Effective method
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In metalogic, mathematical logic, and computability theory, an effective method[1] or effective procedure is a finite-time, deterministic procedure for solving a problem from a specific class.[2][3] An effective method is sometimes also called a mechanical method or procedure.[4] 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.[5]
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. As this is not a mathematical statement, it cannot be proven by a mathematical proof.[citation needed]
See also
- Decidability (logic)
- Decision problem
- Effective results in number theory
- Function problem
- Model of computation
- Recursive set
- Undecidable problem
References
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- ^ Template:Hunter 1996
- ^ Whether or not a process with random interior processes (not including the input) is an algorithm is debatable. Rogers opines that: "a computation is carried out in a discrete stepwise fashion, without the use of continuous methods or analog devices ... carried forward deterministically, without resort to random methods or devices, e.g., dice" (Rogers 1987:2).
- ^ Page Module:Citation/CS1/styles.css has no content.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.
- ^ Page Module:Citation/CS1/styles.css has no content.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, Template:ISBN, pp. 233 ff., esp. p. 231.
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