ELEMENTARY

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: ${\displaystyle f(m,x_{1},\ldots ,x_{n})=\sum \limits _{i=0}^{m}g(i,x_{1},\ldots ,x_{n})}$ is elementary recursive if g is elementary recursive.
3. Bounded product: ${\displaystyle f(m,x_{1},\ldots ,x_{n})=\prod \limits _{i=0}^{m}g(i,x_{1},\ldots ,x_{n})}$ is elementary recursive if g is elementary recursive.

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

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.

Also known as Skolem elementary functions.[3][4]

Whereas elementary recursive functions have potentially more than exponential growth, the lower elementary recursive functions have polynomial growth.

The class of lower elementary functions has a description in terms of composition of simple functions analogous to that we have for elementary functions.[4][5] Namely, a polynomial-bounded function is lower elementary if and only if it can be expressed using a composition of the following functions: projections, ${\displaystyle n+1}$, ${\displaystyle nm}$, ${\displaystyle n\,{\stackrel {.}{-}}\,m}$, ${\displaystyle n\wedge m}$, ${\displaystyle \lfloor n/m\rfloor }$, one exponential function (${\displaystyle 2^{n}}$ or ${\displaystyle n^{m}}$) with the following restriction on the structure of formulas: the formula can have no more than two floors with respect to an exponent (for example, ${\displaystyle xy(z+1)}$ has 1 floor, ${\displaystyle (x+y)^{yz+x}+z^{x+1}}$ has 2 floors, ${\displaystyle 2^{2^{x}}}$ has 3 floors). Here ${\displaystyle n\wedge m}$ is a bitwise AND of n and m.

In descriptive complexity, ELEMENTARY is equal to the class of high order queries.[6] 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, ${\displaystyle {\mathsf {NTIME}}\left(2^{2^{\cdots {2^{O(n)}}}}\right)=\exists {}{\mathsf {HO}}^{i}}$, where ⋯ indicates a tower of i exponentiations and ${\displaystyle \exists {}{\mathsf {HO}}^{i}}$ is the class of queries that begin with existential quantifiers of ith order and then a formula of (i − 1)th order.

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

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