the Ackermann function, named after Wilhelm Ackermann, is one of the simplest and earliest-discovered examples of a total computable function that is...

work in mathematical logic and the Ackermann function, an important example in the theory of computation. Ackermann was born in Herscheid, Germany, and...

grows phenomenally fast as a function of n, far faster than any primitive recursive function or the Ackermann function, for example.[citation needed]...

recursive functions. However, not every total recursive function is a primitive recursive function—the most famous example is the Ackermann function. Other...

Wilhelm Ackermann Ackermann function Ackermann ordinal Ackermann set theory Ackermann steering geometry, in mechanical engineering Ackermann's formula...

3=g_{3}(2)=g_{2}^{2}(1)=g_{2}(g_{2}(1))=f^{f(1)}(1)=f^{a^{b}}(1)} The Ackermann function can be expressed using Conway chained arrow notation: A ( m , n )...

total time O(mα(n)), where α(n) is the extremely slow-growing inverse Ackermann function. Disjoint-set forests do not guarantee this performance on a per-operation...

function, which is provable total but not primitive recursive, is the Ackermann function: since it is recursively defined, it is indeed easy to prove its computability...

faster than exponential functions, but much more slowly than double exponential functions. However, tetration and the Ackermann function grow faster. See Big...

primitive recursive function is total recursive, but not all total recursive functions are primitive recursive. The Ackermann function A(m,n) is a well-known...

Lamé function Mathieu function Mittag-Leffler function Painlevé transcendents Parabolic cylinder function Arithmetic–geometric mean Ackermann function: in...

Turing-complete language and can express all computable functions. For example, it can express the Ackermann function, which (not being primitive recursive) cannot...

Sudan function is an example of a function that is recursive, but not primitive recursive. This is also true of the better-known Ackermann function. In...

Wikimedia Commons has media related to tetration. Ackermann function Big O notation Double exponential function Hyperoperation Iterated logarithm Symmetric...

rule part of the definition, as in Knuth's up-arrow version of the Ackermann function: a [ n ] b = a [ n − 1 ] ( a [ n ] ( b − 1 ) ) , n ≥ 1 {\displaystyle...

α(V)) for this loop, where α is the extremely slowly growing inverse Ackermann function. This part of the time bound is much smaller than the time for the...

the pentation function may also be obtained from the values in the fourth row of the table of values of a variant of the Ackermann function: if A ( n ,...

recursive function is dominated by fω, which is a variant of the Ackermann function. For n ≥ 3, the set E n {\displaystyle {\mathcal {E}}^{n}} in the...

ISBN 978-3-319-68397-3. Halmos, Chapter 11 Rubtsov, C.A.; Romerio, G.F. (2004). "Ackermann's Function and New Arithmetical Operations" (PDF). Paul R. Halmos (1968). Naive...

increasing) function; in particular, Ackermann function. Simple function: a real-valued function over a subset of the real line, similar to a step function. Measurable...

include divide-and-conquer algorithms such as Quicksort, and functions such as the Ackermann function. All of these algorithms can be implemented iteratively...

enormously large – bounds that grow exponentially, or even as fast as the Ackermann function are not uncommon. In some small niche cases, upper and lower bounds...

recursive functions such as the Ackermann function. It dominates every computable function provably total in Peano arithmetic, which includes functions such...

mathematical functions, the analysis of some puzzles and games, and some pathological examples which can be modelled as the Ackermann function. A Latin square...

the Ackermann function. Raphael M. Robinson called functions of two natural number variables G(n, x) double recursive with respect to given functions, if...

by a function involving the first four hyperoperators;. Then, f ω ( x ) {\displaystyle f_{\omega }(x)} is comparable to the Ackermann function, f ω +...

on Borůvka's and runs in O(E α(E,V)) time, where α is the inverse Ackermann function. These randomized and deterministic algorithms combine steps of Borůvka's...

edge additions in O(m α(m, n)) total time, where α is the inverse Ackermann function. This time bound is proved to be optimal. Uzi Vishkin and Robert Tarjan...

extremely large numbers: Knuth's up-arrow notation/hyperoperators/Ackermann function, including tetration Conway chained arrow notation Steinhaus-Moser...

the rapidly growing Ackermann function A(n, n). (In fact, f ( n ) > A ( n , n ) {\displaystyle f(n)>A(n,n)} for all n.) The function f can also be expressed...