# primitive recursive function

**primitive recursive function** A function that can be obtained from certain initial functions by a finite number of applications of composition and primitive recursion. The initial functions are normally the zero function, successor function, and projection (or generalized identity) functions, where all functions are defined on the nonnegative integers. Primitive recursive functions are total functions, defined in a simple way by induction. There is also a notion of *primitive recursive set*, namely one whose characteristic function is primitive recursive.

The arithmetic functions of addition and multiplication are examples of primitive recursive functions. Indeed most of the functions and sets on natural numbers that we wish to compute are primitive recursive.

The idea can be generalized: for example, a primitive recursive function on lists satisfies a definition analogous to the one given above, with the successor function adding an element to the front of a list.

See also recursive function.

# primitive recursion

**primitive recursion** In the study of effective computability, a particular way of defining a new function in terms of other simpler ones. The functions involved are functions over the nonnegative integers. Primitive recursion is then the process of defining a function *f* of *n*+1 variables in the following manner: *f*(*x*_{1},*x*_{2},…*x _{n}*,0) =

*g*(

*x*

_{1},

*x*

_{2},…

*x*),

_{n}*f*(

*x*

_{1},

*x*

_{2},…

*x*,

_{n}*y*+1) =

*h*(

*x*

_{1},

*x*

_{2},…

*x*,

_{n}*y*,

*f*(

*x*

_{1},…

*x*,

_{n}*y*))

where

*g*and

*h*are functions of

*n*and

*n*+2 variables respectively. See also primitive recursive function.

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