The Grzegorczyk hierarchy, named after the Polish logician Andrzej Grzegorczyk, is a hierarchy of functions used in computability theory. 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 levels of the hierarchy grow slower than functions in the higher levels.

Definition

First we introduce an infinite set of functions, denoted Ei for some natural number i. We define E_0 is the addition function, and E_1 is a unary function which squares its argument and adds two. Then, for each n greater than 1,, i.e. the x-th iterate of E_{n-1} evaluated at 2. From these functions we define the Grzegorczyk hierarchy. , the n-th set in the hierarchy, contains the following functions: In other words, is the closure of set with respect to function composition and limited recursion (as defined above).

Properties

These sets clearly form the hierarchy because they are closures over the B_n's and. They are strict subsets. In other words because the hyperoperation H_n is in but not in. x mod y , etc. Notably, both the function U and the characteristic function of the predicate T from the Kleene normal form theorem are definable in a way such that they lie at level of the Grzegorczyk hierarchy. This implies in particular that every recursively enumerable set is enumerable by some -function.

Relation to primitive recursive functions

The definition of is the same as that of the primitive recursive functions,, except that recursion is limited ( for some j in ) and the functions (E_k)_{k<n} 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, thus and the sets partition the set of primitive recursive functions,. Meyer and Ritchie introduced another hierarchy subdividing the primitive recursive functions, based on the nesting depth of loops needed to write a LOOP program that computes the function. For a natural number i, let denote the set of functions computable by a LOOP program with and commands nested no deeper than i levels. Fachini and Maggiolo-Schettini showed that coincides with for all integers i>1. p.63

Extensions

The Grzegorczyk hierarchy can be extended to transfinite ordinals. Such extensions define a fast-growing hierarchy. To do this, the generating functions E_\alpha must be recursively defined for limit ordinals (note they have already been recursively defined for successor ordinals by the relation ). If there is a standard way of defining a fundamental sequence \lambda_m, whose limit ordinal is \lambda, then the generating functions can be defined. However, this definition depends upon a standard way of defining the fundamental sequence. suggests a standard way for all ordinals α < ε0. The original extension was due to Martin Löb and Stan S. Wainer and is sometimes called the Löb–Wainer hierarchy.