Computable Function - Definition

Definition

The class of computable functions can be defined in many equivalent models of computation, including

• Turing machines
• μ-recursive functions
• Lambda calculus
• Post machines (Post–Turing machines and tag machines).
• Register machines

Although those models use different representations for the functions, their inputs and their outputs, translations exist between any two models. In the remainder of this article, functions from natural numbers to natural numbers are used (as is the case for, e.g., the μ-recursive functions).

Each computable function f takes a fixed, finite number of natural numbers as arguments. Because the functions are partial in general, they may not be defined for every possible choice of input. If a computable function is defined for a certain input, then it returns a single natural number as output (this output can be interpreted as a list of numbers using a pairing function). These functions are also called partial recursive functions. In computability theory, the domain of a function is taken to be the set of all inputs for which the function is defined.

A function which is defined for all possible arguments is called total. If a computable function is total, it is called a total computable function or total recursive function.

The notation f(x₁, ..., x_k)↓ indicates that the partial function f is defined on arguments x₁, ..., x_k, and the notation f(x₁, ..., x_k) = y indicates that f is defined on the arguments x₁, ..., x_k and the value returned is y. The case that a function f is undefined for arguments x₁, ..., x_k is denoted by f(x₁, ..., x_k)↑ .