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, which means multiexpansion is comparable to \(a\rightarrow a\rightarrow b\rightarrow 3\) in Chained Arrow Notation and \((a\{3,3\}b)\) in Notation Array Notation.

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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