ELEMENTARY

In computational complexity theory, the complexity class 'ELEMENTARY' is the union of the classes in the exponential hierarchy.
:

m{ELEMENTARY} & = &
m{EXP}cup
m{2EXP}cup
m{3EXP}cupcdots \
& = &
m{DTIME}(2^{n})cup
m{DTIME}(2^{2^{n}})cup

m{DTIME}(2^{2^{2^{n}}})cupcdots
end{matrix}

The name was coined 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 which are not in ELEMENTARY. We know
:LOWER-ELEMENTARY $subsetneq$ EXPTIME $subsetneq$ ELEMENTARY $subsetneq$ PR
Whereas ELEMENTARY contains bounded applications of exponentiation (for example, $$
m{O}(2^{2^n})), PR allows more general hyper operators (for example, $$
m{O}(nuparrowuparrow n), using Knuth's up-arrow notation) which are not contained in ELEMENTARY.

Contents

Definition

Lower elementary recursive functions

Relationship to primitive recursion

See also

References

Definition

The definitions of elementary recursive functions are the same as for primitive recursive functions, except that primitive recursion is replaced by bounded summation and bounded product. All functions work over the natural numbers. The basic functions, all of them elementary recursive, are:
# 'Zero function': returns zero. E.g., ''f''() = 0.
# 'Successor function'. E.g., ''f''(''x'') = ''x'' + 1. Oftentimes this is denoted by ''S'', as in ''S''(''x''). Via repeated application of a successor function, one can achieve addition.
# 'Projection functions': these are used for ignoring arguments. For example, ''f''(''a'', ''b'') = ''a'' is a projection function.
From these basic functions, we can build other elementary recursive functions.
# 'Composition': applying values from some elementary recursive function as an argument to another elementary recursive function. In ''f''(''x''₁, ..., ''x''_n) = ''h''(''g''₁(''x''₁, ..., ''x''_n), ..., ''g''_m(''x''₁, ..., ''x''_n)) is elementary recursive if ''h'' is elementary recursive and each ''g''_i is elementary recursive.
# 'Bounded summation': $f(m,\; x\_1,\; ldots,\; x\_n)\; =\; sumlimits\_\{i=0\}^mg(i,\; x\_1,\; ldots,\; x\_n)$ is elementary recursive if ''g'' is elementary recursive.
# 'Bounded product': $f(m,\; x\_1,\; ldots,\; x\_n)\; =\; prodlimits\_\{i=0\}^mg(i,\; x\_1,\; ldots,\; x\_n)$ is elementary recursive if ''g'' is elementary recursive.

Lower elementary recursive functions

''Lower elementary recursive'' functions follow the definitions as above, except that bounded product is disallowed. That is, a lower elementary recursive function must be a zero, successor, or projection function, a composition of other lower elementary recursive functions, or the bounded sum of another lower elementary recursive function.
Whereas elementary recursive functions have potentially exponential growth, and comprise the exponential hierarchy, the lower elementary recursive functions have polynomial growth.

Relationship to primitive recursion

The definitions for elementary recursive functions and primitive recursive functions are identical, except that in lieu of primitive recursion, elementary recursion offers bounded sums and products. Bounded sums and products offer a more restricted means of repeatedly applying some function, and indeed the elementary recursive functions form a strict subset of the primitive recursive functions.

See also

★ Primitive recursive function

★ EXPTIME

References

★ Rose, H.E., "Subrecursion: Functions and hierarchies", Oxford University Press, New York, USA, 1984. ISBN 0-19-853189-3