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In computational complexity theory, the complexity class ${\displaystyle {\mathsf {ELEMENTARY}}}$ 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 ${\displaystyle 2^{n}}$ for a bounded number ${\displaystyle k}$ of iterations, $${\displaystyle \left.{\begin{matrix}2^{\scriptstyle 2^{\scriptstyle 2^{\scriptstyle \cdot ^{\scriptstyle \cdot ^{\scriptstyle \cdot ^{\scriptstyle n}}}}}}\end{matrix}}\right\}k.}$$

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

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

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 ${\displaystyle {\mathsf {ELEMENTARY}}}$, and cannot compute languages beyond this complexity class. The time hierarchy theorem implies that ${\displaystyle {\mathsf {ELEMENTARY}}}$ 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 ${\displaystyle {\mathsf {ELEMENTARY}}}$.

References

1. "ELEMENTARY", Complexity Zoo, retrieved 2024-11-03
2. Friedman, Harvey (1999), "Some decision problems of enormous complexity" (PDF), 14th Annual IEEE Symposium on Logic in Computer Science, Trento, Italy, July 2-5, 1999, {IEEE} Computer Society, pp. 2–12, doi:10.1109/LICS.1999.782577, ISBN 0-7695-0158-3, MR 1942515
3. Engelfriet, Joost (1991), "Iterated stack automata and complexity classes", Information and Computation, 95 (1): 21–75, doi:10.1016/0890-5401(91)90015-T, MR 1133778

• Complexity classes