View source for Grzegorczyk hierarchy

{{Short description|Functions in computability theory}}
The '''Grzegorczyk hierarchy''' ({{IPAc-en|g|r|ɛ|'|g|ɔːr|tʃ|ə|k}}, {{IPA|pl|ɡʐɛˈɡɔrt͡ʂɨk}}), named after the Polish logician [[Andrzej Grzegorczyk]], is a hierarchy of functions used in [[computability theory]].{{sfn|Wagner|Wechsung|1986|p=43}} Every [[function (mathematics)|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 ''E<sub>i</sub>'' for some [[natural number]] ''i''. We define 
:<math> 
\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}
</math>
 
<math>E_0</math> is the addition function, and <math>E_1</math> is a unary function which squares its argument and adds two. Then, for each ''n'' greater than 1, <math>E_n(x)=E^{x}_{n-1}(2)</math>, i.e. the ''x''-th [[iterated function|iterate]] of <math>E_{n-1}</math> evaluated at 2.

From these functions we define the Grzegorczyk hierarchy. <math>\mathcal{E}^n</math>, the ''n''-th set in the hierarchy, contains the following functions:
# ''E<sub>k</sub>'' for ''k'' < ''n''
# the zero function (''Z''(''x'') = 0);
# the [[successor function]] (''S''(''x'') = ''x'' + 1);
# the [[projection function]]s (<math> p_i^m(t_1, t_2, \dots, t_m) = t_i </math>);
# the (generalized) [[function composition|compositions of functions]] in the set (if ''h'', ''g''<sub>1</sub>, ''g''<sub>2</sub>, ... and ''g''<sub>m</sub>  are in <math>\mathcal{E}^n</math>, then <math> f(\bar{u}) = h(g_1(\bar{u}), g_2(\bar{u}), \dots, g_m(\bar{u})) </math> is as well);<ref group=note name="tuple notation">Here <math>\bar{u}</math> represents a [[tuple]] of inputs to ''f''. The notation <math>f(\bar{u})</math> means that ''f'' takes some arbitrary [[arity|number of arguments]] and if <math>\bar{u} = (x, y, z)</math>, then <math>f(\bar{u}) = f(x, y, z)</math>. In the notation <math>f(t, \bar{u})</math>, the first argument, ''t'', is specified explicitly and the rest as the arbitrary tuple <math>\bar{u}</math>. Thus, if <math>\bar{u} = (x, y, z)</math>, then <math>f(t, \bar{u}) = f(t, x, y, z)</math>. This notation allows composition and limited recursion to be defined for functions, ''f'', of any number of arguments.</ref> and
# the results of limited (primitive) recursion applied to functions in the set, (if ''g'', ''h'' and ''j'' are in <math>\mathcal{E}^n</math> and <math>f(t, \bar{u}) \leq j(t, \bar{u})</math> for all ''t'' and <math>\bar{u}</math>, and further <math>f(0, \bar{u}) = g(\bar{u})</math> and <math>f(t+1, \bar{u}) = h(t,\bar{u},f(t,\bar{u}))</math>, then ''f'' is in <math>\mathcal{E}^n</math> as well).<ref group=note name="tuple notation"/>

In other words, <math>\mathcal{E}^n</math> is the [[closure (mathematics)|closure]] of set <math>B_n = \{Z, S, (p_i^m)_{i \le m}, E_k : k < n\}</math> with respect to function composition and limited recursion (as defined above).

== Properties ==

These sets clearly form the hierarchy
:<math> \mathcal{E}^0 \subseteq \mathcal{E}^1 \subseteq \mathcal{E}^2 \subseteq \cdots </math>
because they are closures over the <math>B_n</math>'s and <math> B_0 \subseteq B_1 \subseteq B_2 \subseteq \cdots</math>.

They are strict subsets.{{sfn|Rose|1984}}{{sfn|Gakwaya|1997}} In other words
:<math> \mathcal{E}^0 \subsetneq \mathcal{E}^1 \subsetneq \mathcal{E}^2 \subsetneq \cdots </math>
because the [[hyperoperation]] <math>H_n</math> is in <math>\mathcal{E}^n</math> but not in <math>\mathcal{E}^{n-1}</math>.

*<math>\mathcal{E}^0</math> includes functions such as ''x''+1, ''x''+2, ...
** Every unary function ''f(x)'' in <math>\mathcal{E}^0</math> is ''upper bounded'' by some ''x''+''n''. However, <math>\mathcal{E}^0</math> also includes more complicated functions like ''x''∸1, ''x''∸''y'' (where ∸ is the [[monus]] sign defined as ''x''∸''y'' = max(''x''-''y'', 0)), {{math|''x'' mod ''y''}}, etc.
*<math>\mathcal{E}^1</math> provides all addition functions, such as ''x''+''y'', 4''x'', ...
*<math>\mathcal{E}^2</math> provides all multiplication functions, such as ''xy'', ''x''<sup>4</sup> 
*<math>\mathcal{E}^3</math> provides all exponentiation functions, such as ''x''<sup>''y''</sup>, 2<sup>2<sup>2<sup>''x''</sup></sup></sup>, and is exactly the [[ELEMENTARY|elementary recursive functions]].
*<math>\mathcal{E}^4</math> provides all [[tetration]] functions, and so on.

Notably, both the function <math>U</math> and the characteristic function of the predicate <math>T</math> from the [[Kleene normal form theorem]] are definable in a way such that they lie at level <math>\mathcal{E}^0</math> of the Grzegorczyk hierarchy. This implies in particular that every recursively enumerable set is enumerable by some <math>\mathcal{E}^0</math>-function.

== Relation to primitive recursive functions ==

The definition of <math>\mathcal{E}^n</math> is the same as that of the [[primitive recursive function]]s, {{sans-serif|PR}}, except that recursion is ''limited'' (<math>f(t, \bar{u}) \leq j(t, \bar{u})</math> for some ''j'' in <math>\mathcal{E}^n</math>) and the functions <math>(E_k)_{k<n}</math> are explicitly included in <math>\mathcal{E}^n</math>. 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. <math> \mathcal{E}^n \subseteq \mathsf{PR} </math>) and thus:
:<math> \bigcup_n{\mathcal{E}^n} \subseteq \mathsf{PR} </math>

It can also be shown that all primitive recursive functions are in some level of the hierarchy,{{sfn|Rose|1984}}{{sfn|Gakwaya|1997}} thus
:<math> \bigcup_n{\mathcal{E}^n} = \mathsf{PR} </math>
and the sets <math> \mathcal{E}^0, \mathcal{E}^1 - \mathcal{E}^0, \mathcal{E}^2 - \mathcal{E}^1, \dots, \mathcal{E}^n - \mathcal{E}^{n-1}, \dots </math> [[partition of a set|partition]] the set of primitive recursive functions, {{sans-serif|PR}}.

Meyer and [[Dennis Ritchie|Ritchie]] introduced another hierarchy subdividing the primitive recursive functions, based on the nesting depth of loops needed to write a [[LOOP (programming language)|LOOP]] program that computes the function. For a natural number <math>i</math>, let <math>\mathcal{L}_i</math> denote the set of functions computable by a LOOP program with <code>LOOP</code> and <code>END</code> commands nested no deeper than <math>i</math> levels.<ref>A. R. Meyer, D. M. Ritchie, "[https://people.csail.mit.edu/meyer/meyer-ritchie.pdf The complexity of loop programs"]. Proceedings A.C.M. National Meeting, 1967.</ref> Fachini and Maggiolo-Schettini showed that <math>\mathcal{L}_i</math> coincides with <math>\mathcal{E}_{i+1}</math> for all integers <math>i>1</math>.<ref>E. Fachini, A. Maggiolo-Schettini, "[A hierarchy of primitive recursive sequence functions]". From ''Informatique théorique'', book 13, no. 1 (1979), pp.49--67.</ref><sup>p.63</sup>

== Extensions ==
{{main|Fast-growing hierarchy}}

The Grzegorczyk hierarchy can be extended to [[Transfinite number|transfinite]] [[Ordinal number|ordinals]]. Such extensions define a [[fast-growing hierarchy]]. To do this, the generating functions <math>E_\alpha</math> must be recursively defined for [[limit ordinal]]s (note they have already been recursively defined for successor ordinals by the relation <math> E_{\alpha+1}(n) = E_\alpha^n(2) </math>). If there is a standard way of defining a ''fundamental sequence'' <math>\lambda_m</math>, whose limit ordinal is <math>\lambda</math>, then the generating functions can be defined <math> E_\lambda(n) = E_{\lambda_n}(n) </math>. However, this definition depends upon a standard way of defining the fundamental sequence. {{harvtxt|Rose|1984}} suggests a standard way for all ordinals ''α'' < [[Epsilon numbers (mathematics)|ε<sub>0</sub>]].

The original extension was due to [[Martin Löb]] and [[Stan S. Wainer]] and is sometimes called the [[Löb–Wainer hierarchy]].{{sfn|Löb|Wainer|1970}}

== See also ==
* [[ELEMENTARY]]
* [[Fast-growing hierarchy]]
* [[Ordinal analysis]]

== Notes ==
{{reflist|group=note}}

== References ==
{{reflist}}

== Bibliography ==
*{{cite book
|last1= Brainerd
|first1= Walter S.
|last2= Landweber
|first2= Lawrence H.
|year= 1974
|title= Theory of computation
|publisher= Wiley
|isbn= 9780471095859
}}

*{{cite journal
|last1= Cichon
|first1= E. A.
|last2= Wainer
|first2= S. S.
|year= 1983
|title= The slow-growing and the Grzegorczyk hierarchies
|journal= [[Journal of Symbolic Logic]]
|issn= 0022-4812
|volume= 48
|issue= 2
|pages= 399–408
|doi= 10.2307/2273557
|jstor= 2273557
|mr= 704094
|s2cid= 1390729
}}

*{{Cite CiteSeerX
|last= Gakwaya
|first= Jean-Sylvestre
|year= 1997
|title= A survey on the Grzegorczyk Hierarchy and its Extension through the BSS Model of Computability
|citeseerx= 10.1.1.69.4621
}}

*{{cite journal
|last= Grzegorczyk
|first= Andrzej
|author-link= Andrzej Grzegorczyk
|year= 1953
|title= Some classes of recursive functions
|url= http://matwbn.icm.edu.pl/ksiazki/rm/rm04/rm0401.pdf
|journal= Rozprawy Matematyczne
|volume= 4| pages = 1–45
}}

*{{cite journal
|last1= Löb
|first1= Martin Hugo
|author-link1= Martin Löb
|last2= Wainer
|first2= Stan S.
|author-link2= Stan S. Wainer
|year= 1970
|title= Hierarchies of Number Theoretic Functions I, II: A Correction
|journal= [[Archiv für mathematische Logik und Grundlagenforschung]]
|volume= 14
|issue= 3–4
|pages= 198–199
|doi=10.1007/bf01991855
|s2cid= 119830535
}}

*{{cite book
|last= Rose
|first= H.E.
|year= 1984
|title= Subrecursion: Functions and hierarchies
|publisher= [[Oxford University Press]]
|isbn= 0-19-853189-3
}}

*{{cite journal
|last1= Wagner
|first1= K.
|last2= Wechsung
|first2= G.
|year= 1986
|title= Computational Complexity
|journal= Mathematics and Its Applications
|volume= 21
|publisher= Springer
|isbn=978-90-277-2146-4
}}

{{ComplexityClasses}}
{{Hyperoperations}}
{{Large numbers}}
[[Category:Computability theory]]
[[Category:Hierarchy of functions]]