In computability theory, a primitive recursive function is, roughly speaking, a function that can be computed by a computer program whose loops are all...

mathematics, primitive recursive set functions or primitive recursive ordinal functions are analogs of primitive recursive functions, defined for sets or ordinals...

recursive functions. However, not every total recursive function is a primitive recursive function—the most famous example is the Ackermann function....

recursive. All primitive recursive functions are total and computable, but the Ackermann function illustrates that not all total computable functions...

a function from a set X to a set Y assigns to each element of X exactly one element of Y. The set X is called the domain of the function and the set Y...

= 2 and S(2) = 3. The successor function is one of the basic components used to build a primitive recursive function. Successor operations are also known...

elementary recursive function, also called an elementary function, or a Kalmár elementary function, is a restricted form of a primitive recursive function, allowing...

the function can be chosen to be injective. The set S is the range of a primitive recursive function or empty. Even if S is infinite, repetition of values...

In mathematical logic, the primitive recursive functionals are a generalization of primitive recursive functions into higher type theory. They consist...

functions. Another example is the Ackermann function, which is recursively defined but not primitive recursive. For definitions of this type to avoid circularity...

and recursive rule, one can generate the set of all natural numbers. Other recursively defined mathematical objects include factorials, functions (e.g...

the set S {\displaystyle S} is computable if and only if the indicator function 1 S {\displaystyle \mathbb {1} _{S}} is computable. Every recursive language...

property. Adding the μ-operator to the primitive recursive functions makes it possible to define all computable functions. Suppose that R(y, x1, ..., xk) is...

smaller instances of the same problem. Recursion solves such recursive problems by using functions that call themselves from within their own code. The approach...

surjective function (also known as surjection, or onto function /ˈɒn.tuː/) is a function f such that, for every element y of the function's codomain, there...

Zermelo–Fraenkel set theory with the axiom of choice excluded. Informally, Zermelo–Fraenkel set theory is intended to formalize a single primitive notion, that...

higher-order functions that analyze a recursive data structure and through use of a given combining operation, recombine the results of recursively processing...

homotopy type theory, a set may be regarded as a homotopy 0-type, with universal properties of sets arising from the inductive and recursive properties of higher...

the primitive recursive functions. The language is derived from the counter-machine model. Like the counter machines the LOOP language comprises a set of...

In mathematics, the domain of a function is the set of inputs accepted by the function. It is sometimes denoted by dom ⁡ ( f ) {\displaystyle \operatorname...

In mathematics, an injective function (also known as injection, or one-to-one function ) is a function f that maps distinct elements of its domain to...

Hall. ISBN 0-13-181629-2. Rosen, Kenneth (2007). "Basic Structures: Sets, Functions, Sequences, and Sums". Discrete Mathematics and Its Applications (Sixth ed...

existence of any primitive recursive function in x × ω → y {\displaystyle x\times \omega \to y} , and in particular in the uncountable function spaces out of...

functions used in computability theory. Every function in the Grzegorczyk hierarchy is a primitive recursive function, and every primitive recursive function...

Primitive recursive arithmetic (PRA) is a quantifier-free formalization of the natural numbers. It was first proposed by Norwegian mathematician Skolem...

dictionary. Course-of-values recursion Recursion (computer science) Primitive recursive function Inline expansion Leaf subroutine Corecursion Like this: if (ls...

In set theory, the complement of a set A, often denoted by A c {\displaystyle A^{c}} (or A′), is the set of elements not in A. When all elements in the...

example the μ-recursive functions obtained from primitive recursion and the μ operator. The terminology for computable functions and sets is not completely...

variables, functions, or even other sets. A set may also be called a collection or family, especially when its elements are themselves sets; this may avoid...

of valid strings of an alphabet. A set of strings which can be enumerated in this manner is called a recursively enumerable language. The Turing machine...