Effectively calculable

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 from a specific class. An effective method is sometimes also called mechanical method or procedure.[2]

A method is called effective for a class of problems iff

  • it consists of a finite number of exact, finite instructions
  • when applied to a problem from its class, it always finishes (terminates) after a finite number of steps
  • when applied to a problem from its class, 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 be followed rigorously to succeed; in particular, it requires no ingenuity to do.[3]

Optionally, one may require that when an effective method is applied to a problem from outside the class for which it is effective, it may halt without result or diverge, but must not return a result as if it were the answer to the problem. Adding this requirement reduces the set of classes for which there is an effective method.

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.

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 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

References

  • S. C. Kleene (1967), Mathematical logic. Reprinted, Dover, 2002, ISBN 0-486-42533-9, pp. 233 ff., esp. p. 231.


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