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
[edit]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
[edit]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
[edit]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.[citation needed]
See also
[edit]- Decidability (logic)
- Decision problem
- Function problem
- Effective results in number theory
- Recursive set
- Undecidable problem
References
[edit]- ^ Hunter, Geoffrey (1996) [1971]. "1.7: The notion of effective method in logic and mathematics". Metalogic: An Introduction to the Metatheory of Standard First-Order Logic. University of California Press (published 1973). ISBN 9780520023567. OCLC 36312727. (accessible to patrons with print disabilities)
- ^ Gandy, Robin (1980). "Church's Thesis and the Principles for Mechanisms". The Kleene Symposium. Studies in Logic and the Foundations of Mathematics. 101: 123–148. doi:10.1016/S0049-237X(08)71257-6. ISBN 978-0-444-85345-5. Retrieved 19 April 2024.
- ^ Copeland, B.J.; Copeland, Jack; Proudfoot, Diane (June 2000). "The Turing-Church Thesis". AlanTuring.net. Turing Archive for the History of Computing. Retrieved 23 March 2013.
- ^ 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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