- Effective method
An effective method (also called an effective procedure) for a class of problems is a method for which each step in the method may be described as a mechanical operation and which, if followed rigorously, and as far as may be necessary, is bound to:
*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. Church's thesisstates that the two notions coincide: any number-theoretic functionthat is effectively calculable is recursively computable. Church's thesis is not a mathematical statement and cannot be proved by a mathematical proof.
A further elucidation of the term "effective method" may include the requirement that, when given a problem from outside the class for which the method is effective, the method may halt or loop forever without halting, but must not return a result as if it were the answer to the problem.
An essential feature of an effective method is that it does not require any
ingenuityfrom any person or machine executing it. [The Cambridge Dictionary of Philosophy, "effective procedure"]
Effective results in number theory
* S. C. Kleene (1967), "Mathematical logic". Reprinted, Dover, 2002, ISBN 0-486-42533-9, pp. 233 ff., esp. p. 231.
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