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| − | '''Multiexpansion '''refers to the function |
+ | '''Multiexpansion '''refers to the binary 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> |
| + | 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]]. |
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| − | == |
+ | ==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 \(\{a,3,2,2\} = a\{a...a\{a\}a...a\}a (A\ a's)\) |
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| + | *\(\{3,2,2,2\} = 3\{\{2\}\}2 = 3\{\{1\}\}3 = \{3,3,1,2\}\) |
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| + | *\(\{4,2,2,2\} = \{4,4,1,2\}\) |
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| + | *\(\{3,3,2,2\} = 3\{\{2\}\}3 = 3\{\{1\}\}3\{\{1\}\}3 = \{3,\{3,3,1,2\},1,2\}\) |
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| + | *\(\{6,4,2,2\} = 6\{\{2\}\}4 = 6\{\{1\}\}6\{\{1\}\}6\{\{1\}\}6 = \{6,\{6,\{6,6,1,2\},1,2\},1,2\}\) |
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| + | |||
| + | == Pseudocode == |
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| + | Below is an example of pseudocode for multiexpansion. |
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| + | |||
| + | '''function''' multiexpansion(''a'', ''b''): |
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| + | ''result'' := ''a'' |
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| + | '''repeat''' ''b'' - 1 '''times''': |
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| + | '''result''' := expansion(''a'', ''result'') |
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| + | '''return''' ''result'' |
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| + | |||
| + | '''function''' expansion(''a'', ''b''): |
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| + | ''result'' := ''a'' |
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| + | '''repeat''' ''b'' - 1 '''times''': |
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| + | '''result''' := hyper(''a'', ''a'', ''result'' + 2) |
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| + | '''return''' ''result'' |
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| + | |||
| + | '''function''' hyper(''a'', ''b'', ''n''): |
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| + | '''if''' ''n'' = 1: |
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| + | '''return''' ''a'' + ''b'' |
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| + | ''result'' := ''a'' |
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| + | '''repeat''' ''b'' - 1 '''times''': |
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| + | ''result'' := hyper(''a'', ''result'', ''n'' - 1) |
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| + | '''return''' ''result'' |
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| + | |||
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| + | == See also == |
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{{ExtendedOps}} |
{{ExtendedOps}} |
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| + | [[ja:乗算膨張]] |
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| + | [[zh:Multiexpansion]] |
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[[Category:Functions]] |
[[Category:Functions]] |
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| + | [[Category:Binary operators]] |
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| + | [[Category:BEAF]] |
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| + | [[Category:Jonathan Bowers]] |
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| + | [[Category:Extended hyper operators]] |
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Revision as of 00:51, 28 November 2020
Multiexpansion refers to the binary 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 \(\{a,3,2,2\} = a\{a...a\{a\}a...a\}a (A\ a's)\)
- \(\{3,2,2,2\} = 3\{\{2\}\}2 = 3\{\{1\}\}3 = \{3,3,1,2\}\)
- \(\{4,2,2,2\} = \{4,4,1,2\}\)
- \(\{3,3,2,2\} = 3\{\{2\}\}3 = 3\{\{1\}\}3\{\{1\}\}3 = \{3,\{3,3,1,2\},1,2\}\)
- \(\{6,4,2,2\} = 6\{\{2\}\}4 = 6\{\{1\}\}6\{\{1\}\}6\{\{1\}\}6 = \{6,\{6,\{6,6,1,2\},1,2\},1,2\}\)
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
Main article: hyper operator
Hyper-operators: successor · addition · multiplication · exponentiation · tetration · pentation · hexation · heptation · octation · enneation · decation (un/doe/tre) · vigintation (doe) · trigintation · centation (tri) · docentation · myriation · circulationBowers' extensions: expansion · multiexpansion · powerexpansion · expandotetration · explosion (multi/power/tetra) · detonation · pentonation
Saibian's extensions: hexonation · heptonation · octonation · ennonation · deconation
Tiaokhiao's extensions: megotion (multi/power/tetra) · megoexpansion (multi/power/tetra) · megoexplosion · megodetonation · gigotion (expand/explod/deto) · terotion · more...