ELEMENTARY

Elementary}} In computational complexity theory, the complexity class $\backslash mathsf\{ELEMENTARY\}$ 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, $$\backslash left.\; \backslash begin\{matrix\}\; 2^\{\backslash scriptstyle\; 2^\{\backslash scriptstyle\; 2^\{\backslash scriptstyle\; \backslash cdot^\{\backslash scriptstyle\; \backslash cdot^\{\backslash scriptstyle\; \backslash cdot^\{\backslash scriptstyle\; n\}\}\}\}\}\}\; \backslash end\{matrix\}\; \backslash right\backslash \}\; k.$$

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

: 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 $\backslash mathsf\{ELEMENTARY\}$, and cannot compute languages beyond this complexity class. The time hierarchy theorem implies that $\backslash mathsf\{ELEMENTARY\}$ 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 $\backslash mathsf\{ELEMENTARY\}$. Provided by Wikipedia