In metalogic, mathematical logic, and computability theory, an effective method or effective procedure is a finite-time, deterministic procedure for solving a problem from a specific class. An effective method is sometimes also called a mechanical method or procedure. Functions for which an effective method exists are sometimes called 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 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 to succeed. In other words, it requires no ingenuity to succeed.

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 functions, Turing machines, λ-calculus) that later were shown to be equivalent. The notion captured by these definitions is known as recursive or effective computability.

The Church–Turing thesis states that the two notions coincide: any number-theoretic function that is effectively calculable is recursively computable.

See also

• Decidability (logic)

• Decision problem

• Effective results in number theory

• Function problem

• Model of computation

• Recursive set

• Undecidable problem

References

• S. C. Kleene (1967), Mathematical logic. Reprinted, Dover, 2002, , pp.233 ff., esp. p.231.