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| − | '''Quick array notation''' is a notation for large numbers made by Wikia user [[User:Nirvana Supermind|Nirvana Supermind]]<ref>[https://integralview.wordpress.com/2020/11/15/nimble-array-notation/ Quick array notation] (Retrieved at (UTC) |
+ | '''Quick array notation''' is a notation for large numbers made by Wikia user [[User:Nirvana Supermind|Nirvana Supermind]]<ref>[https://integralview.wordpress.com/2020/11/15/nimble-array-notation/ Quick array notation] (Retrieved at (UTC) 6:30 22/11/2020)</ref>. The notation is currently composed of 1 extension |
, which is as follows: |
, which is as follows: |
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| − | * Linear array notation<ref name=" |
+ | * Linear array notation<ref name="newest">[https://integralview.wordpress.com/2020/11/15/linear-array-notation/ Linear array notation] (Retrieved at UTC 6:30 22/11/2020)</ref><ref name="first">[https://integralview.wordpress.com/2020/11/15/linear-array-notation/ Linear array notation] (Retrieved at (UTC) 10:30 20/11/2020)</ref> |
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| − | Unfortunately, versions of them are '''ill-defined''', as we will explain later. |
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== Linear array notation == |
== Linear array notation == |
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| + | '''Linear array notation''' is the first part of quick array notation. |
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| − | '''Linear array notation''' is the first part of quick array notation. Similar to the creator's other notations [[Rampant Array Notation]] and [[Extensible Illion System]], '''the creator tends to insist the well-definedness even after errors are pointed out and to remove the description related to the ill-definedness.'''<ref name="incomplete">[https://googology.wikia.org/wiki/Quick_array_notation?type=revision&diff=303900&oldid=303267 A difference page] of this article. (The creator removed this article from the category of ill-defined numbers, and said "Since it is properly defined now, it is not incomplete".)</ref><ref name="removement 1">[https://googology.wikia.org/wiki/Quick_array_notation?type=revision&diff=303920&oldid=303905 A difference page] of this article. (The creator removed the descriptions of the issues, and wrote "If none of the 3 rules above applies, start the process shown below starting from the third entry inside the brackets:", although it was not clarified in the source.)</ref> Therefore we deeply explain the histories of this notation and this article. |
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| − | Since there are common errors in the original version<ref name="first" /> and other versions<ref name="second" /><ref name="third" />, we start with the explanation of the original version first. However, the creator insists that the reason is '''a false'''.<ref name="typo 2">[https://googology.wikia.org/wiki/Talk:Quick_array_notation?type=revision&diff=303943&oldid=303930 A difference page] of the talk page of this arctile. (The creator said ""The reason you say "Since there are common errors in the original version and other versions, we start with the explanation of the original version first." is a false because the current notation '''is well-defined'''"" and "'''all mistakes you mentioned are typos, errors and mistakes I made, that do not contribute to the well-definedness or ill-definedness of a notation.'''")</ref> |
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| − | |||
| − | ==== Definition ==== |
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| ⚫ | |||
| − | # Q<a,b,c,#> = x<a,x<a,b-1,c,#>,c-1,#> for b>0 |
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| ⚫ | Here # denotes a portion of the array, which can be empty, and x denotes some number |
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| − | This notation is intended to reach ω<sup>ω</sup> in the fast-growing hierarchy by the creator. Although the creator insisted "it is '''most likely well-defined'''",<ref>[https://googology.wikia.org/wiki/Quick_array_notation?type=revision&diff=302968&oldid=302967 A difference page] of this article. (The creator wrote "This notation most likely reaches ω<sup>ω</sup>".)</ref> it is '''ill-defined''' mainly because x, which is called "it", is undefined. See "Issues section" below for the detail. |
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| − | ==== Issues ==== |
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| − | |||
| ⚫ | Let us compute Q<0,0,0>. The resulting expression is x<0,0> for some x, because the third rule is applicable. Since there is no rule applicable to x<0,0>, we need to start the exceptional process. In Case A, the process refers to "it". Since the terminology of "it" is quite ambiguous, there are two possibilities: |
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| ⚫ | |||
| ⚫ | |||
| ⚫ | In the first interpretation, there is no clarification of what entry the process refers to first. Perhaps it should be the first entry of the array following the outer iterator x. Since the first entry is 0, Case A is applicable. Then the current entry becomes the next 0, and hence Case A is applicable again. However, there is no entry next to the current entry, and hence the process does not terminate in a usual way. |
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| ⚫ | In the second interpretation, then Case A does not change it. Therefore the process for x<0,0> causes an infinite loop if the undefined x is 0. In any case, the reference to the undefined value x is invalid. As a result, x<0,0> and hence Q<0,0,0> are |
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| − | However, the creator insists that the errors are just typos.<ref name="typo">[https://googology.wikia.org/wiki/Talk:Quick_array_notation?type=revision&diff=303924&oldid=303906 A difference page] of the talk page of this article. (The creator said "What is the point of saying a notation is ill-defined because there was a typo or two from my XS-based draft?".)</ref> Although another user pointed out that the description of x must not be just a typo but a theoretic error at the talk page<ref>[https://googology.wikia.org/wiki/Talk:Quick_array_notation?oldid=303127 The first version] of the talk page of this article. (A user wrote "What was x? The creator insists that the issues on the original definition are based on typos, but I cannot understand what are typos for the undefined value x, called the outer iterator and it. It is unreasonable to conclude that it was just a typo, because the removement of the whole notion of the outer iterator drastically changes the description. Unless a reasoning is given, I will edit the explanation as if the issues were only based on typos.".)</ref>, the creator ignored the issue. |
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| − | ==== Examples ==== |
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| − | Q<0,0,0> = x<0,0> ('''ill-defined''') |
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| − | Unfortunately, this example was removed by the creator from the article.<ref name="removement 1" /> |
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| + | === Current version === |
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| − | Next, we explain the second version, which is given after the issues on the original definition were pointed out.<ref name="second">[https://integralview.wordpress.com/2020/11/15/linear-array-notation/ Linear array notation] (Retrieved at (UTC) 12:00 21/11/2020)</ref> |
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| − | An old version of the notation had reference to an undefined value x called the outer iterator and some typos that made it '''ill-defined''',<ref name="first" /> but the creator has fixed several of them since then.<ref name="second" /> There is a more in-depth explanation of the issues in the "Original version" section. The main difference from the original definition is that the new definition does not use x. However, as we explained above, the creator insists that the errors in the original definition are just typos, even though the creator removed the named notion, i.e. the outer iterator x.<ref name="typo" /><ref name="typo 2" /> |
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==== Definition ==== |
==== Definition ==== |
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Q<a,b,c,d,e…> |
Q<a,b,c,d,e…> |
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| + | The rules for it are: |
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| − | The domain is perhaps intended to be the set of all arrays of non-negative integers, although the second definition<ref name="second" /> has the same typo "Where are nonnegative integers" of "Where a, b, c, and so on are nonnegative integers" or something like that as the original definition<ref name="first" /> explained above, although the creator insisted all errors are fixed.<ref>[https://googology.wikia.org/wiki/Quick_array_notation?type=revision&diff=303109&oldid=302995 A difference page] of this article. (The creator wrote "An old version of the notation had some typos that made it ill-defined, but the creator has fixed them since then" and "All information here only applies to the old version of the notation".)</ref> The rules for it are: |
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# Q<> = Q<a> = 0 |
# Q<> = Q<a> = 0 |
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# Q<#,0> = Q<#> |
# Q<#,0> = Q<#> |
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# Q<a,1,#> = a for b>0 |
# Q<a,1,#> = a for b>0 |
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| − | # Q<a,b,c,#> = Q<a,Q<a,b-1,c,#>,c-1,#> for b>0 |
+ | # Q<a,b,c,#> = Q<a,Q<a,b-1,c,#>,c-1,#> for b,c>0 |
| − | Here # denotes a portion of the array, which can be empty |
+ | Here # denotes a portion of the array, which can be empty. If there are two or more distinct rules to apply to a single expression, the uppermost-numbered rule which is applicable and whose result is a valid expression will be applied. If none of the 3 rules above applies, start the process shown below starting from the third entry inside the brackets: |
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| − | It is still written "If none of the 3 rules above applies, start the process shown below:", although there are five rules again. |
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* Case A: If the entry is 0, jump to the next entry. |
* Case A: If the entry is 0, jump to the next entry. |
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** Terminate the process. Repeat if the rules still do not apply. |
** Terminate the process. Repeat if the rules still do not apply. |
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| + | The creator stated that this extension most likely reaches ω<sup>ω</sup> in the [[fast-growing hierarchy]]. |
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| − | The entry to which the exceptional process refers first is not clear, similarly to the original version<ref name="first" /> explained above. Even if we fix the entry in a reasonable way, it is still '''ill-defined'''. See "Issues" section below for the detail. |
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| + | An old version of the notation had some typos that made it ill-defined, but the creator has fixed them since then<ref name="first" />. There is a more in-depth explanation of this in the "Original version" section. |
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| − | ==== Issues ==== |
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| − | Since it is written "In other words, from the 3rd entry on, we search for the first non-0 entry." right after the description of the exceptional process in the source, there are two possibilities: |
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| − | # Following the explanation in the definition, the entry to which the process refers first is the third entry. |
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| − | # Ignoring the explanation in the definition, the entry to which the process refers first is the first entry. |
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| − | Therefore the definition is ambiguous. If we assume the first possibility, we can solve Q<1,0,1>. On the other hand, if we assume the second possibility, the entry to which the process refers first is the first entry 1, and hence Case B is applied to it. However, there is no previous entry, while the rule refers to it as 0 in order to replace it to the seoncd entry b = 2. Therefore Case B does not make sense. As a reuslt, Q<1,0,1> is '''ill-defined''' in that case. |
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| − | In addition, consider Q<1,2,0,1>. Since there is no restriction of c in rule 5 unlike b, rule 5 is applicable to it. Then the result Q<1,Q<1,1,0,1>,-1,1> is '''an invalid expression'''. Therefore Q<1,2,0,1> is '''ill-defined'''. |
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| − | As a conclusion, the second version is '''ill-defined''', too. However, the creator insists that this version is also '''well-defined'''.<ref name="incomplete" /> Later, the creator removed the description of the issues from this article, and wrote "start the process shown below starting from the third entry inside the brackets" as if it were clarified in the source.<ref name="removement 1" /> As we have explained, even if the entry to which the process refers first is the third entry, it is still '''ill-defined''', though. |
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==== Example ==== |
==== Example ==== |
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Q<3,2,1> |
Q<3,2,1> |
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| − | = Q<3,Q<3,1,1>,0> |
+ | = Q<3,Q<3,1,1>,0> |
| − | = Q<3,Q<3,1,1>> |
+ | = Q<3,Q<3,1,1>> |
| − | = Q<3,3> |
+ | = Q<3,3> |
| − | = 3*3 |
+ | = 3*3 |
= 9 |
= 9 |
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| ⚫ | |||
| − | Q<1,0,1> is '''ill-defined''', if the entry to which the exceptional process refers first is the first entry. |
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| + | All information here only applies to the old version of the notation. |
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| − | Q<1,2,0,1> = Q<1,Q<1,1,0,1>,-1,1> ('''ill-defined'''). |
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==== Definition ==== |
==== Definition ==== |
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| ⚫ | |||
| − | The main difference from the second definition is that the entry to which the process refers first is clarified as "If none of the 3 rules above applies, start the process shown below starting from the third number inside the brackets:". |
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| − | + | # Q<> = Q<a> = 0 |
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| ⚫ | |||
| ⚫ | |||
| ⚫ | |||
| ⚫ | |||
| ⚫ | Here # denotes a portion of the array, which can be empty, and x denotes some number. In the original source, it is written "the outer number x is called the iterator, and it". If there are two or more distinct rules to apply to a single expression, the uppermost-numbered rule which is applicable and whose result is a valid expression will be applied. |
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| − | In particular, the typos "Where are nonnegative integers", "b" in the fourth rule, and "the 3 rules" are not fixed, and the issue of Q<1,2,0,1> is not solved, although the creator insisted that it is '''most likely well-defined'''<ref>[https://googology.wikia.org/wiki/Quick_array_notation?type=revision&diff=303928&oldid=303905 A difference page] of this article. (The creator wrote "The creator stated that this extension most likely reaches ω<sup>ω</sup>".)</ref> and '''was ill-defined'''.<ref>[https://googology.wikia.org/wiki/Category_talk:Numbers_by_Nirvana_Supermind?type=revision&diff=303918&oldid=303908 A difference page] of the talk page of [[Category:Numbers by Nirvana Supermind]]. (The creator said "I have scrapped those notations and have stopped working on them in the favor of [[quick array notation]] which was well-defined." refering to [[Rampant Array Notation]] and [[Extensible Illion System]].)</ref> According to the creator, all typos in the definition do not contribute to the '''ill-definedness'''.<ref name="typo 2" /> |
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| ⚫ | |||
| − | ==== Examples ==== |
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| ⚫ | |||
| − | Q<3,2,1> = Q<3,Q<3,1,1>,0> = Q<3,Q<3,1,1>> = Q<3,3> = 3*3 = 9 |
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| ⚫ | |||
| ⚫ | |||
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| + | This notation is intended to reach ω<sup>ω</sup> in the fast-growing hierarchy by the creator, but is ill-defined mainly because x, which is called "it", is undefined. |
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| − | Q<1,2,0,1> = Q<1,Q<1,1,0,1>,-1,1> ('''ill-defined''') |
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| − | === |
+ | ==== Example ==== |
| ⚫ | Let us compute Q<0,0,0>. The resulting expression is x<0,0> for some x, because the third rule is applicable. Since there is no rule applicable to x<0,0>, we need to start the exceptional process. In Case A, the process refers to "it". Since the terminology of "it" is quite ambiguous, there are two possibilities: |
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| − | Although the creator does not care about typos and those versions have errors, it is easy to solve the issues of the '''ill-definedness''' by using complete case classification, as we did in the article of [[Extensible Illion System]]. |
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| ⚫ | |||
| ⚫ | |||
| ⚫ | In the first interpretation, there is no clarification of what entry the process refers to first. Perhaps it should be the first entry of the array following the outer iterator x. Since the first entry is 0, Case A is applicable. Then the current entry becomes the next 0, and hence Case A is applicable again. However, there is no entry next to the current entry, and hence the process does not terminate in a usual way. |
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| ⚫ | In the second interpretation, then Case A does not change it. Therefore the process for x<0,0> causes an infinite loop if the undefined x is 0. In any case, the reference to the undefined value x is invalid. As a result, x<0,0> and hence Q<0,0,0> are ill-defined. In particular, the analysis by the creator is wrong. |
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| − | ==== Definition ==== |
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| ⚫ | |||
| − | For any array @ of non-negative integers, we define a non-negative integer Q<@> in the following recursive way: |
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| + | There is also a second old version of this notaion in an attempt to fix it.<ref name="second">[https://integralview.wordpress.com/2020/11/15/linear-array-notation/ Linear array notation] retrieved on 02:44, 23 November 2020 (UTC)</ref>. It was the same as the current definition noted above but had another typo that made it ill-defined. |
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| − | # If @ is of length < 2, then Q<@> := 0. |
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| − | # If @ = "a,b" for non-negative integers a and b, then Q<@> := a*b. |
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| − | # Otherwise, if @ = "#,0" for an array # of non-negative integers, then Q<@> := Q<#>. |
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| − | # Otherwise, if @ = "a,1,#" for a non-negative integer a and an array # of non-negative integers, then Q<@> := a. |
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| − | # Otherwise, if @ = "a,b,c,#" for a non-negative integer a, positive integers b and c, and an array # of non-negative integers, then Q<@> := Q<a,Q<a,b-1,c,#>,c-1,#>. |
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| − | # Otherwise, Q<@> := Q<@'>, where @' is the array of non-negative integers defined in the following way: |
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| − | ## Denote by L the length of @. |
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| − | ## Denote by @<sub>i</sub> the i-th entry of @ for each positive integer i≦L. |
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| − | ## Denote by k≦L the least positive integer satifying k≧3 and @<sub>k</sub> > 0. |
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| − | ## Denote by # the array of copies of @<sub>1</sub> of length k-2. |
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| − | ## Denote by @<sub>>k</sub> the array given as the final segment of @ of length L-k, i.e. the array given by removing the first k entries from @. |
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| − | ## Set @' := "#,@<sub>2</sub>,@<sub>k</sub>-1,@<sub>>k</sub>". |
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| − | Since we have assumed that c in rule 5 is possitive unlike the official versions, rule 5 is not applicable to Q<1,2,0,1> any more. Instead, rule 6 is applicable to it, and the resulting expression is Q<1,1,2,0>. |
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| − | ==== Examples ==== |
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| − | |||
| − | Q<1,2,0,1> = Q<1,1,2,0> = Q<1,1,2> = 1 |
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== Sources == |
== Sources == |
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[[Category:Notations]] |
[[Category:Notations]] |
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[[Category:Functions]] |
[[Category:Functions]] |
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| − | [[Category:Incomplete]] |
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| − | [[Category:Numbers by Nirvana Supermind]] |
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Revision as of 00:10, 24 November 2020
Quick array notation is a notation for large numbers made by Wikia user Nirvana Supermind[1]. The notation is currently composed of 1 extension , which is as follows:
Linear array notation
Linear array notation is the first part of quick array notation.
Current version
All information here only applies to the current version of the notation, which is given after the issues on the original definition were pointed out.
Definition
The linear notation has the following form:
Q<a,b,c,d,e…>
The rules for it are:
- Q<> = Q<a> = 0
- Q<a,b> = a*b
- Q<#,0> = Q<#>
- Q<a,1,#> = a for b>0
- Q<a,b,c,#> = Q<a,Q<a,b-1,c,#>,c-1,#> for b,c>0
Here # denotes a portion of the array, which can be empty. If there are two or more distinct rules to apply to a single expression, the uppermost-numbered rule which is applicable and whose result is a valid expression will be applied. If none of the 3 rules above applies, start the process shown below starting from the third entry inside the brackets:
- Case A: If the entry is 0, jump to the next entry.
- Case B: If the entry is not 0, then:
- Replace the “0,n” with “b,n-1” where n is this entry and b is the second entry of the array.
- Replace all entries before the “b,n-1” with a (the first entry).
- Terminate the process. Repeat if the rules still do not apply.
The creator stated that this extension most likely reaches ωω in the fast-growing hierarchy.
An old version of the notation had some typos that made it ill-defined, but the creator has fixed them since then[3]. There is a more in-depth explanation of this in the "Original version" section.
Example
Q<3,2,1>
= Q<3,Q<3,1,1>,0>
= Q<3,Q<3,1,1>>
= Q<3,3>
= 3*3
= 9
Original version
All information here only applies to the old version of the notation.
Definition
The domain is perhaps intended to be the set of all arrays of non-negative integers, although the original definition has a typo "Where are nonnegative integers" of "Where a, b, c, and so on are nonnegative integers" or something like that.[3] It has the following rules:
- Q<> = Q<a> = 0
- Q<a,b> = a*b
- Q<#,0> = x<#>
- Q<a,1,#> = a for b>0
- Q<a,b,c,#> = x<a,x<a,b-1,c,#>,c-1,#> for b>0
Here # denotes a portion of the array, which can be empty, and x denotes some number. In the original source, it is written "the outer number x is called the iterator, and it". If there are two or more distinct rules to apply to a single expression, the uppermost-numbered rule which is applicable and whose result is a valid expression will be applied.
If none of the 3 rules above applies, start the process shown below:
- Case A: If it’s 0, jump to the next entry.
- Case B: If the entry is not 0, then:
- Replace the “0,n” with “b,n-1” where n is this entry and b is the second entry of the array.
- Replace all entries before the “b,n-1” with a (the first entry).
- Terminate the process. Repeat if the rules still do not apply.
This notation is intended to reach ωω in the fast-growing hierarchy by the creator, but is ill-defined mainly because x, which is called "it", is undefined.
Example
Let us compute Q<0,0,0>. The resulting expression is x<0,0> for some x, because the third rule is applicable. Since there is no rule applicable to x<0,0>, we need to start the exceptional process. In Case A, the process refers to "it". Since the terminology of "it" is quite ambiguous, there are two possibilities:
- The "it" means the current entry to which the process refers.
- The "it" means x.
In the first interpretation, there is no clarification of what entry the process refers to first. Perhaps it should be the first entry of the array following the outer iterator x. Since the first entry is 0, Case A is applicable. Then the current entry becomes the next 0, and hence Case A is applicable again. However, there is no entry next to the current entry, and hence the process does not terminate in a usual way.
In the second interpretation, then Case A does not change it. Therefore the process for x<0,0> causes an infinite loop if the undefined x is 0. In any case, the reference to the undefined value x is invalid. As a result, x<0,0> and hence Q<0,0,0> are ill-defined. In particular, the analysis by the creator is wrong.
Second version
There is also a second old version of this notaion in an attempt to fix it.[4]. It was the same as the current definition noted above but had another typo that made it ill-defined.
Sources
- ↑ Quick array notation (Retrieved at (UTC) 6:30 22/11/2020)
- ↑ Linear array notation (Retrieved at UTC 6:30 22/11/2020)
- ↑ 3.0 3.1 3.2 Linear array notation (Retrieved at (UTC) 10:30 20/11/2020)
- ↑ Linear array notation retrieved on 02:44, 23 November 2020 (UTC)