In logic and mathematics – especially metalogic and computability theory – an effective method (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.
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 ingenuity from any person or machine executing it.
Famous quotes containing the words effective and/or method:
“As in political revolutions, so in paradigm choice—there is no standard higher than the assent of the relevant community. To discover how scientific revolutions are effected, we shall therefore have to examine not only the impact of nature and of logic, but also the techniques of persuasive argumentation effective within the quite special groups that constitute the community of scientists.”
—Thomas S. Kuhn (b. 1922)
“In child rearing it would unquestionably be easier if a child were to do something because we say so. The authoritarian method does expedite things, but it does not produce independent functioning. If a child has not mastered the underlying principles of human interactions and merely conforms out of coercion or conditioning, he has no tools to use, no resources to apply in the next situation that confronts him.”
—Elaine Heffner (20th century)