Grzegorczyk hierarchy

First we introduce an infinite set of functions, denoted Ei for some natural number i. We define : \begin{array}{lcl} E_0(x,y) & = & x + y \\ E_1(x) & = & x^2 + 2 \\ E_{n+2}(0) & = & 2 \\ E_{n+2}(x+1) & = & E_{n+1}(E_{n+2}(x)) \\ \end{array} 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, E_n(x)=E^{x}_{n-1}(2), i.e. the x-th iterate of E_{n-1} evaluated at 2. From these functions we define the Grzegorczyk hierarchy. \mathcal{E}^n, the n-th set in the hierarchy, contains the following functions: • Ek for k p_i^m(t_1, t_2, \dots, t_m) = t_i ); • the (generalized) compositions of functions in the set (if h, g1, g2, ... and gm are in \mathcal{E}^n, then f(\bar{u}) = h(g_1(\bar{u}), g_2(\bar{u}), \dots, g_m(\bar{u})) is as well); and • the results of limited (primitive) recursion applied to functions in the set, (if g, h and j are in \mathcal{E}^n and f(t, \bar{u}) \leq j(t, \bar{u}) for all t and \bar{u}, and further f(0, \bar{u}) = g(\bar{u}) and f(t+1, \bar{u}) = h(t,\bar{u},f(t,\bar{u})), then f is in \mathcal{E}^n as well). In other words, \mathcal{E}^n is the closure of set B_n = \{Z, S, (p_i^m)_{i \le m}, E_k : k with respect to function composition and limited recursion (as defined above). == Properties ==

These sets clearly form the hierarchy : \mathcal{E}^0 \subseteq \mathcal{E}^1 \subseteq \mathcal{E}^2 \subseteq \cdots because they are closures over the B_n's and B_0 \subseteq B_1 \subseteq B_2 \subseteq \cdots. They are strict subsets. In other words : \mathcal{E}^0 \subsetneq \mathcal{E}^1 \subsetneq \mathcal{E}^2 \subsetneq \cdots because the hyperoperation H_n is in \mathcal{E}^n but not in \mathcal{E}^{n-1}. • \mathcal{E}^0 includes functions such as x+1, \; x+2, \; \ldots ::Every unary function f(x) in \mathcal{E}^0 is upper bounded by some x+n. ::However, \mathcal{E}^0 also includes more complicated functions like x \mathbin{\dot{-}} 1, x \mathbin{\dot{-}} y, x \bmod y, \; \ldots • \mathcal{E}^1 provides all addition functions, such as x+y, \; 4x, \; \ldots • \mathcal{E}^2 provides all multiplication functions, such as xy, \; x^4, \; \ldots • \mathcal{E}^3 provides all exponentiation functions, such as x^y, \; 2^{2^{2^x}}, and is exactly the elementary recursive functions. • \mathcal{E}^4 provides all tetration functions, and so on. 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 \mathcal{E}^0 of the Grzegorczyk hierarchy. This implies in particular that every computably enumerable set is enumerable by some \mathcal{E}^0-function. == Relation to primitive recursive functions ==

The definition of \mathcal{E}^n is the same as that of the primitive recursive functions, , except that recursion is limited (f(t, \bar{u}) \leq j(t, \bar{u}) for some j in \mathcal{E}^n) and the functions (E_k)_{k are explicitly included in \mathcal{E}^n. 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. \mathcal{E}^n \subseteq \mathsf{PR} ) and thus: : \bigcup_n{\mathcal{E}^n} \subseteq \mathsf{PR} It can also be shown that all primitive recursive functions are in some level of the hierarchy, thus : \bigcup_n{\mathcal{E}^n} = \mathsf{PR} and the sets \mathcal{E}^0, \mathcal{E}^1 - \mathcal{E}^0, \mathcal{E}^2 - \mathcal{E}^1, \dots, \mathcal{E}^n - \mathcal{E}^{n-1}, \dots 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 \mathcal{L}_i denote the set of functions computable by a LOOP program with LOOP and END commands nested no deeper than i levels. Fachini and Maggiolo-Schettini showed that \mathcal{L}_i coincides with \mathcal{E}_{i+1} for all integers i>1. == 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 E_{\alpha+1}(n) = E_\alpha^n(2) ). 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 E_\lambda(n) = E_{\lambda_n}(n) . However, this definition depends upon a standard way of defining the fundamental sequence. suggests a standard way for all ordinals α 0. The original extension was due to Martin Löb and Stan S. Wainer and is sometimes called the Löb–Wainer hierarchy. == See also ==