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	PlanetMath: Ackermann function
		Ackermann's function is an example of a recursive function that is not primitive recursive, but is instead
		Later this was simplified by Rosza Peter to a function of two variables, similar to the one given above.
		This is version 6 of Ackermann function, born on 2002-03-23, modified 2004-03-30.
	planetmath.org /encyclopedia/AckermannFunction.html (161 words)

	Ackermann function - Wikipedia, the free encyclopedia
		Ackermann originally considered a function A(m, n, p) of three variables, the p-fold iterated exponentiation of m with n, or m → n → p as expressed using the Conway chained arrow notation.
		Ackermann proved that A is a recursive function, a function a computer with unbounded memory can calculate, but it is not a primitive recursive function, a class of functions including almost all familiar functions such as addition and factorial.
		The Ackermann function, due to its definition in terms of extremely deep recursion, can be used as a benchmark of a compiler's ability to optimize recursion.
	en.wikipedia.org /wiki/Ackermann_function (1936 words)

	Ackermann biography
		Ackermann received his doctoral degree in 1925 with a thesis Begründung des "tertium non datur" mittels der Hilbertschen Theorie der Widerspruchsfreiheit written under Hilbert and was a proof of the consistency of arithmetic without induction.
		Ackermann was also the main contributor to the development of the logical system known as the epsilon calculus, originally due to Hilbert.
		A(x, y, z) was simplified to a function P(x, y) of 2 variables by Rozsa Peter whose initial condition was simplified by Raphael Robinson.
	www-groups.dcs.st-and.ac.uk /~history/Biographies/Ackermann.html (249 words)

	CMPSCI 601 Q&A for HW#4, Spring 2004
		In the mu-operator, the function f was a primitive recursive function?
		So you have these functions which by the IH are g.r., and you want to show that these new functions are g.r., probably by forming them out of the loop-body functions using the mu-operator.
		The mu operator acts on a function of arity k+1 and returns a function of arity k, much as a quantifier acts on a formula with k+1 variables and creates a formula with k free variables.
	www.cs.umass.edu /~barring/cs601/qa/4.html (2067 words)

	UMBC CMSC331 Homework Fall 2004
		Ackermann function is a simple, easily defined recursive function that takes two integers as arguments and returns an integer.
		Most values of the Ackermann function are so large that they cannot be feasibly computed, and in fact their decimal expansions cannot even be stored in the entire physical universe.
		Ackermann's function is mainly used in the theory of computation as an example of a recursive function that is not primitive recursive and therefore difficult to compute.
	www.csee.umbc.edu /331/current/homework/hw5 (801 words)

	Hilbert Levitz's Research Interests (Site not responding. Last check: 2007-10-09)
		A recursive universal function for the primitive recursive functions is constructed.
		Ackermann's function is the classical example of a total computable function which is not primitive recursive.
		A proof of the existence of a recursive universal function for the class of primitive recursive functions is given.
	www.cs.fsu.edu /~levitz/research.html (432 words)

	Partial recursive functions (Site not responding. Last check: 2007-10-09)
		Primitive Recursive Functions are a subset of Total Recursive Functions with the restriction that only primitive recursion is used a finite number of times and recursion uses zero and the successor function.
		This can be extended to partial recursive functions over the integers and over the rational numbers, ratio of two integers, but can not be extended to the set of real numbers.
		y=f(x) is not a partial recursive function when x and y are from the set of real numbers and f(x) is defined as the square root of x, also written as the value of y that satisfies y2 = x or y2 - x = 0.
	cs.wwc.edu /~aabyan/Logic/Book/book/node151.html (294 words)

	Thoughts on the Ackermann Function
		The Ackermann function is a killer: It looks very simple, yet it plays a major role in computer science and computational complexity theory.
		If you think of the first function as the initialization for line 0 of the array, it becomes obvious that this function doesn't contain any calculations of the actual values, only of their indices.
		Mathematically speaking, that's true, but suppose we want to write a program that calculates the value of the Ackermann function, and let C be the chosen language.
	kosara.net /thoughts/ackermann.html (845 words)

	Large Numbers at MROB
		This function, appropriately enough, is also the "successor" function used as the primitive computational element in algorithms defined in the Church theory of computation, which includes the original Ackermann function.
		A recursive function first described by W. Ackermann in 1928 to demonstrate a property of computability in the field of mathematics, and also used more recently as an example of pathological recursive functions in computer science.
		While it is true that a1(x) grows just as fast as the ack-h() function, and therefore serves as a good way of defining large numbers as a function of one variable, actually computing those numbers involves the recursive definition of the function.
	home.earthlink.net /~mrob/pub/math/largenum-3.html (2346 words)

	Dotzel: The Ackermann function (12 KB)
		The so-called Ackermann function was developed by Wilhelm Ackermann (1896 to 1962) in the year 1928.
		Given any function f(x) or f(x,y) which is PR, there exists a value for the Ackermann function's first argument for which its function value is always greater than the value of f.
		The first argument x determines the complexity or kind of the function, the Ackermann function represents, whereas the second argument y is the iteration count for the function determined by the first argument.
	www.modulaware.com /mdlt08.htm (955 words)

	Ackermann (print-only) (Site not responding. Last check: 2007-10-09)
		In 1928, Ackermann observed that A(x, y, z), the z-fold iterated exponentiation of x with y, is an example of a recursive function which is not primitive recursive.
		It is the latter which occurs as Ackermann's function in today's textbooks.
		Among Ackermann's later work are consistency proofs for set theory (1937), full arithmetic (1940) and type free logic (1952).
	www-groups.dcs.st-and.ac.uk /history/Printonly/Ackermann.html (255 words)

	Wilhelm Ackermann - Wikipedia, the free encyclopedia
		Wilhelm Friedrich Ackermann (March 29, 1896, Herscheid municipality, Germany – December 24, 1962 Lüdenscheid, Germany) was a German mathematician best known for the Ackermann function, an important example in the theory of computation.
		Ackermann was awarded the Ph.D. by the University of Goettingen in 1925 for his thesis Begründung des "tertium non datur" mittels der Hilbertschen Theorie der Widerspruchsfreiheit, which was a consistency proof of arithmetic apparently without full Peano induction (although it did use e.g.
		Ackermann went on to construct consistency proofs for set theory (1937), full arithmetic (1940), type-free logic (1952), and a new axiomatization of set theory (1956).
	en.wikipedia.org /wiki/Wilhelm_Ackermann (250 words)

	Nested Recursion
		This note was prompted by Nick Forde's version of a Joy operator which computes the Ackermann function which is one of the few well-known functions that use nested recursion.
		This note gives definitions of (the common simplification of) Ackermann's function, of McCarthy's 91-function, of a function producing a Hamiltonian path over a hypercube, of a function producing Gray sequences, and of the Hanoi problem.
		The Ackermann function widely quoted in the literature is actually a simplification of the function originally defined by Ackermann to show that there are terminating recursive functions that are not primitive recursive.
	www.latrobe.edu.au /philosophy/phimvt/joy/jp-nestrec.html (2595 words)

	Ackermann's function (Site not responding. Last check: 2007-10-09)
		Note: In 1928, Wilhelm Ackermann observed that A(x,y,z), the z-fold iterated exponentiation of x with y, is an example of a recursive function which is not primitive recursive.
		A(x,y,z) was simplified to a function of 2 variables by Rózsa Péter in 1935.
		Many people have given other versions of Ackermann's function, some of which are not simply a restating of this one.
	www.nist.gov /dads/HTML/ackermann.html (165 words)

	[No title]
		It defines the Ackermann function in terms of the Ackermann generalized exponential.
		The Ackermann hierarchy is a sequence of functions {f_i(x)} defined inductively by f_i(1)=2 and f_{i+1}(x+1)=f_i(f_{i+1}(x)) and f_1(x)=2x.
		Now the Ackermann *function* is a function of one variable defined by Ackermann(n)=f_n(n).
	www.math.niu.edu /~rusin/known-math/99/ackermann (463 words)

	EACSL (Site not responding. Last check: 2007-10-09)
		Eligible for the 2007 Ackermann Award are PhD dissertations in topics specified by the EACSL and LICS conferences, which were formally accepted as PhD theses at a university or equivalent institution between 1.1.2005 and 31.12.2006.
		Wilhelm Ackermann was born on March 29, 1896 and died on December 24, 1962.
		Ackermann was also the main contributor to the logical system known as the epsilon calculus, originally due to Hilbert.
	www.dimi.uniud.it /~eacsl/award.html (506 words)

	Ackermann
		/* * Filename: * * ackermann.c * * Description: * * Ackermann's function "is an example of a recursive function which * is not primitive recursive".
		It is interesting from the point of * view of benchmarking because it "grows faster than any primitive * recursive function" and gives us a lot of nested function calls * for little effort.
		* * Credits: * * Ackermann's function is named for Wilhelm Ackermann, a * mathematical logician who worked Germany during the first half * if the 20th century.
	www.xgc.com /benchmarks/ackermann_c.htm (158 words)

	Ackermann Functions (Site not responding. Last check: 2007-10-09)
		An Ackermann function has the characteristic that the length of the sequence of numbers generated by the function cannot be computed directly from the input value.
		This Ackermann has the characteristic that it eventually converges on 1.
		A few examples follow in which the starting value is shown in square brackets followed by the sequence of values that are generated, followed by the length of the sequence in curly braces:
	acm.uva.es /p/v3/371.html (312 words)

	Wilhelm Ackermann (Site not responding. Last check: 2007-10-09)
		Wilhelm Ackermann received his doctoral degree in 1925 with a thesis written under Hilbert.
		In 1928, Ackermann observed that A(x,y,z), the z-fold iterated exponentiation of x with y, is an example of a recursive function which is not primitive recursive.
		A(x,y,z) was simplified to a function P(x,y) of 2 variables by Rosza Peter whose initial condition was further simplified by Raphael Robinson.
	www.stetson.edu /~efriedma/periodictable/html/Ac.html (172 words)

	Optimizing Ackermann's Function by Incrementalization (Site not responding. Last check: 2007-10-09)
		This paper describes a formal derivation of an optimized Ackermann's function following a general and systematic method based on incrementalization.
		This eliminates repeated subcomputations in executions that follow the straightforward recursive definition of Ackermann's function, yielding an optimized program that is drastically faster and takes extremely little space.
		This case study uniquely shows the power and limitation of the incrementalization method, as well as both the iterative and recursive nature of computation underlying the optimized Ackermann's function.
	www.cs.sunysb.edu /~stoller/PEPM2003.html (101 words)

	Large Numbers -- Long Notes at MROB
		The Meyer and Ritchie version (1967) of Ackermann's function may be defined recursively on the nonnegative integers by
		The Z. Manna version (1974) of Ackermann's function may be defined recursively on the nonnegative integers by
		Since everyone else has a version of Ackermann's function, it should cause little or no harm if we also define a version.
	home.earthlink.net /~mrob/pub/math/ln-2deep.html (558 words)

	[No title] (Site not responding. Last check: 2007-10-09)
		; The Ackermann function is the fastest-growing simple recursion function.
		; In magnitude, G and ackermann are essentially identical.
		n) ((ackermann (sub1 m)) 1)] [else ((ackermann (sub1 m)) ((ackermann m) (sub1 n)))])))) ;"G" function as described in class.
	www.cs.indiana.edu /usr/local/www/classes/c311/ackermann.ss (117 words)

	CSC 4170 Ackermann's Function
		Ackermann's function is an example of a function that is mu-recursive but not primitive recursive.
		Ackermann's function is one of the few things I actually remember from the recursive function theory course I took many long years ago.
		Instead of sitting there doodling, bring in a copy of Ackermann's function and see how far you can get with it.
	www.seas.upenn.edu /~cit596/notes/dave/church7.html (167 words)

	NSDL Metadata Record -- Ackermann Function -- from MathWorld
		The Ackermann function is the simplest example of a well defined total function which is computable but not primitive recursive, providing a counterexample to the belief in the early 1900s that every computable function was also primitive recursive (Dotzel 1991).
		It grows faster than an exponential function, or even a multiple exponential function.
		The Ackermann function A(x,y) is defined for integer x and y by A(x,y)\equiv\cases{ y+1 & if x=0\cr A(x-1, 1) & if y=0\cr A(x-1, A(x,...
	nsdl.org /mr/696894 (169 words)

	More on Ackermanns Function
		A few months ago, I posted an algorithm for Ackermann's function, but I did not give a very good background or explanation for it.
		This function is very important in computer science because it helps answer the question of what can and cannot be computed on a computer.
		The special properties of the Ackermann function are a consequence of it's phenominal rate of growth.
	www.fortunecity.com /skyscraper/false/780/ack.html (151 words)

	[No title]
		They are used by compiler vendors as a standard for compiler assessment, and are selected to cover interesting aspects of the compiler and target computer.
		Using a clever yet simple formula, the Ackermann benchmark is able to execute millions of non-redundant calls, while using a modest amount of stack space.
		These programs were run on the ERC32 simulator, with a 10 MHz clock, zero wait states on data read and write, and one wait state on instruction fetch.
	www.xgc.com /benchmarks/benchmarks.htm (358 words)

	Ackermann's Function (Site not responding. Last check: 2007-10-09)
		Computes Ackermann's function, A(x, y), for x = 3 and y = N. This function is a simple example of a total function that is computable but not primitive recursive.
		function calls, and reaches a recursive depth of 2
		In particular, it tests your code generation for function calls.
	www.cs.cornell.edu /Courses/cs412/2001sp/hw/bench/ack.html (113 words)

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