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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]].<ref>[http://www.polytope.net/hedrondude/array.htm]</ref> Or ''a'' [[Expansion|expanded]] to ''a'' for ''b'' times. They are solved from right to left.
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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]].<ref>[http://www.polytope.net/hedrondude/array.htm Array Notation by Jonathan Bowers]</ref> Or ''a'' [[Expansion|expanded]] to ''a'' for ''b'' times. They are solved from right to left.
   
 
In the [[fast-growing hierarchy]], \(f_{\omega+2}(n)\) corresponds to multiexpandal growth rate.
 
In the [[fast-growing hierarchy]], \(f_{\omega+2}(n)\) corresponds to multiexpandal growth rate.

Revision as of 15:08, 16 September 2013

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] Or a expanded to a for b times. They are solved from right to left.

In the fast-growing hierarchy, \(f_{\omega+2}(n)\) corresponds to multiexpandal growth rate.

Examples

  • {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

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

See also

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