Computable Function

Definition

$f:\backslash subseteq\; \backslash mathbb\{N\}\; \backslash to\; \backslash mathbb\{N\}$

• recursive functions
• lambda calculus
• Markov algorithms

• Turing machines
• Post systems
• register machines

Notes

$g:\backslash subseteq\; \backslash mathbb\{N\}^k\; \backslash to\; \backslash mathbb\{N\}$

$f:\backslash subseteq\; \backslash mathbb\{N\}\; \backslash to\; \backslash mathbb\{N\}$

Examples

• constant function f : N^k→ N, f(n₁,...n_k) := n
• addition f : N²→ N, f(n₁,n₂) := n₁ + n₂
• greatest common divisor
• Bzout's identity, a linear Diophantine equation

Properties

• The set of computable functions is countable.
• Given two computable functions f and g then f+g, fg and fog are computable functions.

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