Conway chained arrow notation: Difference between revisions

Revision as of 05:15, 29 August 2003

Revision as of 20:37, 31 August 2003

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As with most combinatorial symbologies, the definition is recursive. Here, ''p'' and ''q'' are positive integers, and ''X'' is a chain (possibly of only one element) substituted textually.

:(1) A chain ''X&rarr;p&rarr;(q+1)'' of more than 2 elements not ending in&nbsp;1 is the same as <br>&nbsp;&nbsp;&nbsp;''X&rarr;(X&rarr;(...X&rarr;(X)&rarr;q...)&rarr;q)&rarr;q'' (with ''p'' copies of ''X'').

:(2) A chain ending in 1 is unchanged by dropping that 1. ''X&rarr;1 = X''

:(3) 2-element chains terminate in [[exponentiation]]. ''p&rarr;q = p^q''

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:''p&rarr;q&rarr;r'' = hyper''(p,r+2,q)'' = ''p^&hellip;^q'' with ''r'' up-arrows.

==ExampleSimple examples==

It is impossible give a fully worked-out '''interesting''' example. We needsince at least 4 elements (are required. However 1-, 2-, and 3-length chains, which beingare subsumed in other notations). 1s do nothing interesting, andare 3&rarr;3&rarr;3&rarr;3expanded ishere muchas greater than Graham'sillustrated numberexamples. Any chain beginning with two 2s stands for 4. That leaves very little room.

''n''

:any single integer ''n'' is just the value n, ie 7 = 7. This does not conflict with the rules since using rule 2 (backwards) we have 7 = 7&rarr;1 = 7¹ = 7.

''p&rarr;q''

:= ''p^q'' (by rule 3)

:Thus 3&rarr;4 = 3⁴ = 81

:Also 123456&rarr;1 = 123456¹ = 123456 (by both rules 2 and 3)

1&rarr;(''any arrowed expression'')

:= 1 since the entire expression eventually reduces to 1^number = 1

24&rarr;3&rarr;2&rarr;2

:= 4&rarr;(4&rarr;(4)&rarr;1)&rarr;1 (by 1) and then, working from the inner parentheses outwards,

:= 24&rarr;(24&rarr;4&rarr;61)&rarr;61 (''supra''remove redundant parentheses [rrp])

:= 3&rarr;24&rarr;(34&rarr;24)&rarr;1 (12)

:= 34&rarr;3(256)&rarr;81 (13)

:= 24&rarr;3256&rarr;81 (2 and 3rrp)

:= 34&rarr;2&rarr;9256 (2 and 3)

:= 1.34078079299e+154 approximately (3)

4&rarr;3&rarr;2 alternatively analysed

:= 24&rarr;3(4&rarr;(24)&rarr;31)&rarr;1 (by 1) and then, removing trailing "&rarr;1",

:= 24&rarr;(24&rarr;2(4)&rarr;71)&rarr;7 (12)

:= 24&rarr;(4&rarr;7(4)) (''supra''2)

:= 4&rarr;(256) (rrp, 3)

:= 1.34078079299e+154 approximately (rrp, 3)

3&rarr;2&rarr;2&rarr;24

:= 2&rarr;(2&rarr;(2&rarr;2&rarr;6)&rarr;6)&rarr;63 (by 1)

:= 2&rarr;2&rarr;3 (rrp)

:= 2&rarr;2&rarr;2 (1, rrp)

:= 2&rarr;2&rarr;1 (1, rrp)

:= 2&rarr;2 (2)

:= 4 (3)

2&rarr;4&rarr;3

:= 2&rarr;(2&rarr;(2&rarr;(2)&rarr;2)&rarr;2)&rarr;2 (by 1)

:= 2&rarr;(2&rarr;(2&rarr;2&rarr;2)&rarr;2)&rarr;2 (rrp)

:= 2&rarr;(2&rarr;(4)&rarr;2)&rarr;2 (previous example)

:= 2&rarr;( 2&rarr;4&rarr;2 )&rarr;2 (rrp) ''(expression expanded in next equation delimited by spaces)''

:= 2&rarr;( 2&rarr;(2&rarr;(2&rarr;(2)&rarr;1)&rarr;1)&rarr;1 )&rarr;2 (1)

:= 2&rarr;(2&rarr;(2&rarr;(2&rarr;2&rarr;1)&rarr;1)&rarr;1)&rarr;2 (rrp)

:= 2&rarr;(2&rarr;(2&rarr;(2&rarr;2)))&rarr;2 (2 repeatedly)

:= 2&rarr;(2&rarr;(2&rarr;(4)))&rarr;2 (3)

:= 2&rarr;(2&rarr;(16))&rarr;2 (3)

:= 2&rarr;256&rarr;2 (3,rrp)

:= 2&rarr;(2&rarr;(2&rarr;(...2&rarr;(2&rarr;(2)&rarr;1)&rarr;1...)&rarr;1)&rarr;1)&rarr;1 (1) with 256 sets of parentheses

:= 2&rarr;(2&rarr;(2&rarr;(...2&rarr;(2&rarr;(2))...)))) (2 repeatedly)

:= 2&rarr;(2&rarr;(2&rarr;(...2&rarr;(4))...)))) (3)

:= 2&rarr;(2&rarr;(2&rarr;(...16...)))) (3)

:= 2&rarr;(very large power of 2) (3 repeatedly)

:= ''very very big number''

2&rarr;3&rarr;2&rarr;2

:= 2&rarr;3&rarr;(2&rarr;3)&rarr;1 (by 1)

:= 2&rarr;3&rarr;8 (2 and 3)

:= 2&rarr;(2&rarr;2&rarr;7)&rarr;7 (1)

:= 2&rarr;4&rarr;7 (''supra'')

:= 2&rarr;(2&rarr;(2&rarr;2&rarr;6)&rarr;6)&rarr;6 (1)

:= 2&rarr;(2&rarr;4&rarr;6)&rarr;6 (''supra'')

:= ''gonna be huge but needs a couple more steps''

3&rarr;2&rarr;2&rarr;2

:= 3&rarr;2&rarr;(3&rarr;2)&rarr;1 (1)

:= 3&rarr;2&rarr;9 (2 and 3)

:= 3&rarr;3&rarr;8 (1)

:= ''huge''

==Simpler More typical examples==

Conway's arrow doesn't help to express [[Graham's number]] <var>G</var> succinctly, but:

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<br>which is the given inequality.

Note that 3&rarr;3&rarr;3&rarr;3 is much greater than Graham's number.

==Simpler examples==

It is impossible give a fully worked-out interesting example. We need at least 4 elements (1-, 2-, 3-length chains being subsumed in other notations). 1s do nothing interesting, and 3&rarr;3&rarr;3&rarr;3 is much greater than Graham's number. Any chain beginning with two 2s stands for 4. That leaves very little room.

2&rarr;3&rarr;2&rarr;2

:= 2&rarr;3&rarr;(2&rarr;3)&rarr;1 (by 1)

:= 2&rarr;3&rarr;8 (2 and 3)

:= 2&rarr;(2&rarr;2&rarr;7)&rarr;7 (1)

:= 2&rarr;4&rarr;7 (''supra'')

:= 2&rarr;(2&rarr;(2&rarr;2&rarr;6)&rarr;6)&rarr;6 (1)

:= 2&rarr;(2&rarr;4&rarr;6)&rarr;6 (''supra'')

:= ''gonna be huge but needs a couple more steps''

3&rarr;2&rarr;2&rarr;2

:= 3&rarr;2&rarr;(3&rarr;2)&rarr;1 (1)

:= 3&rarr;2&rarr;9 (2 and 3)

:= 3&rarr;3&rarr;8 (1)

:= ''huge''

==See also==