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Before the precise definition of computable function mathematician often used the informal term effectively computable.

1 Definition

Generally a computable function is a partial function

The class of computable functions is equivalent to the class of functions defined by

• recursive functions
• lambda calculus
• Markov algorithms

Alternatively they can be defined as those algorithms that can be calculated by

• Turing machines
• Post systems
• register machines

2 Notes

Sometimes, for reasons of clarity, we write a computable function as

We can easily encode g into a new function

using a pairing function.

3 Examples

• constant function f : N^k→ N, f(n₁,...n_k) := n
• addition f : N²→ N, f(n₁,n₂) := n₁ + n₂
• greatest common divisor
• Bézout's identityIn number theory, Bezout's identity named after Etienne Bezout, is a linear diophantine equation. It states that if a and b are integers with greatest common divisor d then there exist integers x and y such that ax + by d''. Numbers x and y as above can b, a linear Diophantine equationIn mathematics, a Diophantine equation is an equation between two polynomials with integer coefficients with any number of unknowns. A Diophantine problem is given as a Diophantine equation, whose solutions are the possible assignments of integers for the

4 Properties

• The setThis article is about sets in mathematics. For other meanings, see Set (disambiguation). Sets are one of the most important and fundamental concepts in modern mathematics. Basic set theory, having only been invented at the end of the 19th century, is now of computable functions is countable.
• Given two computable functions f and g then f+g, fg and fog are computable functions.