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Multiexpansion

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Multiexpansion refers to the function \(a \{\{2\}\} b = \{a,b,2,2\} = \underbrace{a \{\{1\}\} a \{\{1\}\} \ldots \{\{1\}\} a \{\{1\}\} a}_{\text{b a's}}\), using BEAF.[1]

In the fast-growing hierarchy, \(f_{\omega+2}(n)\) is comparable to multiexpandal growth rate. That means two things: one is that Multiexpansion is comparable to Chained Arrow Notation and that Multiexpansion can be comparable in Notation Array by (a{3,3}b).

ExamplesEdit

  • {a,3,2,2} = a{{1}}a{{1}}a. This is equal to a expanded to (a expanded to a). Let A = a{a{a...a{a{a}a}a...a}a}a (a a's), then {a,3,2,2} = a{a...a{a}a...a}a (A a's)
  • {3,2,2,2} = 3{{2}}2 = 3{{1}}3 = {3,3,1,2}
  • {4,2,2,2} = {4,4,1,2}
  • {3,3,2,2} = 3{{2}}3 = 3{{1}}3{{1}}3 = {3,{3,3,1,2},1,2}
  • {4,3,2,2} = {4,{4,4,1,2},1,2}

Pseudocode Edit

Below is an example of pseudocode for multiexpansion.

function multiexpansion(a, b):
    result := a
    repeat b - 1 times:
        result := expansion(a, result)
    return result

function expansion(a, b):
    result := a
    repeat b - 1 times:
        result := hyper(a, a, result + 2)
    return result

function hyper(a, b, n):
    if n = 1:
        return a + b
    result := a
    repeat b - 1 times:
        result := hyper(a, result, n - 1)
    return result

Sources Edit

  1. Array Notation by Jonathan Bowers

See also Edit

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