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

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

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

function, a computable partial function from natural numbers to natural numbers Primitive recursive function, a function which can be computed with loops...

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...

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

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

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

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

computation of a value of a function requires only the previous value; for example, for a 1-ary primitive recursive function g the value of g(n+1) is computed...

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

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

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

= 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...

construct such as a recursive function call, it is no longer capable of full μ-recursion, but only primitive recursion. Ackermann's function is the canonical...

arithmetically definable functions is closed under primitive recursion, and therefore includes all primitive recursive functions. The β function was introduced...

that allow the use of primitive recursive functions, as now the quantifiers may be bounded by any primitive recursive function of the arguments. The Σ...

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

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

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 may be...

indirection – and thereby compute the sub-class of primitive recursive functions – by using a primitive recursive "operator" called "definition by cases" (defined...

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

LOOP is a simple register language that precisely captures the primitive recursive functions. The language is derived from the counter-machine model. Like...

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

of recursive functions by use of the IF-THEN-ELSE construction common to computer science, together with four of the operators of primitive recursive functions:...

of a function is a value provided to obtain the function's result. It is also called an independent variable. For example, the binary function f ( x...

differentiating. Prime number Infinitude of the prime numbers Primitive recursive function Principle of bivalence no propositions are neither true nor false...

\left|2^{S}\right|=2^{n}=\sum _{k=0}^{n}{\binom {n}{k}}} If S is a finite set, then a recursive definition of P(S) proceeds as follows: If S = {}, then P(S) = { {} }...

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

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