Effective method

In logic and mathematics – especially metalogic and computability theory – an effective method^[1] (also called an effective procedure) is a procedure which takes some class of problems and reduces the solution to a set of steps which:

always give some answer rather than ever give no answer;
always give the right answer and never give a wrong answer;
always be completed in a finite number of steps, rather than in an infinite number;
work for all instances of problems of the class.

An effective method for calculating the values of a function is an algorithm; functions with an effective method are sometimes called effectively calculable.

Several independent efforts to give a formal characterization of effective calculability led to a variety of proposed definitions (general recursion, Turing machines, λ-calculus) that later were shown to be equivalent; the notion captured by these definitions is known as (recursive) computability.

The Church–Turing thesis states that the two notions coincide: any number-theoretic function that is effectively calculable is recursively computable. This is not a mathematical statement and cannot be proven by a mathematical proof.

Optionally, one may require that when an effective method is applied to a problem from outside the class for which it is effective, it may halt without result or continue forever without halting, but must not return a result as if it were the answer to the problem (c.f. divergence).

An essential feature of an effective method is that it does not require any ingenuity from any person or machine executing it.^[2]

References [edit]

^ Hunter, Geoffrey, Metalogic: An Introduction to the Metatheory of Standard First-Order Logic, University of California Press, 1971
^ The Cambridge Dictionary of Philosophy, effective procedure

S. C. Kleene (1967), Mathematical logic. Reprinted, Dover, 2002, ISBN 0-486-42533-9, pp. 233 ff., esp. p. 231.
Copeland, B.J. (2000), "The Turing-Church Thesis", AlanTuring.net (Turing Archive for the History of Computing), retrieved March 23, 2013

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