recursive function


recursive function

      in logic and mathematics, a type of function or expression predicating some concept or property of one or more variables, which is specified by a procedure that yields values or instances of that function by repeatedly applying a given relation or routine operation to known values of the function. The theory of recursive functions was developed by the 20th-century Norwegian Thoralf Albert Skolem, a pioneer in metalogic, as a means of avoiding the so-called paradoxes of the infinite that arise in certain contexts when “all” is applied to functions that range over infinite classes; it does so by specifying the range of a function without any reference to infinite classes of entities.

      Recursion can be intuitively illustrated by taking some familiar concept such as “human”—or the function “x is human.” Instead of defining this concept or function by its qualities and dispositions, one might say: “Adam and Eve are human; and any offspring of theirs is human; and any offspring of offspring . . . of their offspring is human.” Here two values of the function “x is human” are mentioned, and a relationship in which they stand to other entities is given. Through this relationship all things that are values of “x is human” are selected by a back reference, or “recursion,” by many steps, to Adam and Eve.

      This recursiveness in a function or concept is closely related to the procedure known as mathematical induction and is mainly of importance in logic and mathematics. For example, “x is a formula of logical system L,” or “x is a natural number,” is frequently defined recursively. These functions are correlated with purely routine operations that may be repeatedly applied to given formulas or numbers, eventually relating them to certain listed values of the functions—e.g., to “P and Q” as one formula or to zero as one natural number—thus avoiding functions that range over infinite classes with the risk of incurring paradoxes. See decision problem.

* * *


Universalium. 2010.

Look at other dictionaries:

  • Recursive function — may refer to: Recursion (computer science), a procedure or subroutine, implemented in a programming language, whose implementation references itself A total computable function, a function which is defined for all possible inputs See also μ… …   Wikipedia

  • recursive function — noun a) Any function whose value may be obtained using a finite number of operations using a precisely specified algorithm b) Any function that uses recursion and can call itself until a certain condition is met …   Wiktionary

  • Primitive recursive function — The primitive recursive functions are defined using primitive recursion and composition as central operations and are a strict subset of the recursive functions (recursive functions are also known as computable functions). The term was coined by… …   Wikipedia

  • Μ-recursive function — In mathematical logic and computer science, the μ recursive functions are a class of partial functions from natural numbers to natural numbers which are computable in an intuitive sense. In fact, in computability theory it is shown that the μ… …   Wikipedia

  • μ-recursive function — In mathematical logic and computer science, the μ recursive functions are a class of partial functions from natural numbers to natural numbers which are computable in an intuitive sense. In fact, in computability theory it is shown that the μ… …   Wikipedia

  • Non-recursive function — might refer to: Recursion (computer science): a procedure or subroutine, implemented in a programming language, whose implementation references itself μ recursive function, defined from a particular formal model of computable functions using… …   Wikipedia

  • Recursive — may refer to:*Recursion *Recursively enumerable language *Recursively enumerable set *Recursive filter *Recursive function *Recursive language *Recursive acronym *Recursive set *Primitive recursive function …   Wikipedia

  • Recursive set — In computability theory, a set of natural numbers is called recursive, computable or decidable if there is an algorithm which terminates after a finite amount of time and correctly decides whether or not a given number belongs to the set. A more… …   Wikipedia

  • recursive — A procedure that is applied once, and then applied to the result of that application, and so on. A recursive definition (definition by induction) defines the result of some operation for 0, and then the result for any number n + 1 in terms of the …   Philosophy dictionary

  • Function (mathematics) — f(x) redirects here. For the band, see f(x) (band). Graph of example function, In mathematics, a function associates one quantity, the a …   Wikipedia


Share the article and excerpts

Direct link
Do a right-click on the link above
and select “Copy Link”

We are using cookies for the best presentation of our site. Continuing to use this site, you agree with this.