Definition
First we introduce an infinite set of functions, denoted Ei for some natural number i. We define and . I.e., E0 is the addition function, and E1 is a unary function which squares its argument and adds two. Then, for each n greater than 1, we define .
From these functions we define the Grzegorczyk hierarchy., the n-th set in the hierarchy, contains the following functions:
- Ek for k < n
- the zero function (Z(x) = 0);
- the successor function (S(x) = x + 1);
- the projection functions ;
- the (generalized) compositions of functions in the set (if h, g1, g2, ... and gm are in, then is as well); and
- the results of limited (primitive) recursion applied to functions in the set, (if g, h and j are in and for all t and, and further and, then f is in as well)
In other words, is the closure of set with respect to function composition and limited recursion (as defined above).
Read more about this topic: Grzegorczyk Hierarchy
Famous quotes containing the word definition:
“It’s a rare parent who can see his or her child clearly and objectively. At a school board meeting I attended . . . the only definition of a gifted child on which everyone in the audience could agree was “mine.””
—Jane Adams (20th century)
“The physicians say, they are not materialists; but they are:MSpirit is matter reduced to an extreme thinness: O so thin!—But the definition of spiritual should be, that which is its own evidence. What notions do they attach to love! what to religion! One would not willingly pronounce these words in their hearing, and give them the occasion to profane them.”
—Ralph Waldo Emerson (1803–1882)
“The definition of good prose is proper words in their proper places; of good verse, the most proper words in their proper places. The propriety is in either case relative. The words in prose ought to express the intended meaning, and no more; if they attract attention to themselves, it is, in general, a fault.”
—Samuel Taylor Coleridge (1772–1834)