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...

= 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...

the class of elementary recursive functions ("Kalmár elementary functions") as a subset of the primitive recursive functions — specifically, those that...

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...

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

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

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...

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...

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...

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

trick) states that any recursively enumerable set of well-formed formulas of a first-order language is (primitively) recursively axiomatizable. This result...

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

mutually recursive functions are primitive recursive, which can be proven by course-of-values recursion, building a single function F that lists the values...

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...

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

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

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...

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...

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

computable in the limit, limit recursive and recursively approximable are also used. One can think of limit computable functions as those admitting an eventually...

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

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...

reverse mathematics (Simpson 2009). Elementary recursive arithmetic (ERA) is a subsystem of primitive recursive arithmetic (PRA) in which recursion is restricted...

can express all computable functions. For example, it can express the Ackermann function, which (not being primitive recursive) cannot be written in BlooP...

science, the reentrant mutex (also known as a recursive mutex or recursive lock) is a synchronization primitive that may be locked multiple times by the same...