elementary

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Dictionary: el·e·men·ta·ry (ĕl'ə-mĕn'tə-rē, -trē) pronunciation

adj.

Of, relating to, or constituting the basic, essential, or fundamental part: an elementary need for love and nurturing.
Of, relating to, or involving the fundamental or simplest aspects of a subject: an elementary problem in statistics.
Of or relating to an elementary school or elementary education: the elementary grades; elementary teachers.

elementarily el'e·men·ta'ri·ly (-tĕr'ə-lē) adv.
elementariness el'e·men'ta·ri·ness n.

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Thesaurus: elementary

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adjective

Of or being an irreducible element: basic, elemental, essential, fundamental, primitive, ultimate, underlying. See surface/depth.
Of or treating the most basic aspects: basal, basic, beginning, rudimental, rudimentary. See simple/complex, start/end.

Antonyms: elementary

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adj

Definition: simple, basic
Antonyms: abstruse, advanced, complex, complicated, compound, difficult, hard, intricate, involved, secondary

Veterinary Dictionary: elementary

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Not resolvable into simpler parts.

e. body — 1. the constituent parts of inclusion bodies. Considered to be virus particles. See also inclusion body.

Chlamydiales.

Word Tutor: elementary

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IN BRIEF: adj. - Easy and not involved or complicated; Of or being the essential or basic part.

Tutor's tip: Learning about the "alimentary" (relating to nutrition) canal is part of the science curriculum in "elementary" (simplest, basic) school.

Wikipedia: ELEMENTARY

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

$\begin{matrix} \rm{ELEMENTARY} & = & \rm{EXP}\cup\rm{2EXP}\cup\rm{3EXP}\cup\cdots \\ & = & \rm{DTIME}(2^{n})\cup\rm{DTIME}(2^{2^{n}})\cup \rm{DTIME}(2^{2^{2^{n}}})\cup\cdots \end{matrix}$

The name was coined by Laszlo Kalmar, 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, $\rm{O}(2^{2^n})$ ), PR allows more general hyper operators (for example, $\rm{O}(n\uparrow\uparrow n)$ , using Knuth's up-arrow notation) which are not contained in ELEMENTARY.

1 Definition
2 Lower elementary recursive functions
3 Relationship to primitive recursion
4 Basis for ELEMENTARY
5 See also
6 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: f(x) = 0.
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.
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(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.
Bounded summation: $f(m, x_1, \ldots, x_n) = \sum\limits_{i=0}^mg(i, x_1, \ldots, x_n)$ is elementary recursive if g is elementary recursive.
Bounded product: $f(m, x_1, \ldots, x_n) = \prod\limits_{i=0}^mg(i, x_1, \ldots, x_n)$ 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.

Basis for ELEMENTARY

The class of elementary functions turns out to coincide with the closure with respect to composition of the projections and one of the following function sets $\{ n+1, n \stackrel{.}{-} m, [n/m], n^{m} \},\; \{ n+m, n \stackrel{.}{-} m, [n/m], 2^{n} \},\; \{ n+m, n^{2}, rem(n,m), 2^{n} \}$ where $n \stackrel{.}{-} m=max(n-m,0).$

References

Rose, H.E., "Subrecursion: Functions and hierarchies", Oxford University Press, New York, USA, 1984. ISBN 0-19-853189-3
Mazzanti, S., "Plain Bases for Classes of Primitive Recursive Functions", Mathematical Logic Quarterly, 48 (2002) 93-104

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Misspellings: elementary

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Common misspelling(s) of elementary

elimentary

Translations: Elementary

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Dansk (Danish)
adj. - elementær, simpel, grund-

idioms:

elementary particle elementpartikel
elementary school underskolen

Nederlands (Dutch)
elementair, basis-

Français (French)
adj. - élémentaire, simple, (Math) élémentaire, (US, GB) (école, enseignement) primaire

idioms:

elementary particle particule élémentaire
elementary school école primaire

Deutsch (German)
adj. - elementar

idioms:

elementary particle Elementarteilchen
elementary school Grundschule

Ελληνική (Greek)
adj. - στοιχειώδης, στοιχειακός

idioms:

elementary particle (φυσ.) στοιχειώδες σωματίδιο
elementary school δημοτικό σχολείο

Italiano (Italian)
elementare

idioms:

elementary particle particella elementare
elementary school scuola elementare

Português (Portuguese)
adj. - elementar, básico, essencial

idioms:

elementary particle partículas (f pl) que compõem o átomo
elementary school escola (f) primária

Русский (Russian)
элементарный, простой, основной, первоначальный, неразложимый

idioms:

elementary particle элементарная частица
elementary school начальная школа

Español (Spanish)
adj. - elemental, básico, fundamental

idioms:

elementary particle partícula elemental
elementary school escuela primaria

Svenska (Swedish)
adj. - elementär, grund- (kem.)

中文（简体）(Chinese (Simplified))
初步的, 基本的

idioms:

elementary particle 基本粒子, 元质点
elementary school 小学

中文（繁體）(Chinese (Traditional))
adj. - 初步的, 基本的

idioms:

elementary particle 基本粒子, 元質點
elementary school 小學

한국어 (Korean)
adj. - 기본의 , 원소의, 단체의

日本語 (Japanese)
adj. - 基本の, 初等の, 初級の

idioms:

elementary particle 素粒子
elementary school 小学校

العربيه (Arabic)
‏(صفه) أولي, ابتدائي‏

עברית (Hebrew)
adj. - ‮יסודי, אלמנטרי, פשוט, בלתי-ניתן לחלוקה (כימיה)‬

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Did you mean: elementary, Elementary (2007 Album by The End), Elementary (Cindy Morgan album), Elementary (2001 Album by Cindy Morgan), Elementary (1977 Album by Wah Wah Watson)

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	meson (Science)
	subatomic particles (Science)
	elementar

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		Dictionary. The American Heritage® Dictionary of the English Language, Fourth Edition Copyright © 2007, 2000 by Houghton Mifflin Company. Updated in 2007. Published by Houghton Mifflin Company. All rights reserved. Read more
		Thesaurus. Roget's II: The New Thesaurus, Third Edition by the Editors of the American Heritage® Dictionary Copyright © 1995 by Houghton Mifflin Company. Published by Houghton Mifflin Company. All rights reserved. Read more
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		Veterinary Dictionary. Saunders Comprehensive Veterinary Dictionary 3rd Edition. Copyright © 2007 by D.C. Blood, V.P. Studdert and C.C. Gay, Elsevier. All rights reserved. Read more
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		Wikipedia. This article is licensed under the GNU Free Documentation License. It uses material from the Wikipedia article "ELEMENTARY". Read more
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elementary

Contents

Definition

Lower elementary recursive functions

Relationship to primitive recursion

Basis for ELEMENTARY

See also

References