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:
“Whoever has a keen eye for profits, is blind in relation to his craft.”
—Sophocles (497–406/5 B.C.)
“Only in a house where one has learnt to be lonely does one have this solicitude for things. One’s relation to them, the daily seeing or touching, begins to become love, and to lay one open to pain.”
—Elizabeth Bowen (1899–1973)
“That primitive head
So ambitiously vast,
Yet so rude in its art,
Is as easily read
For the woes of the past
As a clinical chart.”
—Robert Frost (1874–1963)
“When Western people train the mind, the focus is generally on the left hemisphere of the cortex, which is the portion of the brain that is concerned with words and numbers. We enhance the logical, bounded, linear functions of the mind. In the East, exercises of this sort are for the purpose of getting in tune with the unconscious—to get rid of boundaries, not to create them.”
—Edward T. Hall (b. 1914)