Googology Wiki
Advertisement
Googology Wiki

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:

Unfortunately, many versions of them are ill-defined, although the creator insisted the well-definedness, as we will explain later.

Linear array notation

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.[6][7][8][9][10][11][12][13] According to the creator, we can freely remove any correct and sourced descriptions because the policy does not forbid it.[8][9][10][13] Of course, since wikia is not a personal website and FANDOM officially recommends us to expand articles rather than removing description, it is obviously wrong.

Thus, this article has a quite special feature of manipulation. Although the creator insists that this feature is unnecessary information,[13] there are few examples in googology which involve a similar situation. Therefore we deeply explain the histories of this notation and this article.

Original version

Since there are common errors in the original version[2] and other versions[3][4], we start with the explanation of the original version first. However, the creator insists that the reason is a false.[14]

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.[2] It has the following rules:

  1. Q<> = Q<a> = 0
  2. Q<a,b> = a*b
  3. Q<#,0> = x<#>
  4. Q<a,1,#> = a for b>0
  5. 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. Maybe the b in the fourth definition is a typo of a. 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.

In addition, it is written "If none of the 3 rules above applies, start the process shown below:", although there are five rules.

  • 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. Although the creator insisted "it is most likely well-defined",[15] it is ill-defined mainly because x, which is called "it", is undefined. See "Issues section" below for the detail.

Issues

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:

  1. The "it" means the current entry to which the process refers.
  2. 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.

However, the creator insists that the errors are just typos.[16] Although another user pointed out that the description of x must not be just a typo but a theoretic error at the talk page[17], the creator ignored the issue.

Examples

Q<0,0,0> = x<0,0> (ill-defined)

Unfortunately, this example was removed by the creator from the article.[7]

Second version

Next, we explain the second version, which is given after the issues on the original definition were pointed out.[3]

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,[2] but the creator has fixed several of them since then.[3] 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.[16][14]

Unfortunately, this section is a target of the creator's removement.[9][10][11][12][13]

Definition

The linear notation has the following form:

Q<a,b,c,d,e…>

The domain is perhaps intended to be the set of all arrays of non-negative integers, although the second definition[3] 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[2] explained above, although the creator insisted all errors are fixed.[18] The rules for it are:

  1. Q<> = Q<a> = 0
  2. Q<a,b> = a*b
  3. Q<#,0> = Q<#>
  4. Q<a,1,#> = a for b>0
  5. Q<a,b,c,#> = Q<a,Q<a,b-1,c,#>,c-1,#> for b>0

Here # denotes a portion of the array, which can be empty. Maybe the b in the fourth definition is a typo of a again. 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.

It is still written "If none of the 3 rules above applies, start the process shown below:", although there are five rules again.

  • 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 entry to which the exceptional process refers first is not clear, similarly to the original version[2] explained above. Even if we fix the entry in a reasonable way, it is still ill-defined. See "Issues" section below for the detail.

Issues

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:

  1. Following the explanation in the definition, the entry to which the process refers first is the third entry.
  2. Ignoring the explanation in the definition, the entry to which the process refers first is the first entry.

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.

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.

As a conclusion, the second version is ill-defined, too. However, the creator insists that this version is also well-defined.[6] 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.[7] As we have explained, even if the entry to which the process refers first is the third entry, it is still ill-defined, though.

Example

Q<3,2,1>

= Q<3,Q<3,1,1>,0> (rule 5)

= Q<3,Q<3,1,1>> (rule 3)

= Q<3,3> (rule 4)

= 3*3 (rule 2)

= 9

Q<1,0,1> is ill-defined, if the entry to which the exceptional process refers first is the first entry.

Q<1,2,0,1> = Q<1,Q<1,1,0,1>,-1,1> (ill-defined).

Third version

We explain the third definition, which is given later after the issues of the second definition are pointed out.[4]

Unfortunately, this section is a target of the creator's removement.[9][10][11][12][13]

Definition

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:".

Issues

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[19] and was well-defined.[20] According to the creator, all typos in the definition do not contribute to the ill-definedness.[14]

Examples

Q<3,2,1> = Q<3,Q<3,1,1>,0> = Q<3,Q<3,1,1>> = Q<3,3> = 3*3 = 9

Q<1,2,0,1> = Q<1,Q<1,1,0,1>,-1,1> (ill-defined)

Alternative version

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.

Unfortunately, this section is a target of the creator's removement.[9][10][11] According to the creator, the version was most likely well-defined.[12][13]

Definition

For any array @ of non-negative integers, we define a non-negative integer Q<@> in the following recursive way:

  1. If @ is of length < 2, then Q<@> := 0.
  2. If @ = "a,b" for non-negative integers a and b, then Q<@> := a*b.
  3. Otherwise, if @ = "#,0" for an array # of non-negative integers, then Q<@> := Q<#>.
  4. Otherwise, if @ = "a,1,#" for a non-negative integer a and an array # of non-negative integers, then Q<@> := a.
  5. 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,#>.
  6. Otherwise, Q<@> := Q<@'>, where @' is the array of non-negative integers defined in the following way:
    1. Denote by L the length of @.
    2. Denote by @i the i-th entry of @ for each positive integer i≦L.
    3. Denote by k≦L the least positive integer satifying k≧3 and @k > 0.
    4. Denote by # the array of copies of @1 of length k-2.
    5. Denote by @>k the array given as the final segment of @ of length L-k, i.e. the array given by removing the first k entries from @.
    6. Set @' := "#,@2,@k-1,@>k".

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>.

Examples

Q<1,2,0,1> = Q<1,1,2,0> = Q<1,1,2> = 1

Fourth version

Since the ill-definedness of Q<1,2,0,1> is pointed out, the creator independently tried to fix the issues, after we gave an alternative definition.[5] However, the creator is trying to remove the alternative definition together with the sections on other ill-defined versions.[9][10][11][12][13] Therefore even if the creator remove the alternative definition again, readers should be very careful that the fourth version is created only after the issues above are pointed out and the alternative definition has been already given.

Since the removement of correct and sourced information and an alternative definition is a quite specific feature of this notation and the creator, we keep the information.

Definition

The main difference from the third version is that the creator added the condition c>0 to rule 5, because we explained how the lack of the condition causes a problem, even though the creator strongly insisted the well-definedness of the third version.[19][20]

Issues

In particular, the typos "Where are nonnegative integers", "b" in the fourth rule, and "the 3 rules" are not fixed. According to the creator, typos do not contributes to the ill-definedness,[14] and even serious errors can be regarded as typos.[16]

Sources

  1. Quick array notation (Retrieved at (UTC) 10:30 20/11/2020)
  2. 2.0 2.1 2.2 2.3 2.4 2.5 Linear array notation (Retrieved at (UTC) 10:30 20/11/2020)
  3. 3.0 3.1 3.2 3.3 3.4 Linear array notation (Retrieved at (UTC) 12:00 21/11/2020)
  4. 4.0 4.1 4.2 Linear array notation (Retrieved at (UTC) 7:45 23/11/2020)
  5. 5.0 5.1 Linear array notation (Retrieved at (UTC) 1:00 27/11/2020)
  6. 6.0 6.1 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".)
  7. 7.0 7.1 7.2 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.)
  8. 8.0 8.1 A difference page of this article. (The creator said "Renember your ""correct information"" is unneeded by a policy, so no one said it can't be removed.", and removed information related to ill-definedness and the alternative definition.)
  9. 9.0 9.1 9.2 9.3 9.4 9.5 A difference page of this article. (The creator said "But that 2/3 of the article is not required by the wikia policy, and is fake description.")
  10. 10.0 10.1 10.2 10.3 10.4 10.5 A difference page of this article. (The creator said ""Do you realize that this is "correct and sourced information" too. In fact your information is fake description, fandom just wants stubs to expand. The notation is well-defined now, especially after complete case classification has been added."")
  11. 11.0 11.1 11.2 11.3 11.4 A difference page of this arctile. (The creator removed many descriptions related to the ill-definedness and the alternative definition by saying "I don't understand,my information is also valid. It features the full notation history.")
  12. 12.0 12.1 12.2 12.3 12.4 A difference page of this article. (The creator drastically removed the contents by saying "Fixed the remaining typos, condensed the information on second and higher versions and alternative version." Also, the creator seriously broke the layout of the article, and explained that the alternative version was most likely well-defined.)
  13. 13.0 13.1 13.2 13.3 13.4 13.5 13.6 A difference page of this article. (The creator drastically removed the contents by saying "But. The. Policy. Does. NOT. Prohibit. Removing. That. Unnessecary. Information. That. Is. Confusing. To. Readers".)
  14. 14.0 14.1 14.2 14.3 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.")
  15. A difference page of this article. (The creator wrote "This notation most likely reaches ωω".)
  16. 16.0 16.1 16.2 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?".)
  17. 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.".)
  18. 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".)
  19. 19.0 19.1 A difference page of this article. (The creator wrote "The creator stated that this extension most likely reaches ωω".)
  20. 20.0 20.1 A difference page of the talk page of. (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.)
Advertisement