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:
“Beauty, like all other qualities presented to human experience, is relative; and the definition of it becomes unmeaning and useless in proportion to its abstractness. To define beauty not in the most abstract, but in the most concrete terms possible, not to find a universal formula for it, but the formula which expresses most adequately this or that special manifestation of it, is the aim of the true student of aesthetics.”
—Walter Pater (1839–1894)
“The very definition of the real becomes: that of which it is possible to give an equivalent reproduction.... The real is not only what can be reproduced, but that which is always already reproduced. The hyperreal.”
—Jean Baudrillard (b. 1929)
“The man who knows governments most completely is he who troubles himself least about a definition which shall give their essence. Enjoying an intimate acquaintance with all their particularities in turn, he would naturally regard an abstract conception in which these were unified as a thing more misleading than enlightening.”
—William James (1842–1910)