Effective method
In logic, mathematics and computer science, especially metalogic and computability theory, an effective method[1] or effective procedure is a procedure for solving a problem by any intuitively 'effective' means from a specific class.[2] An effective method is sometimes also called a mechanical method or procedure.[3]
Definition
The definition of an effective method involves more than the method itself. In order for a method to be called effective, it must be considered with respect to a class of problems. Because of this, one method may be effective with respect to one class of problems and not be effective with respect to a different class.
A method is formally called effective for a class of problems when it satisfies these 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.[4]
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 an algorithm. Functions for which an effective method exists are sometimes called effectively calculable.
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. As this is not a mathematical statement, it cannot be proven by a mathematical proof.
See also
- Decidability (logic)
- Decision problem
- Function problem
- Effective results in number theory
- Recursive set
- Undecidable problem
References
- ↑ Hunter, Geoffrey, Metalogic: An Introduction to the Metatheory of Standard First-Order Logic, University of California Press, 1971
- ↑ Gandy, Robin (1980). Church's Thesis and the Principles for Mechanisms.
- ↑ Copeland, B.J.; Copeland, Jack (June 2000). "The Turing-Church Thesis". AlanTuring.net. Turing Archive for the History of Computing. http://www.alanturing.net/turing_archive/pages/reference%20articles/The%20Turing-Church%20Thesis.html.
- ↑ 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.
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