Colorfulness

In computational complexity theory, the complexity class ELEMENTARY of elementary recursive functions is the union of the classes in the exponential hierarchy.

$\begin{array}{ccc}\mathrm{E}\mathrm{L}\mathrm{E}\mathrm{M}\mathrm{E}\mathrm{N}\mathrm{T}\mathrm{A}\mathrm{R}\mathrm{Y}& =& \mathrm{E}\mathrm{X}\mathrm{P}\cup \mathrm{2}\mathrm{E}\mathrm{X}\mathrm{P}\cup \mathrm{3}\mathrm{E}\mathrm{X}\mathrm{P}\cup \cdots \\ & =& \mathrm{D}\mathrm{T}\mathrm{I}\mathrm{M}\mathrm{E}(2^n)\cup \mathrm{D}\mathrm{T}\mathrm{I}\mathrm{M}\mathrm{E}(2^{2^n})\cup \mathrm{D}\mathrm{T}\mathrm{I}\mathrm{M}\mathrm{E}(2^{2^{2^n}})\cup \cdots \end{array}$

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 $⊊$ R

Whereas ELEMENTARY contains bounded applications of exponentiation (for example, $O(2^{2^n})$), PR allows more general hyper operators (for example, tetration) which are not contained in ELEMENTARY.

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:

1. Zero function. Returns zero: f(x) = 0.
2. Successor function: f(x) = x + 1. Often this is denoted by S, as in S(x). Via repeated application of a successor function, one can achieve addition.
3. Projection functions: these are used for ignoring arguments. For example, f(a, b) = a is a projection function.
4. Subtraction function: f(x, y) = x - y if y < x, or 0 if y ≥ x. This function is used to define conditionals and iteration.

From these basic functions, we can build other elementary recursive functions.

1. Composition: applying values from some elementary recursive function as an argument to another elementary recursive function. In f(x₁, ..., x_n) = h(g₁(x₁, ..., x_n), ..., g_m(x₁, ..., x_n)) is elementary recursive if h is elementary recursive and each g_i is elementary recursive.
2. Bounded summation: $f(m,x_1,\dots ,x_n)=\sum _{i=0}^{m}g(i,x_1,\dots ,x_n)$ is elementary recursive if g is elementary recursive.
3. Bounded product: $f(m,x_1,\dots ,x_n)=\prod _{i=0}^{m}g(i,x_1,\dots ,x_n)$ 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.

Basis for ELEMENTARY

The class of elementary functions coincides with the closure with respect to composition of the projections and one of the following function sets: $\{n+1,n\stackrel{.}{-}m,\lfloor n/m\rfloor ,n^m\}$, $\{n+m,n\stackrel{.}{-}m,\lfloor n/m\rfloor ,2^n\}$, $\{n+m,n^2,nmodm,2^n\}$, where $n\stackrel{.}{-}m=\max \{n-m,0\}$ is the subtraction function defined above.

Descriptive characterization

In descriptive complexity, ELEMENTARY is equal to the class of high order queries. This means that every language in the ELEMENTARY complexity class can be written as a high order formula that is true only for the elements on the language. More precisely, $\mathrm{N}\mathrm{T}\mathrm{I}\mathrm{M}\mathrm{E}(2^{2^{\cdots 2^{O(n)}}})=\mathrm{\exists }H{O}^i$, where $\cdots$ indicates a tower of $i$ exponentiations and $\mathrm{\exists }H{O}^i$ is the class of queries that begin with existential quantifiers of $i$th order and then a formula of $(i-1)$th order.

See also

• Elementary function arithmetic
• Primitive recursive function
• Grzegorczyk hierarchy
• EXPTIME

References

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

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