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:
“It would be disingenuous, however, not to point out that some things are considered as morally certain, that is, as having sufficient certainty for application to ordinary life, even though they may be uncertain in relation to the absolute power of God.”
—René Descartes (1596–1650)
“There is a constant in the average American imagination and taste, for which the past must be preserved and celebrated in full-scale authentic copy; a philosophy of immortality as duplication. It dominates the relation with the self, with the past, not infrequently with the present, always with History and, even, with the European tradition.”
—Umberto Eco (b. 1932)
“The price we pay for the complexity of life is too high. When you think of all the effort you have to put in—telephonic, technological and relational—to alter even the slightest bit of behaviour in this strange world we call social life, you are left pining for the straightforwardness of primitive peoples and their physical work.”
—Jean Baudrillard (b. 1929)
“Mark the babe
Not long accustomed to this breathing world;
One that hath barely learned to shape a smile,
Though yet irrational of soul, to grasp
With tiny finger—to let fall a tear;
And, as the heavy cloud of sleep dissolves,
To stretch his limbs, bemocking, as might seem,
The outward functions of intelligent man.”
—William Wordsworth (1770–1850)