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| In [[computational complexity theory]], the [[complexity class]] '''ELEMENTARY''' of '''elementary recursive functions''' is the union of the classes in the [[exponential hierarchy]].
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| : <math> \begin{matrix}
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| \mathrm{ELEMENTARY} & = & \mathrm{EXP}\cup\mathrm{2EXP}\cup\mathrm{3EXP}\cup\cdots \\
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| & = & \mathrm{DTIME}(2^{n})\cup\mathrm{DTIME}(2^{2^{n}})\cup
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| \mathrm{DTIME}(2^{2^{2^{n}}})\cup\cdots
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| \end{matrix}
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| </math>
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| The name was coined by [[László Kalmár]], in the context of [[Computable function|recursive functions]] and [[Undecidable problem|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
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| :LOWER-ELEMENTARY <math>\subsetneq</math> [[EXPTIME]] <math>\subsetneq</math> ELEMENTARY <math>\subsetneq</math> [[PR (complexity)|PR]] <math>\subsetneq</math> [[R (complexity)|R]] | |
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| Whereas ELEMENTARY contains bounded applications of [[exponentiation]] (for example, <math>O(2^{2^n})</math>), [[PR (complexity)|PR]] allows more general [[hyper operator]]s (for example, [[tetration]]) which are not contained in ELEMENTARY.
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| ==Definition==
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| The definitions of elementary recursive functions are the same as for [[primitive recursive function]]s, 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:
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| # '''Zero function'''. Returns zero: ''f''(x) = 0.
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| # '''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.
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| # '''Projection functions''': these are used for ignoring arguments. For example, ''f''(''a'', ''b'') = ''a'' is a projection function.
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| # '''Subtraction function''': ''f''(''x'', ''y'') = ''x'' - ''y'' if ''y'' < ''x'', or 0 if ''y'' ≥ ''x''. This function is used to define conditionals and iteration.
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| From these basic functions, we can build other elementary recursive functions.
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| # '''Composition''': applying values from some elementary recursive function as an argument to another elementary recursive function. In ''f''(''x''<sub>1</sub>, ..., ''x''<sub>n</sub>) = ''h''(''g''<sub>1</sub>(''x''<sub>1</sub>, ..., ''x''<sub>n</sub>), ..., ''g''<sub>m</sub>(''x''<sub>1</sub>, ..., ''x''<sub>n</sub>)) is elementary recursive if ''h'' is elementary recursive and each ''g''<sub>i</sub> is elementary recursive.
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| # '''Bounded summation''': <math>f(m, x_1, \ldots, x_n) = \sum\limits_{i=0}^mg(i, x_1, \ldots, x_n)</math> is elementary recursive if ''g'' is elementary recursive.
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| # '''Bounded product''': <math>f(m, x_1, \ldots, x_n) = \prod\limits_{i=0}^mg(i, x_1, \ldots, x_n)</math> is elementary recursive if ''g'' is elementary recursive.
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| == Lower elementary recursive functions ==
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| ''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.
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| Whereas elementary recursive functions have potentially exponential growth, and comprise the [[exponential hierarchy]], the lower elementary recursive functions have polynomial growth.
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| == Basis for ELEMENTARY ==
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| The class of elementary functions coincides with the closure with respect to composition of the projections and one of the following function sets: <math>\{ n+1, n \stackrel{.}{-} m, \lfloor n/m \rfloor, n^{m} \}</math>, <math>\{ n+m, n \stackrel{.}{-} m, \lfloor n/m\rfloor, 2^{n} \}</math>, <math>\{ n+m, n^{2}, n \bmod m, 2^{n} \}</math>, where <math>n \stackrel{.}{-} m = \max\{n-m, 0\}</math> is the subtraction function defined above.<ref>Mazzanti, S., "Plain Bases for Classes of Primitive Recursive Functions", Mathematical Logic Quarterly, 48 (2002) 93-104 </ref> | |
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| == Descriptive characterization ==
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| In [[descriptive complexity]], ELEMENTARY is equal to the class of [[HO (complexity)|high order queries]].<ref>{{Citation | author = Lauri Hella and José María Turull-Torres | title =Computing queries with higher-order logics | journal =Theoretical Computer Science | volume = 355 | edition= (what is called "number" in bibtex) | issue = 2 | year = 2006 | pages = 197–214 | issn =0304-3975 | publisher =Elsevier Science Publishers Ltd. | place = Essex, UK | url = http://portal.acm.org/citation.cfm?id=1142890.1142897}}</ref> 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, <math>\mathrm{NTIME}(2^{2^{\cdots{2^{O(n)}}}}) = \exists{}HO^i</math>, where <math>\cdots</math> indicates a tower of <math>i</math> exponentiations and <math>\exists{}HO^i</math> is the class of queries that begin with existential quantifiers of <math>i</math>th order and then a formula of <math>(i-1)</math>th order.
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| == See also ==
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| * [[Elementary function arithmetic]]
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| * [[Primitive recursive function]]
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| * [[Grzegorczyk hierarchy]]
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| * [[EXPTIME]]
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| == References ==
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| {{reflist}}
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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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| {{ComplexityClasses}}
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| [[Category:Complexity classes]]
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| [[Category:Computability theory]]
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