The Grzegorczyk hierarchy (pronounced: [ɡʐɛˈɡɔrt͡ʂɨk]), named after the Polish logician Andrzej Grzegorczyk, is a hierarchy of functions used in computability theory (Wagner and Wechsung 1986:43). Every function in the Grzegorczyk hierarchy is a primitive recursive function, and every primitive recursive function appears in the hierarchy at some level. The hierarchy deals with the rate at which the values of the functions grow; intuitively, functions in lower level of the hierarchy grow slower than functions in the higher levels.
Contents
Definition
First we introduce an infinite set of functions, denoted Ei for some natural number i. We define
From these functions we define the Grzegorczyk hierarchy.
- Ek for k < n
- the zero function (Z(x) = 0);
- the successor function (S(x) = x + 1);
- the projection functions (
p i m ( t 1 , t 2 , … , t m ) = t i - the (generalized) compositions of functions in the set (if h, g1, g2, ... and gm are in
E n f ( u ¯ ) = h ( g 1 ( u ¯ ) , g 2 ( u ¯ ) , … , g m ( u ¯ ) ) is as well); and - the results of limited (primitive) recursion applied to functions in the set, (if g, h and j are in
E n f ( t , u ¯ ) ≤ j ( t , u ¯ ) for all t andu ¯ f ( 0 , u ¯ ) = g ( u ¯ ) andf ( t + 1 , u ¯ ) = h ( t , u ¯ , f ( t , u ¯ ) ) , then f is inE n
In other words,
Properties
These sets clearly form the hierarchy
because they are closures over the
They are strict subsets (Rose 1984; Gakwaya 1997). In other words
because the hyper operation
Notably, both the function
Relation to primitive recursive functions
The definition of
It is clear from this fact that all functions in any level of the Grzegorczyk hierarchy are primitive recursive functions (i.e.
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
Extensions
The Grzegorczyk hierarchy can be extended to transfinite ordinals. Such extensions define a fast-growing hierarchy. To do this, the generating functions
The original extension was due to Martin Löb and Stan S. Wainer (1970) and is sometimes called the Löb–Wainer hierarchy.