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Arrow notation or up-arrow notation is a widely used notation for the hyper operators, devised by Donald Knuth in 1976 to represent large numbers. It is defined by the following rules: \begin{eqnarray} a \uparrow^1 b &=& a^b \\ a \uparrow^n 1 &=& a \\ a \uparrow^{n + 1} (b + 1) &=& a \uparrow^n (a \uparrow^{n + 1} b) \\ \end{eqnarray} Specifically, \(a \uparrow b\) is exponentiation, \(a \uparrow\uparrow b\) is tetration, \(a \uparrow\uparrow\uparrow b\) is pentation, and so forth. In ASCII, these are written a^b, a^^b, a^^^b, ...

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rdfs:label
  • Arrow notation
rdfs:comment
  • Arrow notation or up-arrow notation is a widely used notation for the hyper operators, devised by Donald Knuth in 1976 to represent large numbers. It is defined by the following rules: \begin{eqnarray} a \uparrow^1 b &=& a^b \\ a \uparrow^n 1 &=& a \\ a \uparrow^{n + 1} (b + 1) &=& a \uparrow^n (a \uparrow^{n + 1} b) \\ \end{eqnarray} Specifically, \(a \uparrow b\) is exponentiation, \(a \uparrow\uparrow b\) is tetration, \(a \uparrow\uparrow\uparrow b\) is pentation, and so forth. In ASCII, these are written a^b, a^^b, a^^^b, ...
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Author
  • Donald Knuth
Year
  • 1976(xsd:integer)
notation
  • \
growthrate
  • \ >* all primitive-recursive functions
abstract
  • Arrow notation or up-arrow notation is a widely used notation for the hyper operators, devised by Donald Knuth in 1976 to represent large numbers. It is defined by the following rules: \begin{eqnarray} a \uparrow^1 b &=& a^b \\ a \uparrow^n 1 &=& a \\ a \uparrow^{n + 1} (b + 1) &=& a \uparrow^n (a \uparrow^{n + 1} b) \\ \end{eqnarray} \(a \uparrow^{n} b\) is a shorthand for \(a \uparrow\uparrow\cdots\uparrow\uparrow b\) with n arrows (where n is a positive integer). So, for example, \(a \uparrow^2 b = a \uparrow\uparrow b\). Arrow notation operators are right-associative; \(a \uparrow b \uparrow c\) always means \(a \uparrow (b \uparrow c)\). Specifically, \(a \uparrow b\) is exponentiation, \(a \uparrow\uparrow b\) is tetration, \(a \uparrow\uparrow\uparrow b\) is pentation, and so forth. In ASCII, these are written a^b, a^^b, a^^^b, ... The function \(f(n) = n \uparrow^n n\) is a fast-growing function that eventually dominates all primitive recursive functions, and can be approximated using the fast-growing hierarchy as \(f_\omega(n)\). Arrow notation has been generalized to other notations. A few notable ones are chained arrow notation, BEAF, and BAN. It has also been compared exactly with Notation Array Notation using the function (a{2, number of arrows}b). Nathan Ho and Wojowu showed that arrow notation terminates — that is, \(a \uparrow^n b\) exists for all \(a,b,n\).
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