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161
Ackermann Functions and Transfinite Ordinals
 Applied Mathematics Letters
, 1995
"... Abstract—A set of binary operators are defined and shown to be equivalent to Ackermann functions. The same set of operators are used to develop a notation for writing the sequence of transfinite ordinals. ..."
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Abstract—A set of binary operators are defined and shown to be equivalent to Ackermann functions. The same set of operators are used to develop a notation for writing the sequence of transfinite ordinals.
Ackermann functions of complex argument
, 2008
"... Existence of analytic extension of the fourth Ackermann function A(4, z) to the complex z plane is supposed. This extension is assumed to remain finite at the imaginary axis. On the base of this assumption, the algorithm is suggested for evaluation of this function. The numerical implementation with ..."
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Existence of analytic extension of the fourth Ackermann function A(4, z) to the complex z plane is supposed. This extension is assumed to remain finite at the imaginary axis. On the base of this assumption, the algorithm is suggested for evaluation of this function. The numerical implementation
The Ackermann functions are not optimal, but by how much
 J. Symbolic Logic
"... By taking a closer look at the construction of an Ackermann function we see that between any primitive recursive degree and its Ackermann modification there is a dense chain of primitive recursive degrees. ..."
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By taking a closer look at the construction of an Ackermann function we see that between any primitive recursive degree and its Ackermann modification there is a dense chain of primitive recursive degrees.
Ackermann Function, or What's So Special about 1969?
"... this article can provide a more definitive description of what's going on. To be specific, let N denote the set ..."
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this article can provide a more definitive description of what's going on. To be specific, let N denote the set
The American Mathematical Monthly Unsolved Problems: A modn Ackermann Function, or What's So Special about 1969?
"... computable), but it grows too fast to be primitive recursive (i.e., computable without using dirty tricks like double recursion or the operator \the least n such that"). A kind of inverse for this function (which grows excruciatingly slowlyit makes something like log(log(n)) look like the U.S. ..."
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.S. national debt by comparison) enters into the e±ciency analysis of some important algorithms, such as keeping track of the components of a graph as new edges are added. If we restrict the range of the Ackermann function to a ¯nite set (with a suitable \mod"i¯cation of its de¯nition), then we might
Ackermann and the Superpowers
, 1995
"... Introduction and denitions The \arrow" or \superpower" notation has been introduced by Knuth [1] as a convenient way of expressing very large numbers. It is based on the innite sequence of operators: +; ; "; We shall see that the arrow notation is closely related to the Ackermann ..."
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to the Ackermann function (see, for instance, [2]. 1.1 The Superpowers Let us begin with the following sequence of integer operators. a n = a + a + + a  {z } n a's a " n = a a a  {z } n a's a 2 " n = a " a " " a  {z } n a's In general we dene a
Efficiency of a Good But Not Linear Set Union Algorithm
, 1975
"... Two types of instructmns for mampulating a family of disjoint sets which partitmn a umverse of n elements are considered FIND(x) computes the name of the (unique) set containing element x UNION(A, B, C) combines sets A and B into a new set named C. A known algorithm for implementing sequences of the ..."
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Cited by 321 (15 self)
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of these mstructmns is examined It is shown that, if t(m, n) as the maximum time reqmred by a sequence of m> n FINDs and n 1 intermixed UNIONs, then kima(m, n) _~ t(m, n) < k:ma(m, n) for some positive constants ki and k2, where a(m, n) is related to a functional inverse of Ackermann's functmn
Optimizing Ackermann's Function by Incrementalization
, 2001
"... This paper describes a formal derivation of an optimized Ackermann's function following a general and systematic method based on incrementalization. The method identifies an appropriate input increment operation and computes the function by repeatedly performing an incremental computation at th ..."
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Cited by 7 (3 self)
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This paper describes a formal derivation of an optimized Ackermann's function following a general and systematic method based on incrementalization. The method identifies an appropriate input increment operation and computes the function by repeatedly performing an incremental computation
Sensitivity Analysis of Minimum Spanning Trees in SubInverseAckermann Time
, 2015
"... We present a deterministic algorithm for computing the sensitivity of a minimum spanning tree (MST) or shortest path tree in O(m logα(m,n)) time, where α is the inverseAckermann function. This improves upon a long standing bound of O(mα(m,n)) established by Tarjan. Our algorithms are based on an e ..."
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Cited by 8 (4 self)
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We present a deterministic algorithm for computing the sensitivity of a minimum spanning tree (MST) or shortest path tree in O(m logα(m,n)) time, where α is the inverseAckermann function. This improves upon a long standing bound of O(mα(m,n)) established by Tarjan. Our algorithms are based
ACKERMANN’S FUNCTION AND NEW ARITHMETICAL OPERATIONS
"... Abstract. Ackermann’s function implies the existence of an infinite spectrum of new arithmetical operations (hyperoperations) belonging to the Grzegorczyk Hierarchy. Two of these hyperoperations, tetration and zeration, together with their inverses are given special attention. Tetration (power tower ..."
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Abstract. Ackermann’s function implies the existence of an infinite spectrum of new arithmetical operations (hyperoperations) belonging to the Grzegorczyk Hierarchy. Two of these hyperoperations, tetration and zeration, together with their inverses are given special attention. Tetration (power
Results 1  10
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