ELEMENTARY

For other meanings, see Elementary.

In computational complexity theory, the complexity class ${\mathsf {ELEMENTARY}}$ consists of the decision problems that can be solved in time bounded by an elementary recursive function. The most quickly-growing elementary functions are obtained by iterating an exponential function such as $2^{n}$ for a bounded number $k$ of iterations, $$\left.{\begin{matrix}2^{\scriptstyle 2^{\scriptstyle 2^{\scriptstyle \cdot ^{\scriptstyle \cdot ^{\scriptstyle \cdot ^{\scriptstyle n}}}}}}\end{matrix}}\right\}k.$$

Thus, ${\mathsf {ELEMENTARY}}$ is the union of the classes

${\begin{aligned}{\mathsf {ELEMENTARY}}&=\bigcup _{k\in \mathbb {N} }k{\mathsf {{\mbox{-}}EXP}}\\&={\mathsf {DTIME}}\left(2^{n}\right)\cup {\mathsf {DTIME}}\left(2^{2^{n}}\right)\cup {\mathsf {DTIME}}\left(2^{2^{2^{n}}}\right)\cup \cdots .\end{aligned}}$

It is sometimes described as iterated exponential time, though this term more commonly refers to time bounded by the tetration function.

This complexity class can be characterized by a certain class of "iterated stack automata", pushdown automata that can store the entire state of a lower-order iterated stack automaton in each cell of their stack. These automata can compute every language in ${\mathsf {ELEMENTARY}}$, and cannot compute languages beyond this complexity class. The time hierarchy theorem implies that ${\mathsf {ELEMENTARY}}$ has no complete problems.

Every elementary recursive function can be computed in a time bound of this form, and therefore every decision problem whose calculation uses only elementary recursive functions belongs to the complexity class ${\mathsf {ELEMENTARY}}$.