solving procedures with certain characteristics}} {{more references|date=May 2025}}

In [[metalogic]], [[mathematical logic]], and [[computability theory]], an '''effective method'''<ref>{{Hunter 1996|section='''1.7''': The notion of effective method in logic and mathematics}}</ref> or '''effective procedure''' is a finite-time, [[Deterministic algorithm|deterministic procedure]] for [[Problem solving|solving a problem]] from a specific class.<ref>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).</ref><ref>{{cite journal |last1=Gandy |first1=Robin |authorlink = Robin Gandy|title=Church's Thesis and the Principles for Mechanisms |journal=The Kleene Symposium |series=Studies in Logic and the Foundations of Mathematics |date=1980 |volume=101 |pages=123–148 |doi=10.1016/S0049-237X(08)71257-6 |isbn=978-0-444-85345-5 | url=https://doi.org/10.1016/S0049-237X(08)71257-6 |access-date=19 April 2024|url-access=subscription }}</ref> An effective method is sometimes also called a ''mechanical'' method or procedure.<ref name=alanturingnet>{{cite web|last=Copeland|first=B.J.|title=The Turing-Church Thesis|url=http://www.alanturing.net/turing_archive/pages/reference%20articles/The%20Turing-Church%20Thesis.html|work=AlanTuring.net|publisher=Turing Archive for the History of Computing|date=June 2000|access-date=23 March 2013|author-link=Jack Copeland |author2=Copeland, Jack |author3=Proudfoot, Diane}}</ref> Functions for which an effective method exists are sometimes called ''[[Computable function|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 [[wikt:finite|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 [[Rigour|rigorously]] to succeed. In other words, it requires no [[Creativity|ingenuity]] to succeed.<ref>The Cambridge Dictionary of Philosophy, ''effective procedure''</ref>

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 function]]s, [[Turing machine]]s, [[λ-calculus]]) that later were shown to be equivalent. The notion captured by these definitions is known as [[Computable function|recursive or effective computability]].

The [[Church–Turing thesis]] states that the two notions coincide: any [[number-theoretic function]] that is effectively calculable is [[computable function|recursively computable]]. As this is not a mathematical statement, it cannot be proven by a [[mathematical proof]].{{cn|date=February 2024}}

==See also== *[[Decidability (logic)]] *[[Decision problem]] *[[Effective results in number theory]] *[[Function problem]] *[[Model of computation]] *[[Recursive set]] *[[Undecidable problem]]

==References == {{reflist}}

* [[Stephen Cole Kleene|S. C. Kleene]] (1967), ''Mathematical logic''. Reprinted, Dover, 2002, {{ISBN|0-486-42533-9}}, pp.&nbsp;233 ff., esp. p.&nbsp;231.

{{Metalogic}}

[[Category:Metalogic]] [[Category:Computability theory]] [[Category:Theory of computation]]