Existence of Fixed-point Combinators
In certain mathematical formalizations of computation, such as the untyped lambda calculus and combinatory logic, every expression can be considered a higher-order function. In these formalizations, the existence of a fixed-point combinator means that every function has at least one fixed point; a function may have more than one distinct fixed point.
In some other systems, for example in the simply typed lambda calculus, a well-typed fixed-point combinator cannot be written. In those systems any support for recursion must be explicitly added to the language. In still others, such as the simply typed lambda calculus extended with recursive types, fixed-point operators can be written, but the type of a "useful" fixed-point operator (one whose application always returns) may be restricted. In polymorphic lambda calculus (System F) a polymorphic fixed-point combinator has type ∀a.(a→a)→a, where a is a type variable.
Read more about this topic: Fixed-point Combinator
Famous quotes containing the words existence of and/or existence:
“We are compelled by the theory of God’s already achieved perfection to make Him a devil as well as a god, because of the existence of evil. The god of love, if omnipotent and omniscient, must be the god of cancer and epilepsy as well.... Whoever admits that anything living is evil must either believe that God is malignantly capable of creating evil, or else believe that God has made many mistakes in His attempts to make a perfect being.”
—George Bernard Shaw (1856–1950)
“Sport in the sense of a mass-spectacle, with death to add to the underlying excitement, comes into existence when a population has been drilled and regimented and depressed to such an extent that it needs at least a vicarious participation in difficult feats of strength or skill or heroism in order to sustain its waning life-sense.”
—Lewis Mumford (1895–1990)