Relation To Primitive Recursive Functions
The definition of is the same as that of the primitive recursive functions, RP, except that recursion is limited ( for some j in ) and the functions are explicitly included in . Thus the Grzegorczyk hierarchy can be seen as a way to limit the power of primitive recursion to different levels.
It is clear from this fact that all functions in any level of the Grzegorczyk hierarchy are primitive recursive functions (i.e. ) and thus:
It can also be shown that all primitive recursive functions are in some level of the hierarchy (Rose 1984; Gakwaya 1997), thus
and the sets partition the set of primitive recursive functions, RP.
Read more about this topic: Grzegorczyk Hierarchy
Famous quotes containing the words relation to, relation, primitive and/or functions:
“Much poetry seems to be aware of its situation in time and of its relation to the metronome, the clock, and the calendar. ... The season or month is there to be felt; the day is there to be seized. Poems beginning “When” are much more numerous than those beginning “Where” of “If.” As the meter is running, the recurrent message tapped out by the passing of measured time is mortality.”
—William Harmon (b. 1938)
“There is a certain standard of grace and beauty which consists in a certain relation between our nature, such as it is, weak or strong, and the thing which pleases us. Whatever is formed according to this standard pleases us, be it house, song, discourse, verse, prose, woman, birds, rivers, trees, room, dress, and so on. Whatever is not made according to this standard displeases those who have good taste.”
—Blaise Pascal (1623–1662)
“It was a purely wild and primitive American sound, as much as the barking of a chickaree, and I could not understand a syllable of it.”
—Henry David Thoreau (1817–1862)
“If photography is allowed to stand in for art in some of its functions it will soon supplant or corrupt it completely thanks to the natural support it will find in the stupidity of the multitude. It must return to its real task, which is to be the servant of the sciences and the arts, but the very humble servant, like printing and shorthand which have neither created nor supplanted literature.”
—Charles Baudelaire (1821–1867)