You are here
Homeelementary recursive function
Primary tabs
elementary recursive function
Informally, elementary recursive functions are functions that can be obtained from functions encountered in elementary schools: addition, multiplication, subtraction, and division, using basic operations such as substitutions and finite summation and product. Before stating the formal definition, the following notations are used:
$\mathcal{F}=\bigcup\{F_{k}\mid k\in\mathbb{N}\}$, where for each $k\in\mathbb{N}\text{, }F_{k}=\{f\mid f\colon\mathbb{N}^{{k}}\to\mathbb{N}\}$.
Definition. The set of elementary recursive functions, or elementary functions for short, is the smallest subset $\mathcal{ER}$ of $\mathcal{F}$ where:
 1.
(addition) $\operatorname{add}\in\mathcal{ER}\cap F_{2}$, given by $\operatorname{add}(m,n):=m+n$;
 2.
(multiplication) $\operatorname{mult}\in\mathcal{ER}\cap F_{2}$, given by $\operatorname{mult}(m,n):=mn$;
 3.
(difference) $\operatorname{diff}\in\mathcal{ER}\cap F_{2}$, given by $\operatorname{diff}(m,n):=mn$;
 4.
 5.
(projection) $p^{k}_{m}\in\mathcal{ER}\cap F_{k}$, where $m\leq k$, given by $p^{k}_{m}(n_{1},\ldots,n_{k}):=n_{m}$;
 6.
$\mathcal{ER}$ is closed under composition: If $\{g_{1},\ldots,g_{m}\}\subseteq\mathcal{ER}\cap F_{{k}}$ and $h\in\mathcal{ER}\cap F_{m}$, then $f\in\mathcal{ER}\cap F_{{k}}$, where
$f(n_{1},\ldots,n_{k})=h(g_{1}(n_{1},\ldots,n_{k}),\ldots,g_{m}(n_{1},\ldots,n_% {k}));$  7.
$\mathcal{ER}$ is closed under bounded sum: if $f\in\mathcal{ER}\cap F_{k}$, then $f_{s}\in\mathcal{ER}\cap F_{k}$, where
$f_{s}(\boldsymbol{x},y):=\sum_{{i=0}}^{y}f(\boldsymbol{x},i);$  8.
$\mathcal{ER}$ is closed under bounded product: if $f\in\mathcal{ER}\cap F_{k}$, then $f_{p}\in\mathcal{ER}\cap F_{k}$, where
$f_{p}(\boldsymbol{x},y):=\prod_{{i=0}}^{y}f(\boldsymbol{x},i).$
Examples.

The initial functions in the definition of primitive recursive functions are elementary:
(a) The zero function $z(x)$ is $\operatorname{diff}(x,x)$.
(b) The constant function $\operatorname{const}_{1}(x):=1$ is $\operatorname{quo}(x,x)$.
(c) The successor function $s(x)$ can be obtained by substituting (by composition) the constant function $\operatorname{const}_{1}$ and the projection function $p_{1}^{1}$, into the addition function $\operatorname{add}(p_{1}^{1}(x),\operatorname{const}_{1}(x))$.

Multiplication $\operatorname{mult}$ in 2 above may be removed from the definition, since
$\operatorname{mult}(x,y)=\operatorname{diff}(f(x,y),p_{1}^{2}(x,y)),\quad\mbox% {where }f(x,y):=\sum_{{i=0}}^{y}p_{1}^{2}(x,i)$ 
Many other basic functions, such as the restricted subtraction, exponential function, the $i$th prime function, are all elementary. One may replace the difference function in 3 above by the restricted subtraction without changing $\mathcal{ER}$.
Remarks

Consider the set $\mathcal{PR}$ of primitive recursive functions. The functions in the first five groups above are all in $\mathcal{PR}$. In addition, $\mathcal{PR}$ is closed under the operations in 6, 7, and 8 above, we see that $\mathcal{ER}\subseteq\mathcal{PR}$, since $\mathcal{ER}$, as defined, is the smallest such set.

Furthermore, $\mathcal{ER}\neq\mathcal{PR}$. For example, the superexponential function, given by $f(x,0)=m$, and $f(x,n+1)=\exp(m,f(x,n))$, where $m>1$, can be shown to be nonelementary.

In addition, it can be shown that $\mathcal{ER}$ is the set of primitive recursive functions that can be obtained from the zero function, the successor function, and the projection functions via composition, and no more than three applications of primitive recursion.

By taking the closure of $\mathcal{ER}$ with respect to unbounded minimization, one obtains $\mathcal{R}$, the set of all recursive functions (partial or total). In fact, by replacing bounded sum and bounded product with unbounded minimization, and the difference function with restricted subtraction, one obtains $\mathcal{R}$.
Mathematics Subject Classification
03D20 no label found Forums
 Planetary Bugs
 HS/Secondary
 University/Tertiary
 Graduate/Advanced
 Industry/Practice
 Research Topics
 LaTeX help
 Math Comptetitions
 Math History
 Math Humor
 PlanetMath Comments
 PlanetMath System Updates and News
 PlanetMath help
 PlanetMath.ORG
 Strategic Communications Development
 The Math Pub
 Testing messages (ignore)
 Other useful stuff
 Corrections