Elementary

In computational complexity theory, the complexity class ELEMENTARY of elementary recursive functions is the union of the classes










E
L
E
M
E
N
T
A
R
Y





=

⋃

k
∈

N



k




-


E
X
P








=


D
T
I
M
E



(

2

n


)

∪


D
T
I
M
E



(

2


2

n




)

∪


D
T
I
M
E



(

2


2


2

n






)

∪
⋯






{\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}}}
The name was coined by László Kalmár, in the context of recursive functions and undecidability; most problems in it are far from elementary. Some natural recursive problems lie outside ELEMENTARY, and are thus NONELEMENTARY. Most notably, there are primitive recursive problems that are not in ELEMENTARY. We know

LOWER-ELEMENTARY ⊊ EXPTIME ⊊ ELEMENTARY ⊊ PR ⊊ RWhereas ELEMENTARY contains bounded applications of exponentiation (for example,



O
(

2


2

n




)


{\displaystyle O(2^{2^{n}})}
), PR allows more general hyper operators (for example, tetration) which are not contained in ELEMENTARY.

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