A recursive definition for the Bézier curve of degree n expresses it as a point-to-point linear combination of a pair of corresponding points in two Bézier curves of degree n − 1.

It is this step that makes the definition recursive.

However the definition expressed by this formula is not recursive.

This comes in contrast with the direct meaning of the notion of semantic consequence, that quantifies over all structures in a particular language, which is clearly not a recursive definition.

The entries of an ordered pair can be other ordered pairs, enabling the recursive definition of ordered n-tuples ( ordered lists of n objects ).

:" Now if Q ( x ) is a partial recursive predicate, there is a decision procedure for Q ( x ) on its range of definition, so the law of the excluded middle or excluded " third " ( saying that, Q ( x ) is either t or f ) applies intuitionistically on the range of definition.

The μ-recursive functions are closely related to primitive recursive functions, and their inductive definition ( below ) builds upon that of the primitive recursive functions.

Von Mises never totally formalized his definition of a proper selection rule for sub-sequences, but in 1940 Alonzo Church defined it as any recursive function which having read the first N elements of the sequence decides if it wants to select element number N + 1.

In their view Church's recursive function definition was too restrictive in that it read the elements in order.

The second, recursive part of the definition represents an ordered pair so that s-exprs are effectively binary trees.

The recursive case of the s-expr definition is traditionally implemented using cons cells.

However, jokes also can have an element of misunderstanding: This parody is the shortest possible example of an erroneous recursive definition of an object, the error being the absence of the termination condition ( or lack of the initial state, if looked at from an opposite point of view ).

This is done by defining a sequence of value functions V < sub > 1 </ sub >, V < sub > 2 </ sub >, ..., V < sub > n </ sub >, with an argument y representing the state of the system at times i from 1 to n. The definition of V < sub > n </ sub >( y ) is the value obtained in state y at the last time n. The values V < sub > i </ sub > at earlier times i = n − 1, n − 2, ..., 2, 1 can be found by working backwards, using a recursive relationship called the Bellman equation.

While the aforementioned recursive definition is correct, it is often tedious to remember and implement.

In programming languages that support anonymous functions, fixed-point combinators allow the definition and use of anonymous recursive functions, i. e. without having to bind such functions to identifiers.

Suppose that g and h are total computable functions that are used in a recursive definition for a function f:

This recursive definition can be converted into a computable function F ( e ) that takes an index for a program e and returns an index F ( e ) such that

The definition in terms of μ-recursive functions as well as a different definition of rekursiv functions by Gödel led to the traditional name recursive for sets and functions computable by a Turing machine.

These models enable some nonrecursive sets of numbers or languages ( including all recursively enumerable sets of languages ) to be " learned in the limit "; whereas, by definition, only recursive sets of numbers or languages could be identified by a Turing machine.

This value is necessary to be consistent with the recursive definition of what a product over a sequence ( or set, given commutativity ) means.

For this reason the recursive descent parser is sometimes prefered over the LALR parser.

The study of which mathematical constructions can be effectively performed is sometimes called recursive mathematics ; the Handbook of Recursive Mathematics ( Ershov et al.

In other cases it is known that any bound must be extraordinarily large, sometimes even larger than any primitive recursive function ; see the Paris-Harrington theorem for an example.

The pair ( q, f ) is sometimes called the recursion data for u, given in the form of a recursive definition:

Unlike those other two codes, however, Elias omega recursively encodes that prefix ; thus, they are sometimes known as recursive Elias codes.

In mathematics, particularly theoretical computer science and mathematical logic, the computable numbers, also known as the recursive numbers or the computable reals, are the real numbers that can be computed to within any desired precision by a finite, terminating algorithm.

* Cascading Style Sheets, also colloquially referred to as a recursive nomenclature CSS Style Sheets, is a language used to describe the style of document presentations in web development

Typically, such caching DNS servers, also called DNS caches, also implement the recursive algorithm necessary to resolve a given name starting with the DNS root through to the authoritative name servers of the queried domain.

Zone file parsing, caching, and recursive resolving are also implemented as separate programs.

Forth also allowed recursive programming, if desired.

Linear search can also be described as a recursive algorithm:

Pascal has its roots in the ALGOL 60 language, but also introduced concepts and mechanisms which ( on top of ALGOL's scalars and arrays ) enabled programmers to define their own complex ( structured ) datatypes, and also made it easier to build dynamic and recursive data structures such as lists, trees and graphs.

The primitive recursive functions are defined using primitive recursion and composition as central operations and are a strict subset of the total µ-recursive functions ( µ-recursive functions are also called partial recursive ).

Rice's theorem states that the decision problem is decidable ( also called recursive or computable ) if and only if or.

It also supports global and local variables, which permits recursive functions and subroutines to be written.

Act II opens with Vladimir singing a recursive round about a dog, which could illustrate the cyclical nature of the play's universe, and also point toward the play's debt to the carnivalesque, music hall traditions, and vaudeville comedy ( this is only one of a number of canine references and allusions in the play ).

Caching name servers are often also recursive name servers — they perform every step necessary to answer any DNS query they receive.

If L is recursively enumerable, then the complement of L is recursively enumerable if and only if L is also recursive.

It is also possible to read " G < sub > T </ sub > is true " in the formal sense that primitive recursive arithmetic proves the implication Con ( T )→ G < sub > T </ sub >, where Con ( T ) is a canonical sentence asserting the consistency of T ( Smoryński 1977 p. 840, Kikuchi and Tanaka 1994 p. 403 )</ ref > but not provable in the theory ( Kleene 1967, p. 250 ).

This condition is also sufficient ; we may compute the coefficients of the inverse series B via the explicit recursive formula

Yudkowsky's research focuses on Artificial Intelligence theory for self-understanding, self-modification, and recursive self-improvement ( seed AI ); and also on artificial-intelligence architectures and decision theories for stably benevolent motivational structures ( Friendly AI, and Coherent Extrapolated Volition in particular ).

It was the development of computability theory ( also known as recursion theory ) that provided a precise explication of the intutitive notion of algorithmic computability, thus making the notion of recursive enumerability perfectly rigorous.

Similarly, an identifier bound to a recursive function is also technically a free variable within its own body but is treated specially.

The method used in this proof can also be used to prove a cut elimination result for Peano arithmetic in a stronger logic than first-order logic, but the consistency proof itself can be carried out in ordinary first-order logic using the axioms of primitive recursive arithmetic and a transfinite induction principle.

A set of natural numbers is said to be a computable set ( also called a decidable, recursive, or Turing computable set ) if there is a Turing machine that, given a number n, halts with output 1 if n is in the set and halts with output 0 if n is not in the set.