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

Whereas ELEMENTARY contains bounded applications of exponentiation (for example, ), PR allows more general hyper operators (for example, tetration) which are not contained in ELEMENTARY. Contents 1 Definition 2 Lower elementary recursive functions 3 Basis for ELEMENTARY 4 Descriptive characterization 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. 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. Composition: applying values from some elementary recursive function as an argument to another elementary recursive function. In f(x1, ..., xn) = h(g1(x1, ..., xn), ..., gm(x1, ..., xn)) is elementary recursive if h is elementary recursive and each gi is elementary recursive. Bounded summation: is elementary recursive if g is elementary recursive. Bounded product: 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 + m,n2,nmod m,2n}, where is the subtraction function defined above.1 Descriptive characterization In descriptive complexity, ELEMENTARY is equal to the class of high order queries.2 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, , where indicates a tower of i exponentiations and is the class of queries that begin with existential quantifiers of ith order and then a formula of (i − 1)th order. See also Elementary function arithmetic Primitive recursive function Grzegorczyk hierarchy EXPTIME References ^ Mazzanti, S., "Plain Bases for Classes of Primitive Recursive Functions", Mathematical Logic Quarterly, 48 (2002) 93-104 ^ Lauri Hella and José María Turull-Torres (2006), "Computing queries with higher-order logics", Theoretical Computer Science (Essex, UK: Elsevier Science Publishers Ltd.) 355 (2): 197–214, ISSN 0304-3975, http://portal.acm.org/citation.cfm?id=1142890.1142897  Rose, H.E., "Subrecursion: Functions and hierarchies", Oxford University Press, New York, USA, 1984. ISBN 0-19-853189-3 v · d · eImportant complexity classes (more) Classes considered feasible DLOGTIME • AC0 • ACC0 • L • SL • RL • NL • NC • SC • P (P-complete) • ZPP • RP • BPP • BQP  Classes suspected to be infeasible UP • NP (NP-complete · NP-hard · co-NP · co-NP-complete) • AM • PH • PP • #P (#P-complete) • IP • PSPACE (PSPACE-complete) Classes considered infeasible EXPTIME • NEXPTIME • EXPSPACE • ELEMENTARY • PR • R • RE • ALL Class hierarchies Polynomial hierarchy • Exponential hierarchy • Grzegorczyk hierarchy • Arithmetic hierarchy Families of complexity classes DTIME • NTIME • DSPACE • NSPACE • Probabilistically checkable proof • Interactive proof system