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| {{about|the arithmetic concept|the group theory concept|Hyperoperation (group theory)}}
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| In [[mathematics]], the '''hyperoperation sequence'''{{#tag:ref|
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| | |
| Sequences similar to the ''hyperoperation sequence'' have historically been referred to by many names, including: the ''[[Ackermann function]]''<ref name=geisler/> (3-argument), the ''Ackermann hierarchy'',<ref name=friedman/> the ''[[Grzegorczyk hierarchy]]''<ref name=campagnola/><ref name=wirz/> (which is more general), ''Goodstein's version of the Ackermann function'',<ref name=goodstein/> ''operation of the nth grade'',<ref name=bennett/> ''z-fold iterated exponentiation of x with y'',<ref name=black/> ''[[Knuth's up-arrow notation|arrow]] operations'',<ref name=littlewood/> ''reihenalgebra''<ref name=mueller/> and ''hyper-n''.<ref name=geisler /><ref name=mueller /><ref name=munafo/><ref name=robbins/><ref name=galidakis/> The most commonly used of any of these terms is the [[Ackermann function]], whose [[Google search]] gives almost a million hits, mostly referring to the 2-argument function.
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| | group="nb"}}
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| is an infinite [[sequence]] of arithmetic operations (called ''hyperoperations'')<ref name=geisler/><ref name=robbins /><ref name=romerioAck/> that starts with the [[unary operation]] of [[Peano postulates#The axioms|successor]], then continues with the [[binary operation]]s of [[addition]], [[multiplication]] and [[exponentiation]], after which the sequence proceeds with further binary operations extending beyond exponentiation, using [[Operator associativity|right-associativity]]. For the operations beyond exponentiation, the ''n''th member of this sequence is named by [[Reuben Goodstein]] after the [[Numerical prefix|Greek prefix]] of ''n'' suffixed with ''-ation'' (such as [[tetration]], [[pentation]])<ref name=goodstein /> and can be written using ''n''-2 arrows in [[Knuth's up-arrow notation]] (if the latter is properly extended to negative arrow-indices for the first three hyperoperations).
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| Each hyperoperation may be understood [[Recursion (computer science)|recursively]] in terms of the previous one by:
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| :<math>a \uparrow^n b = a \uparrow^{n-1} \left(a \uparrow^{n-1}(...(a \uparrow^{n-1}a))...)\right)</math> with ''b'' occurrences of ''a'' on the right hand side of the equation
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| It may also be defined according to the recursion rule part of the definition, as in Knuth's up-arrow version of the [[Ackermann function]]:
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| :<math>a \uparrow^n b = a \uparrow^{n-1} \left(a \uparrow^n (b-1)\right)</math>
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| This recursion rule is common to many variants of hyperoperations (see [[#Generalization|below]]).
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| == Definition ==
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| The ''hyperoperation sequence'' is the [[sequence]] of [[binary operation]]s <math>H_n: \mathbb{N} \times \mathbb{N} \rightarrow \mathbb{N}\,\!</math> indexed by <math>n \in \mathbb{N}</math>, defined [[recursion|recursively]] as follows:
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| :<math>
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| H_n(a, b) =
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| \begin{cases}
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| b + 1 & \text{if } n = 0 \\
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| a &\text{if } n = 1, b = 0 \\
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| 0 &\text{if } n = 2, b = 0 \\
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| 1 &\text{if } n \ge 3, b = 0 \\
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| H_{n-1}(a, H_n(a, b - 1)) & \text{otherwise}
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| \end{cases}\,\!
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| </math>
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| (Note that for ''n'' = 0, the [[binary operation]] essentially reduces to a [[unary operation]] by ignoring the first argument.)
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| | |
| For ''n'' = 0, 1, 2, 3, this definition reproduces the basic arithmetic operations of [[Peano postulates#The axioms|successor]] (which is a unary operation), [[addition]], [[multiplication]], and [[exponentiation]], respectively, as
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| :<math>H_0(a, b) = b + 1\,\!,</math>
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| :<math>H_1(a, b) = a + b\,\!,</math>
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| :<math>H_2(a, b) = a \cdot b\,\!,</math>
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| :<math>H_3(a, b) = a^{b}\,\!,</math>
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| and for ''n'' ≥ 4 it extends these basic operations beyond exponentiation to what can be written in [[Knuth's up-arrow notation]] as
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| :<math>H_4(a, b) = a\uparrow\uparrow{b}\,\!,</math>
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| :<math>H_5(a, b) = a\uparrow\uparrow\uparrow{b}\,\!,</math>
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| :...
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| :<math>H_n(a, b) = a\uparrow^{n-2}b \text{ for } n \ge 3\,\!,</math>
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| :...
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| Knuth's notation could be extended to negative indices ≥ -2 in such a way as to agree with the entire hyperoperation sequence, except for the lag in the indexing:
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| :<math>H_n(a, b) = a \uparrow^{n-2}b\text{ for } n \ge 0.\,\!</math>
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| The hyperoperations can thus be seen as an answer to the question "what's next" in the [[sequence]]: [[Peano postulates#The axioms|successor]], [[addition]], [[multiplication]], [[exponentiation]], [[tetration]], and so on. Noting that | |
| * <math>a + b = 1 + (a + (b - 1)),\,\!</math>
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| * <math>a \cdot b = a + (a \cdot (b - 1)),\,\!</math>
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| * <math>a ^ b = a \cdot (a ^ {(b - 1)}),\,\!</math>
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| * <math>a\uparrow\uparrow{b}\ = a \uparrow (a\uparrow\uparrow{(b - 1)}),\,\!</math>
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| the relationship between basic arithmetic operations is illustrated, allowing the higher operations to be defined naturally as above. The parameters of the hyperoperation hierarchy are sometimes referred to by their analogous exponentiation term;<ref name=romerioTerms>
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| {{cite web
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| | author= G. F. Romerio
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| | title= Hyperoperations Terminology
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| | url= http://math.eretrandre.org/tetrationforum/attachment.php?aid=208
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| | publisher= [http://math.eretrandre.org/tetrationforum/ Tetration Forum]
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| | date= 2008-01-21
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| | accessdate=2009-04-21}}</ref> so ''a'' is the '''''base''''', ''b'' is the '''''exponent'''''
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| (or ''hyperexponent''),<ref name=galidakis /> and ''n'' is the '''''rank''''' (or ''grade'').<ref name=bennett />
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| In common terms, the hyperoperations are ways of compounding numbers that increase in growth based on the iteration of the previous hyperoperation. The concepts of successor, addition, multiplication and exponentiation are all hyperoperations; the successor operation (producing ''x''+1 from ''x'') is the most primitive, the addition operator specifies the number of times 1 is to be added to itself to produce a final value, multiplication specifies the number of times a number is to be added to itself, and exponentiation refers to the number of times a number is to be multiplied by itself.
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| == Examples ==
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| This is a list of the first seven hyperoperations.
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| | |
| {| class="wikitable"
| |
| |-
| |
| ! ''n''
| |
| ! Operation
| |
| ! Definition
| |
| ! Names
| |
| ! Domain
| |
| |-
| |
| ! 0
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| | <math>1 + b</math>
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| | <math>{ 1 + {\underbrace{1 + 1 + 1 + \cdots + 1}_{b}} }</math>
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| | hyper0, increment, [[successor function|successor]], zeration
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| | ''b'' arbitrary
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| |-
| |
| ! 1
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| | <math>a + b</math>
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| | <math>{ a + {\underbrace{1 + 1 + 1 + \cdots + 1}_{b}} }</math>
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| | hyper1, [[addition]]
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| | arbitrary
| |
| |-
| |
| ! 2
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| | <math>a\cdot b</math>
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| | <math>{ {\underbrace{a + a + a + \cdots + a}} \atop{b} }</math>
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| | hyper2, [[multiplication]]
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| | arbitrary
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| |-
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| ! 3
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| | <math>a \uparrow b = a^b</math>
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| | <math>{ {\underbrace{a \cdot a \cdot a \cdot a \cdot \ldots \cdot a}} \atop{b} }</math>
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| | hyper3, [[exponentiation]]
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| | ''a'' > 0, ''b'' real, or ''a'' non-zero, ''b'' an integer, with some multivalued extensions to complex numbers
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| |-
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| ! 4
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| | <math>a \uparrow\uparrow b</math>
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| | <math>{ {\underbrace{a \uparrow (a \uparrow (a \uparrow \cdots \uparrow a))...)}} \atop{b} }</math>
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| | hyper4, [[tetration]]
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| | ''a'' and ''b'' integers > 0 (with some proposed extensions)
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| |-
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| ! 5
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| | <math>a \uparrow\uparrow\uparrow b</math> or <math>a \uparrow^3 b</math>
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| | <math>{ {\underbrace{a \uparrow\uparrow (a \uparrow\uparrow (\cdots \uparrow\uparrow a))...)}} \atop{b} }</math>
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| | hyper5, [[pentation]]
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| | ''a'' and ''b'' integers > 0
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| |-
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| ! 6
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| | <math>a \uparrow^4 b</math>
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| | <math>{ {\underbrace{a \uparrow^3 (a \uparrow^3 (\cdots \uparrow^3 a))...)}} \atop{b} }</math>
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| | hyper6, hexation
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| | ''a'' and ''b'' integers > 0
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| |}
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| | |
| See also [[Knuth's up-arrow notation#Tables of values|Tables of values]].
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| == History ==
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| One of the earliest discussions of hyperoperations was that of Albert Bennett<ref name=bennett /> in 1914, who developed some of the theory of ''commutative hyperoperations'' (see [[#Commutative hyperoperations|below]]). About 12 years later, [[Wilhelm Ackermann]] defined the function
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| <math>\phi(a, b, n)\,\!</math><ref name=ackOrig>{{cite journal
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| | author=Wilhelm Ackermann
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| | journal=[[Mathematische Annalen]]
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| | title=Zum Hilbertschen Aufbau der reellen Zahlen
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| | year=1928 | volume=99 | pages=118–133
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| | doi=10.1007/BF01459088}}</ref>
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| which somewhat resembles the hyperoperation sequence.
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| | |
| In his 1947 paper,<ref name=goodstein/> [[R. L. Goodstein]] introduced the specific sequence of operations that are now called ''hyperoperations'', and also suggested the Greek names [[tetration]], pentation, hexation, etc., for the extended operations beyond exponentiation (because they correspond to the indices 4, 5, 6, etc.). As a three-argument function, e.g., <math>G(n,a,b) = H_n(a,b)\,\!</math>, the hyperoperation sequence as a whole is seen to be a version of the original [[Ackermann function]] <math>\phi(a,b,n)\,\!</math> — [[computable function|recursive]] but not [[primitive recursive]] — as modified by Goodstein to incorporate the primitive [[successor function]] together with the other three basic operations of arithmetic ([[addition]], [[multiplication]], [[exponentiation]]), and to make a more seamless extension of these beyond exponentiation.
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| The original three-argument [[Ackermann function]] <math>\phi\,\!</math> uses the same recursion rule as does Goodstein's version of it (i.e., the hyperoperation sequence), but differs from it in two ways. First, <math>\phi(a,b,n)\,\!</math> defines a sequence of operations starting from addition (''n'' = 0) rather than the [[successor function]], then multiplication (''n'' = 1), exponentiation (''n'' = 2), etc. Secondly, the initial conditions for <math>\phi\,\!</math> result in <math>\phi(a, b, 3) = a \uparrow\uparrow (b + 1)\,\!</math>,
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| thus differing from the hyperoperations beyond exponentiation.<ref name=black/><ref>{{cite web
| |
| | author= Robert Munafo
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| | title= Versions of Ackermann's Function
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| | url= http://www.mrob.com/pub/math/ln-2deep.html
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| | work= Large Numbers at MROB
| |
| | date= 1999-11-03
| |
| | accessdate=2009-04-17}}</ref><ref>{{cite web
| |
| | author= J. Cowles and T. Bailey
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| | title= Several Versions of Ackermann's Function
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| | url= http://www.cs.utexas.edu/users/boyer/ftp/nqthm/nqthm-1992/examples/basic/peter.events
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| | publisher= Dept. of Computer Science, University of Wyoming, Laramie, WY
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| | date= 1988-09-30
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| | accessdate=2009-04-17}}</ref> The significance of the ''b'' + 1 in the previous expression is that <math>\phi(a,b,3)\,\!</math> = <math>a^{a^{\cdot^{\cdot^{\cdot^a}}}}\,\!</math>, where ''b'' counts the number of ''operators'' (exponentiations), rather than counting the number of ''operands'' ("a"s) as does the ''b'' in <math>a\uparrow\uparrow b\,\!</math>, and so on for the higher-level operations. (See the [[Ackermann function]] article for details.)
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| == Notations ==
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| | |
| This is a list of notations that have been used for hyperoperations.
| |
| | |
| {| class="wikitable"
| |
| |-
| |
| ! Name
| |
| ! Notation equivalent to <math>H_n(a, b)\,\!</math>
| |
| ! Comment
| |
| |-
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| | [[Knuth's up-arrow notation]]
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| | <math>a \uparrow^{n-2} b\,\!</math>
| |
| | Used by Knuth<ref name=knuth>{{cite journal
| |
| | author= Donald E. Knuth
| |
| | title= Mathematics and Computer Science: Coping with Finiteness
| |
| | journal= Science
| |
| |date=Dec 1976
| |
| | volume= 194
| |
| | issue= 4271
| |
| | pages= 1235–1242
| |
| | url= http://www.sciencemag.org/cgi/content/abstract/194/4271/1235
| |
| | accessdate=2009-04-21
| |
| | doi= 10.1126/science.194.4271.1235
| |
| | pmid= 17797067}}</ref> (for ''n'' ≥ 2), and found in several reference books.<ref>{{cite book
| |
| | author = Daniel Zwillinger
| |
| | title = CRC standard mathematical tables and formulae, 31st Edition
| |
| | publisher = CRC Press
| |
| | year = 2002
| |
| | pages= 4
| |
| | isbn = 1-58488-291-3 }}</ref><ref>{{cite book
| |
| | author = Eric W. Weisstein
| |
| | title = CRC concise encyclopedia of mathematics, 2nd Edition
| |
| | publisher = CRC Press
| |
| | year = 2003
| |
| | pages= 127–128
| |
| | isbn = 1-58488-347-2 }}</ref>
| |
| |-
| |
| | Goodstein's notation
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| | <math>G(n, a, b)\,\!</math>
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| | Used by [[Reuben Goodstein]].<ref name=goodstein />
| |
| |-
| |
| | Original [[Ackermann function]]
| |
| | <math>
| |
| \begin{matrix}
| |
| \phi(a, b, n-1) \ \text{ for } 1 \le n \le 3 \\
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| \phi(a, b-1, n-1) \ \text{ for } n > 3
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| \end{matrix}\,\!
| |
| </math>
| |
| | Used by [[Wilhelm Ackermann]].<ref name=ackOrig />
| |
| |-
| |
| | [[Ackermann function|Ackermann–Péter function]]
| |
| | <math>A(n, b - 3) + 3 \ \text{for } a = 2\,\!</math>
| |
| | This corresponds to hyperoperations for base 2.
| |
| |-
| |
| | Nambiar's notation
| |
| | <math>a \otimes^n b\,\!</math>
| |
| | Used by Nambiar<ref>{{cite journal
| |
| | author= K. K. Nambiar
| |
| | title= Ackermann Functions and Transfinite Ordinals
| |
| | journal= Applied Mathematics Letters
| |
| | year= 1995
| |
| | volume= 8
| |
| | issue= 6
| |
| | pages= 51–53
| |
| | doi= 10.1016/0893-9659(95)00084-4}}</ref>
| |
| |-
| |
| | Box notation
| |
| | <math>a {\,\begin{array}{|c|}\hline{\!n\!}\\\hline\end{array}\,} b\,\!</math>
| |
| | Used by Rubtsov and Romerio.<ref name=romerioAck/><ref name=romerioTerms/>
| |
| |-
| |
| | Superscript notation
| |
| | <math>a {}^{(n)} b\,\!</math>
| |
| | Used by Robert Munafo.<ref name=munafo/>
| |
| |-
| |
| | Subscript notation
| |
| | <math>a {}_{(n)} b\,\!</math>
| |
| | Used for lower hyperoperations by Robert Munafo.<ref name=munafo/>
| |
| |-
| |
| | Square bracket notation
| |
| | <math>a[n]b\,\!</math>
| |
| | Used in many online forums; convenient for ASCII.
| |
| |-
| |
| | [[Conway chained arrow notation]]
| |
| | <math>a \to b \to (n-2) </math>
| |
| | Used by [[John Horton Conway]]
| |
| |}
| |
| | |
| == Generalization ==
| |
| | |
| For different initial conditions or different recursion rules, very different operations can occur. Some mathematicians refer to all variants as examples of hyperoperations.
| |
| | |
| In the general sense, a '''hyperoperation hierarchy''' <math>(S,\,I,\,F)</math> is a [[Indexed family|family]] <math>(F_n)_{n \in I}</math> of [[binary operation]]s on <math>S</math>, [[Index set|indexed]] by a set <math>I</math>, such that there exists <math>i, j, k \in I</math> where
| |
| * <math>F_i(a, b) = a + b</math> ([[addition]]),
| |
| * <math>F_j(a, b) = a\cdot b</math> ([[multiplication]]), and
| |
| * <math>F_k(a, b) = a^b</math> ([[exponentiation]]).
| |
| Also, if the last condition is relaxed (i.e. there is no [[exponentiation]]), then we may also include the commutative hyperoperations, described below. Although one could list each hyperoperation explicitly, this is generally not the case. Most variants only include the [[successor function]] (or [[addition]]) in their definition, and redefine [[multiplication]] (and beyond) based on a single recursion rule that applies to all ranks. Since this is part of the definition of the hierarchy, and not a property of the hierarchy itself, it is difficult to define formally.
| |
| | |
| There are many possibilities for hyperoperations that are different from Goodstein's version. By using different initial conditions for <math>F_n(a, 0)</math> or <math>F_n(a, 1)</math>, the iterations of these conditions may produce different hyperoperations above exponentiation, while still corresponding to addition and multiplication. The modern definition of hyperoperations includes <math>F_n(a, 0) = 1</math> for all <math>n \ge 3</math>, whereas the variants below include <math>F_n(a, 0) = a</math>, and <math>F_n(a, 0) = 0</math>.
| |
| | |
| An open problem in hyperoperation research is whether the hyperoperation hierarchy <math>(\mathbb{N}, \mathbb{N}, F)</math> can be generalized to <math>(\mathbb{C}, \mathbb{C}, F)</math>, and whether <math>(\mathbb{C}, F_n)</math> forms a [[quasigroup]] (with restricted domains).
| |
| | |
| === Variant starting from ''a'' ===
| |
| {{Main|Ackermann function}}
| |
| | |
| In 1928, [[Wilhelm Ackermann]] defined a 3-argument function <math>\phi(a, b, n)</math> which gradually evolved into a 2-argument function known as the [[Ackermann function]]. The ''original'' Ackermann function <math>\phi</math> was less similar to modern hyperoperations, because his initial conditions start with <math>\phi(a, 0, n) = a</math> for all <math>n > 2</math>. Also he assigned addition to <math>(n = 0)</math>, multiplication to <math>(n = 1)</math> and exponentiation to <math>(n = 2)</math>, so the initial conditions produce very different operations for tetration and beyond.
| |
| | |
| {| class="wikitable"
| |
| |-
| |
| ! ''n''
| |
| ! Operation
| |
| ! Comment
| |
| |-
| |
| ! 0
| |
| | <math>F_0(a, b) = a + b</math>
| |
| |
| |
| |-
| |
| ! 1
| |
| | <math>F_1(a, b) = a\cdot b</math>
| |
| |
| |
| |-
| |
| ! 2
| |
| | <math>F_2(a, b) = a^b</math>
| |
| |
| |
| |-
| |
| ! 3
| |
| | <math>F_3(a, b) = a \uparrow\uparrow (b + 1)</math>
| |
| | An offset form of [[tetration]]. The iteration of this operation is much different than the [[Iterated function|iteration]] of tetration.
| |
| |-
| |
| ! 4
| |
| | <math>F_4(a, b) = (x \to a \uparrow\uparrow (x + 1))^b(a)</math>
| |
| | Not to be confused with [[pentation]].
| |
| |}
| |
| | |
| Another initial condition that has been used is <math>A(0, b) = 2 b + 1</math> (where the base is constant <math>a=2</math>), due to Rózsa Péter, which does not form a hyperoperation hierarchy.
| |
| | |
| === Variant starting from 0 ===
| |
| In 1984, C. W. Clenshaw and F. W. J. Olver began the discussion of using hyperoperations to prevent computer [[Floating point|floating-point]]
| |
| overflows.<ref name=clenshawolver>{{cite journal
| |
| | author= C.W. Clenshaw and F.W.J. Olver
| |
| | title= Beyond floating point
| |
| | journal= Journal of the ACM
| |
| |date=Apr 1984
| |
| | volume= 31
| |
| | issue= 2
| |
| | pages= 319–328
| |
| | url= http://portal.acm.org/citation.cfm?id=62.322429
| |
| | accessdate=2009-04-21
| |
| | doi= 10.1145/62.322429}}</ref>
| |
| Since then, many other authors<ref name=holmes>{{cite journal
| |
| | author= W. N. Holmes
| |
| | title= Composite Arithmetic: Proposal for a New Standard
| |
| | journal= Computer
| |
| |date=Mar 1997
| |
| | volume= 30
| |
| | issue= 3
| |
| | pages= 65–73
| |
| | url= http://portal.acm.org/citation.cfm?id=620661
| |
| | accessdate=2009-04-21
| |
| | doi= 10.1109/2.573666}}</ref><ref>{{cite web
| |
| | author= R. Zimmermann
| |
| | title= Computer Arithmetic: Principles, Architectures, and VLSI Design
| |
| | url= http://www.iis.ee.ethz.ch/~zimmi/publications/comp_arith_notes.pdf
| |
| | publisher= Lecture notes, Integrated Systems Laboratory, ETH Zürich
| |
| | year= 1997
| |
| | accessdate=2009-04-17}}</ref><ref>{{cite web
| |
| | author= T. Pinkiewicz and N. Holmes and T. Jamil
| |
| | title= Design of a composite arithmetic unit for rational numbers
| |
| | url= http://ieeexplore.ieee.org/xpl/freeabs_all.jsp?arnumber=845571
| |
| | publisher= Proceedings of the IEEE
| |
| | year= 2000
| |
| | pages= 245–252
| |
| | accessdate=2009-04-17}}</ref>
| |
| have renewed interest in the application of hyperoperations to [[Floating point|floating-point]] representation.
| |
| While discussing [[tetration]], Clenshaw ''et al.'' assumed the initial condition <math>F_n(a, 0) = 0</math>, which makes yet another hyperoperation hierarchy. Just like in the previous variant, the fourth operation is very similar to [[tetration]], but offset by one.
| |
| | |
| {| class="wikitable"
| |
| |-
| |
| ! ''n''
| |
| ! Operation
| |
| ! Comment
| |
| |-
| |
| ! 1
| |
| | <math>F_1(a, b) = a + b</math>
| |
| |
| |
| |-
| |
| ! 2
| |
| | <math>F_2(a, b) = a\cdot b = e^{\ln(a) + \ln(b)}</math>
| |
| |
| |
| |-
| |
| ! 3
| |
| | <math>F_3(a, b) = a^b</math>
| |
| |
| |
| |-
| |
| ! 4
| |
| | <math>F_4(a, b) = a \uparrow\uparrow (b - 1)</math>
| |
| | An offset form of [[tetration]]. The iteration of this operation is much different than the [[Iterated function|iteration]] of [[tetration]].
| |
| |-
| |
| ! 5
| |
| | <math>F_5(a, b) = (x \to a \uparrow\uparrow (x - 1))^b(0)</math>
| |
| | Not to be confused with [[pentation]].
| |
| |}
| |
| | |
| === Commutative hyperoperations ===
| |
| | |
| Commutative hyperoperations were considered by Albert Bennett as early as 1914,<ref name=bennett /> which is possibly the earliest remark about any hyperoperation sequence. Commutative hyperoperations are defined by the recursion rule
| |
| :<math>F_{n+1}(a, b) = \exp(F_n(\ln(a), \ln(b)))</math>
| |
| which is symmetric in ''a'' and ''b'', meaning all hyperoperations are commutative. This sequence does not contain [[exponentiation]], and so does not form a hyperoperation hierarchy.
| |
| | |
| {| class="wikitable"
| |
| |-
| |
| ! ''n''
| |
| ! Operation
| |
| ! Comment
| |
| |-
| |
| ! 0
| |
| | <math>F_0(a, b) = \ln(e^{a} + e^{b})</math>
| |
| |
| |
| |-
| |
| ! 1
| |
| | <math>F_1(a, b) = a + b</math>
| |
| |
| |
| |-
| |
| ! 2
| |
| | <math>F_2(a, b) = a\cdot b = e^{\ln(a) + \ln(b)}</math>
| |
| | This is due to the [[Logarithm#Properties of the logarithm|properties of the logarithm]].
| |
| |-
| |
| ! 3
| |
| | <math>F_3(a, b) = a^{\ln(b)} = e^{\ln(a)\ln(b)}</math>
| |
| | A [[Commutativity|commutative]] form of [[exponentiation]].
| |
| |-
| |
| ! 4
| |
| | <math>F_4(a, b) = e^{e^{\ln(\ln(a))\ln(\ln(b))}}</math>
| |
| | Not to be confused with [[tetration]].
| |
| |}
| |
| | |
| === Balanced hyperoperations ===
| |
| | |
| Balanced hyperoperations, first considered by Clément Frappier in 1991,<ref name=frappier>{{cite journal
| |
| | author= C. Frappier
| |
| | title= Iterations of a kind of exponentials
| |
| | journal= Fibonacci Quarterly
| |
| | year= 1991
| |
| | volume= 29
| |
| | issue= 4
| |
| | pages= 351–361}}</ref> are based on the iteration of the function <math>x^x</math>, and are thus related to [[Steinhaus-Moser notation]]. The recursion rule used in balanced hyperoperations is
| |
| :<math>F_{n+1}(a, b) = (x \to F_n(x, x))^{\log_2(b)}(a)</math>
| |
| which requires continuous [[iteration]], even for [[integer]] b.
| |
| | |
| {| class="wikitable"
| |
| |-
| |
| ! ''n''
| |
| ! Operation
| |
| ! Comment
| |
| |-
| |
| ! 0
| |
| |
| |
| | Rank 0 does not exist.<ref group="nb">
| |
| If there was a rank 0 balanced hyperoperation <math>f(a, b)</math>, then addition would be <math>a + b = (x \to f(x, x))^{\log_2(b)}(a)</math>. Substituting <math>b = 1</math> in this equation gives <math>a + 1 = (x \to f(x, x))^{0}(a) = a</math> which is a contradiction.
| |
| </ref>
| |
| |-
| |
| ! 1
| |
| | <math>F_1(a, b) = a + b</math>
| |
| |
| |
| |-
| |
| ! 2
| |
| | <math>F_2(a, b) = a\cdot b = a 2^{\log_2(b)}</math>
| |
| |
| |
| |-
| |
| ! 3
| |
| | <math>F_3(a, b) = a^b = a^{2^{\log_2(b)}}</math>
| |
| | This is [[exponentiation]].
| |
| |-
| |
| ! 4
| |
| | <math>F_4(a, b) = (x \to x^x)^{\log_2(b)}(a)</math>
| |
| | Not to be confused with [[tetration]].
| |
| |}
| |
| | |
| === Lower hyperoperations ===
| |
| | |
| An alternative for these hyperoperations is obtained by evaluation from left to right. Since
| |
| * <math>a+b = (a+(b-1))+1</math>
| |
| * <math>a\cdot b = (a\cdot (b-1))+a</math>
| |
| * <math>a^b = (a^{(b-1)})\cdot a</math>
| |
| define (with ° or subscript)
| |
| <math>a_{(n+1)}b = (a_{(n+1)}(b-1))_{(n)}a</math>
| |
| with
| |
| <math>a_{(1)}b = a+b</math>,
| |
| <math>a _ {(2)} 0 = 0</math>, and
| |
| <math>a _ {(n)} 0 = 1</math>
| |
| for
| |
| <math>n>2</math>
| |
| | |
| But this suffers a kind of collapse,
| |
| failing to form the "power tower" traditionally expected of hyper4:
| |
| <math>a_{(4)}b = a^{(a^{(b-1)})}</math>
| |
| | |
| How can <math>a^{(n)}b</math> be so different from <math>a_{(n)}b</math> for ''n>3''? This is because of a [[symmetry]] called [[associativity]] that's ''defined into'' + and × (see [[field (mathematics)|field]]) but which ^ lacks. Let's demonstrate this lack of associativity in exponentiation, which differentiates the higher and lower hyperoperations. Take for example the product: <math>2\cdot3\cdot4</math>. This expression unambiguously evaluates to 24. However, if we replace the multiplication symbols with those of exponentiation, the expression becomes ambiguous. Do we mean <math>(2^3)^4</math> or <math>2^{(3^4)}</math>? There is a big difference, since the former expression can be rewritten as <math>2^{12}</math> while the latter is <math>2^{81}</math>. In other words, left associative folds of the exponential operator on sequences do not coincide with right associative folds, the latter usually resulting in larger numbers. It is more apt to say the two ''(n)''s were decreed to be the same for ''n<4''. (On the other hand, one can object that the field operations were defined to mimic what had been "observed in nature" and ask why "nature" suddenly objects to that symmetry…)
| |
| | |
| The other degrees do not collapse in this way, and so this family has some interest of its own as '''lower''' (perhaps '''lesser''' or '''inferior''') hyperoperations. With hyperfunctions greater than three, it is also '''lower''' in the sense that the answers you get are actually often a lot lower than the answers you get when using the standard method. | |
| | |
| {| class="wikitable"
| |
| |-
| |
| ! ''n''
| |
| ! Operation
| |
| ! Comment
| |
| |-
| |
| ! 0
| |
| | <math>b + 1</math>
| |
| | increment, successor, zeration
| |
| |-
| |
| ! 1
| |
| | <math>F_1(a, b) = a + b</math>
| |
| |
| |
| |-
| |
| ! 2
| |
| | <math>F_2(a, b) = a\cdot b</math>
| |
| |
| |
| |-
| |
| ! 3
| |
| | <math>F_3(a, b) = a^b</math>
| |
| | This is [[exponentiation]].
| |
| |-
| |
| ! 4
| |
| | <math>F_4(a, b) = a^{(a^{(b-1)})}</math>
| |
| | Not to be confused with [[tetration]].
| |
| |-
| |
| ! 5
| |
| | <math>F_5(a, b) = (x \to x^{x^{(a-1)}})^{b-1}(a)</math>
| |
| | Not to be confused with [[pentation]].
| |
| |}
| |
| | |
| ==Coincidence of hyperoperations==
| |
| | |
| Hyperoperations <math> H_i </math> and <math> H_j </math> are said to coincide on <math> (a, b) </math> when <math> H_i(a, b) = H_j(a, b) </math>. For example, for all <math> i, j > 1 </math>, i.e. all hyperoperations above addition, <math> H_i(a, 1) = H_j(a, 1) = a </math>. Similarly, <math> H_i(1, a) = H_j(1, a) = 1 </math>, but in this case both addition and multiplication must be excluded. A point at which ''all'' hyperoperations coincide (excluding the unary successor function which does not really belong as a binary operation) is (2, 2) i.e. for all <math> i = 1, 2, ... </math> we have that <math>H_i(2, 2) = 4 </math>. There is a connection between the arity of these functions i.e. two and this point of coincidence: since the second argument of a hyperoperation is the length of the list on which to fold the previous operation, and this is 2, we get that the previous operation is folded over a list of length two, which amounts to applying it to the pair represented by that list. Also, since the first argument is itself 2, and this is duplicated in the recursion, we arrive again at the pair (2, 2) with each recursion. This happens until we get to 2 + 2 = 4.
| |
| | |
| To be more precise, we have that <math> 2 \uparrow^n 2 </math> = <math> fold (\uparrow^{n - 1}, [2, 2]) </math> = <math> 2 \uparrow^{n - 1} 2 </math>. Note that the unit of <math> \uparrow^{n - 1} </math> need not be supplied to fold when the list has length > 1. To demonstrate this recursion by means of an example we take <math> 2^2 </math>, which is two by itself twice i.e. <math> 2\cdot2 </math>. This, in turn is two plus itself twice i.e. <math> 2 + 2 </math>. At +, the recursion terminates and we are left with four.
| |
| | |
| ==See also==
| |
| * [[Large numbers]]
| |
| | |
| == Notes ==
| |
| <references group="nb" />
| |
| | |
| == References ==
| |
| {{reflist|refs=
| |
| | |
| ,<ref name=bennett>{{cite journal
| |
| | author= Albert A. Bennett
| |
| | title= Note on an Operation of the Third Grade
| |
| | journal= Annals of Mathematics. Second Series
| |
| |date=Dec 1915
| |
| | volume= 17
| |
| | issue= 2
| |
| | pages= 74–75
| |
| | jstor= 2007124
| |
| | doi= 10.2307/2007124}}</ref>
| |
| | |
| ,<ref name=black>{{cite web
| |
| | author= Paul E. Black
| |
| | title= Ackermann's function
| |
| | url= http://www.itl.nist.gov/div897/sqg/dads/HTML/ackermann.html
| |
| | work= [http://www.itl.nist.gov/div897/sqg/dads/ Dictionary of Algorithms and Data Structures]
| |
| | publisher= U.S. National Institute of Standards and Technology (NIST)
| |
| | date= 2009-03-16
| |
| | accessdate=2009-04-17}}</ref>
| |
| | |
| <ref name=campagnola>{{cite journal
| |
| | author= Manuel Lameiras Campagnola and [[Cris Moore|Cristopher Moore]] and José Félix Costa
| |
| | title= Transfinite Ordinals in Recursive Number Theory
| |
| | journal= Journal of Complexity
| |
| |date=Dec 2002
| |
| | volume= 18
| |
| | issue= 4
| |
| | pages= 977–1000
| |
| | url= http://www.sciencedirect.com/science?_ob=ArticleURL&_udi=B6WHX-482XFM6-4&_user=10&_rdoc=1&_fmt=&_orig=search&_sort=d&view=c&_acct=C000050221&_version=1&_urlVersion=0&_userid=10&md5=f4e3ffc2a28e8abd16cde236197fd487
| |
| | accessdate=2009-04-17
| |
| | doi=10.1006/jcom.2002.0655}}</ref>
| |
| | |
| <ref name=friedman>{{cite journal
| |
| | author= Harvey M. Friedman
| |
| | title= Long Finite Sequences
| |
| | journal= Journal of Combinatorial Theory, Series A
| |
| |date=Jul 2001
| |
| | volume= 95
| |
| | issue= 1
| |
| | pages= 102–144
| |
| | url= http://www.sciencedirect.com/science?_ob=ArticleURL&_udi=B6WHS-45RFJ9C-5&_user=10&_rdoc=1&_fmt=&_orig=search&_sort=d&view=c&_version=1&_urlVersion=0&_userid=10&md5=8097ac57c9dbe05b99fef6a95309f1df
| |
| | accessdate=2009-04-17
| |
| | doi= 10.1006/jcta.2000.3154}}</ref>
| |
| | |
| <ref name=galidakis>{{cite web
| |
| | author= I. N. Galidakis
| |
| | title= Mathematics
| |
| | url= http://ioannis.virtualcomposer2000.com/math/index.html
| |
| | year= 2003
| |
| | accessdate=2009-04-17}}</ref>
| |
| | |
| <ref name=geisler>{{cite web
| |
| | author= Daniel Geisler
| |
| | title= What lies beyond exponentiation?
| |
| | url= http://tetration.org/
| |
| | year= 2003
| |
| | accessdate=2009-04-17}}</ref> (3-argument), the
| |
| | |
| <ref name=goodstein>{{cite journal
| |
| | author= R. L. Goodstein
| |
| | title= Transfinite Ordinals in Recursive Number Theory
| |
| | journal= Journal of Symbolic Logic
| |
| |date=Dec 1947
| |
| | volume= 12
| |
| | issue= 4
| |
| | pages= 123–129
| |
| | doi= 10.2307/2266486
| |
| | jstor= 2266486}}</ref>
| |
| | |
| ,<ref name=littlewood>{{cite journal
| |
| | author= J. E. Littlewood
| |
| | title= Large Numbers
| |
| | journal= Mathematical Gazette
| |
| |date=Jul 1948
| |
| | volume= 32
| |
| | issue= 300
| |
| | pages= 163–171
| |
| | doi= 10.2307/3609933
| |
| | jstor= 3609933}}</ref>
| |
| | |
| <ref name=mueller>{{cite web
| |
| | author= Markus Müller
| |
| | title= Reihenalgebra
| |
| | url= http://www.mpmueller.net/reihenalgebra.pdf
| |
| | year= 1993
| |
| | accessdate=2009-04-17}}</ref>
| |
| | |
| <ref name=munafo>{{cite web
| |
| | author= Robert Munafo
| |
| | title= Inventing New Operators and Functions
| |
| | url= http://www.mrob.com/pub/math/largenum-3.html
| |
| | work= Large Numbers at MROB
| |
| | date= November 1999
| |
| | accessdate=2009-04-17}}</ref>
| |
| | |
| <ref name=robbins>{{cite web
| |
| | author= A. J. Robbins
| |
| | title= Home of Tetration
| |
| | url= http://tetration.itgo.com/index.html
| |
| | date= November 2005
| |
| | accessdate=2009-04-17}} {{Dead link|date=October 2009}}</ref>
| |
| | |
| <ref name=romerioAck>{{cite web
| |
| | author= C. A. Rubtsov and G. F. Romerio
| |
| | title= Ackermann's Function and New Arithmetical Operation
| |
| | url= http://forum.wolframscience.com/showthread.php?s=&threadid=956
| |
| | date= December 2005
| |
| | accessdate=2009-04-17}}</ref>
| |
| | |
| <ref name=wirz>{{cite web
| |
| | author= Marc Wirz
| |
| | title= Characterizing the Grzegorczyk hierarchy by safe recursion
| |
| | url= http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.42.3374
| |
| | publisher= CiteSeer
| |
| | year= 1999
| |
| | accessdate=2009-04-21}}</ref>
| |
| }}
| |
| | |
| {{Navbox
| |
| | name=Hyperoperations
| |
| | title=Hyperoperations
| |
| | group1 = Primary
| |
| | list1 = [[Addition]]{{·}} [[Multiplication]]{{·}} [[Exponentiation]]{{·}} [[Tetration]]{{·}} [[Pentation]]
| |
| | group2 = Inverse, group 1
| |
| | list2 = [[Subtraction]]{{·}} [[Division (mathematics)|Division]]{{·}} [[nth root]]{{·}} [[Super-root]]
| |
| | group3 = Inverse, group 2
| |
| | list3 = [[Subtraction]]{{·}} [[Division (mathematics)|Division]]{{·}} [[Logarithm]]{{·}} [[Super-logarithm]]
| |
| }}
| |
| | |
| [[Category:Arithmetic]]
| |
| [[Category:Large numbers]]
| |