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'''Alphabet notation''' is a notation created by Wikia user [[User:Nirvana Supermind|Nirvana Supermind]].<ref name="body">{{Cite web|last=Nirvana Supermind|url=https://integralview.wordpress.com/2020/12/22/alphabet-notation/|title=Alphabet Notation}} (Retrieved at UTC 12:00 13/01/2020)</ref> It is a based on recursion, and inputs a string of English letters. It has only this part currently:
+
'''Alphabet notation''' is a notation created by Wikia user [[User:Nirvana Supermind|Nirvana Supermind]].<ref>{{Cite web|last=Nirvana Supermind|url=https://integralview.wordpress.com/2020/12/22/alphabet-notation/|title=Alphabet Notation}} (Retrieved at UTC 12:00 13/01/2020)</ref><ref name=":0">{{Cite web|last=Nirvana Supermind|url=https://integralview.wordpress.com/2021/01/12/basic-alphabet-notation/|title=Basic Alphabet Notation}} (Retrieved at UTC 12:00 13/01/2020)</ref><ref name="current">{{Cite web|last=Nirvana Supermind|url=https://integralview.wordpress.com/2021/01/12/basic-alphabet-notation/|title=Basic Alphabet Notation}} (Retrieved at Wed, 13 Jan 2021 22:44:39 GMT) </ref>. It is a based on recursion, and inputs a string of English letters. It has only this part currently:
   
 
* Basic Alphabet Notation
* Basic Alphabet Notation<ref name=":0">{{Cite web|last=Nirvana Supermind|url=https://integralview.wordpress.com/2021/01/12/basic-alphabet-notation/|title=Basic Alphabet Notation}} (Retrieved at UTC 12:00 13/01/2020)</ref><ref name="current">{{Cite web|last=Nirvana Supermind|url=https://integralview.wordpress.com/2021/01/12/basic-alphabet-notation/|title=Basic Alphabet Notation}} (Retrieved at Wed, 13 Jan 2021 22:44:39 GMT) </ref>
 
   
The creator clarifies that he or she intends to create at least six other parts:<ref name="body" />
+
The creator clarifies that he or she intends to create at least six other parts:<ref name=":0" />
   
 
* Basic Cascading Alphabet Notation
 
* Basic Cascading Alphabet Notation
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Note that the creator is using (single) capital letters to represent variables, and single small letters to mean the actual letters. For example, "A" in this article means a variable instead of the capital 'A', while "a" in this article means the actual letter 'a'.
 
Note that the creator is using (single) capital letters to represent variables, and single small letters to mean the actual letters. For example, "A" in this article means a variable instead of the capital 'A', while "a" in this article means the actual letter 'a'.
   
== Feature ==
+
== Basic Alphabet Notation ==
   
 
=== Current definition ===
Similar to other articles on notations by the creator, there are many common features in this article. Especially, '''the creator tends to ignore other contributors by removing or replacing descriptions written by them without giving any discussions in the talk page.''' Indeed, the creator replaced the mathjax by the math tag,
 
<ref name="removal 1">[https://googology.wikia.org/wiki/Alphabet_notation?type=revision&diff=312922&oldid=312826 A difference page] of this article. (The creator removed a related information on the difference between actual values and intended values, added fake descriptions such as "all information here only applies to the old version of the notation.", replaced the mathjax by math tags, and so on.)</ref> although he or she has been essentially requested to stop it by other users.<ref>[https://googology.wikia.org/wiki/User_talk:Nirvana_Supermind?diff=next&oldid=302179 A difference page] of the talk page of the creator. (The creator was taught how to change the way to personally display mathematical formulae instead of replacing the mathjax by math tags.)</ref> Even after a request not to replace mathjax by math tags is clarified,<ref>[https://googology.wikia.org/wiki/Talk:Alphabet_notation?oldid=312927 The first version] of the talk page of this article. (The creator was taught not to replace the mathjax by math tags.)</ref> the creator repeats the replacement instead of changing the personal setting which allows the page to be personally displayed in a way following the preference.<ref name="removal 2">[https://googology.wikia.org/wiki/Alphabet_notation?type=revision&diff=312930&oldid=312928 A difference page] of this article. (The creator readded fake descriptions such as "all information here only applies to the old version of the notation." again even though another user explained the incorrectness because of the existence of common factors such as an issue, reordered sections even though the other user explained that the ordering should be kept because they share common explanations, and renamed the section even though the renaming had been reverted by the other user, who originally wrote the section, replaced the mathjax by math tags, even though the creator is requested not to do so without giving discussion in the talk page, by saying '''Great, so now the description of the section is also "fake blah-blah", even though it 1:1 matches up with the history of the source... Some things never change.'''.)</ref><ref name="removal 3">[https://googology.wikia.org/wiki/Alphabet_notation?type=revision&diff=312945&oldid=312932 A difference page] of this article. (The creator repeated the change, by saying '''Removing math tags that aren't rendering and, following the suggestion on "several" in case another commonality is found, clarifying how "two or more" is normal.'''.)</ref> '''The creator never listens others, and always persists the own preference, as the histories of the articles on all other notations by the creator, i.e. [[Rampant Array Notation]], [[Extensible Illion System]], [[Quick array notation]], and [[Infra Notation]].'''
 
   
 
Several informations here only applies to the current version of the notation, which is given after the issues on the original definition were pointed out.
More awfully, '''the creator tend to manipulate articles by adding unsourced descriptions without proofs and removing sourced descriptions.''' According to the creator, any sources are meaningless.<ref name="removal 2" />. Indeed, the creator wrote "all information here only applies to the old version of the notation.", even though there are many common informations such as the issue clearly pointed out in this article, the most part of the definition, and so on. Later, the creator agreed that the information was wrong.<ref name="fair">[https://googology.wikia.org/wiki/Talk:Alphabet_notation?type=revision&diff=312944&oldid=312939 A difference page] of the talk page of this article. (The creator said '''Fair enough, I will change it to "several informations". However, it was not a fake information, just a mistake.''')</ref> Although another user had clearly pointed out the incorrectness of the information<ref>[https://googology.wikia.org/wiki/Alphabet_notation?type=revision&diff=312925&oldid=312922 A difference page] of this article. (A user clarified '''"Changing the section order in order to solve the fake description "all information blah-blah".''')</ref> and the creator even reverted the corrections,<ref name="removal 2" /><ref name="removal 3" /> the creator insists the information was just a mistake but not a fake.<ref name="fair" /> For details on the creator's manipulation and removement, see the main articles on the other notations.
 
   
 
The expressions in this notation are of this form:
Also, the creator tend to write wrong expectations on the well-definedness or the growth rate by saying '''most likely'''. Unfortunately, this notation is not a counterexample. The creator stated that the notation '''most likely''' has double-exponential growth rate, but it is elementary to check the incorrectness of the expectation, as we will explain in [[#Growth rate|Growth rate]] section. For details on the creator's '''most likely''' expectations, see the main articles on the other notations.
 
   
 
(ABCDEFGHIJKLMN…)
== Basic Alphabet Notation ==
 
   
 
Here the “ABCDEFGHIJKLMN…” are a sequence of small letters in the Latin alphabet. The wrapping braces are simply to distinguish the expressions from actual words. () is also a valid expression. An example of a valid expression is (abc).
Since the current version shares one issue with the original version and the current version is essentially based on the alternative definition in [[#Alternative definition|Alternative definition]] section, we explain the original definition first.
 
  +
 
Terminology because they make the definition easier to write:
 
<br />
  +
 
# ord(A) for the letter A is defined as 1 if A = "a" 2 if A = "b" 3 if A = "c" 4 if A = "d" etc.
 
# len(A) for the expression A is defined as is the number of letters in A.
 
# P(A) for the integer A is defined as the Ath prime (zero-based index). So P(0) = 2.
  +
 
Note that P(A) is ill-defined when A = -1. Every expression output a large number. To solve a (possibly empty) expression, we need some rules as follows:
  +
  +
 
# () = 1
 
# (#A) = (#)P(len(#))<sup>ord(A)</sup>
  +
  +
  +
 
Here # denotes a substring of the current expression. It can also be empty. If there are multiple rules to apply to a single expression, the uppermost-numbered rule which is applicable and whose result is a valid expression will be applied.
  +
 
==== Examples ====
 
(abc) = 2250
   
 
=== Original definition ===
 
=== Original definition ===
Several information here only applies to the old version of the notation.
+
All information here only applies to the old version of the notation.
   
 
The expressions in this notation are of this form:
 
The expressions in this notation are of this form:
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==== Issues ====
 
==== Issues ====
   
 
The description "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."<ref name=":0" /> is weird, because there is no valid expression to which two rules are applicable. Moreover, there are no more rules, while the creator expresses "two or '''more rules'''". But this is no problem with this, as the creator specified '''or''' rather than "and", meaning that the "two" is inclusive.
'''Unfortunately, the last paragraph of this section is the target of the removal by the creator.'''<ref name="removal 1" /><ref name="removal 2" /> Since we should not hide an actual feature of the issues, we keep the information.
 
 
The description "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."<ref name=":0" /> is weird, because there is no valid expression to which two rules are applicable. Moreover, there are no more rules, while the creator expresses "two or '''more rules'''".
 
 
According to the creator, it is not an issue because it does not cause a mathematical logical error.<ref name="removal 3/> However, the point is the weirdness of the redundant explanation rather than the logical error. For example, if we define a number \(X\) as "\(1\) if \(1\) is positive and \(0\) if \(1\) is non-positive", then there is no logical error but everybody except for the creator feels weird.
 
   
 
Moreover, (abc) is intended to coincide with 2250, according to the creator.<ref name=":0" /> However, the actual value should be computed in the following way:
 
Moreover, (abc) is intended to coincide with 2250, according to the creator.<ref name=":0" /> However, the actual value should be computed in the following way:
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: = (a)P(1+1)<sup>2</sup>7<sup>3</sup> = (a)5<sup>2</sup>7<sup>3</sup> = ()P(len()+1)<sup>ord(a)</sup>5<sup>2</sup>7<sup>3</sup> = ()P(0+1)<sup>1</sup>5<sup>2</sup>7<sup>3</sup>
 
: = (a)P(1+1)<sup>2</sup>7<sup>3</sup> = (a)5<sup>2</sup>7<sup>3</sup> = ()P(len()+1)<sup>ord(a)</sup>5<sup>2</sup>7<sup>3</sup> = ()P(0+1)<sup>1</sup>5<sup>2</sup>7<sup>3</sup>
 
: = ()3<sup>1</sup>5<sup>2</sup>7<sup>3</sup> = 1×3<sup>1</sup>5<sup>2</sup>7<sup>3</sup> = 25725
 
: = ()3<sup>1</sup>5<sup>2</sup>7<sup>3</sup> = 1×3<sup>1</sup>5<sup>2</sup>7<sup>3</sup> = 25725
Therefore the original definition is not compatible with the intended behaviour. Since the creator tends to change either one of the original definition or the intended value, it is not special for a notation by the creator. For examples of the differences of actual values and intended values of notations by the creator, see [[Extensible Illion System#Example]] and [[Infra Notation#Example 3]].
+
Therefore the original definition is not compatible with the intended behaviour.
  +
  +
However, the new definition has fixed some of these issues, and is compatible with the intended value.
  +
   
 
==== Examples ====
 
==== Examples ====
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(abc) = 25725
 
(abc) = 25725
   
=== Alternative definition ===
+
=== P進大好きbot's definition ===
   
It is quite elementary to solve the issues explained above: We have only to remove the weird description of the application of two or more rules and change rule 2. In order to make the solution clearer, we explain the precise alternative formulation.
+
[[User:P進大好きbot]] created a fixed definition of the notation, before the creator updated their definition. It is quite elementary to solve the issues of the old notation: We have only to remove the weird description of the application of two or more rules and change rule 2. In order to make the solution clearer, we explain the precise alternative formulation.
   
Let \(\mathbb{N}\) denote the set of non-negative integers, and \(T\) the set of formal strings consisting of small letters in the Latin alphabet. For an \(A \in T\), we denote by \(\textrm{len}(A)\) the length of \(A\). For an \(\alpha \in T\) of length \(1\), we denote by \(\textrm{ord}(\alpha)\) the positive integer corresponding to the ordinal numeral of α with respect to the usual ordering of small letters in the Latin alphabet. For example, we have \(\textrm{ord}(a) = 1\), \(\textrm{ord}(b) = 2\), and \(\textrm{ord}(c) = 3\). For an \(n \in \mathbb{N}\), we denote by \(P(n)\) the \((1+n)\)-th prime number. For example, we have \(P(0) = 2\), \(P(1) = 3\), and \(P(2) = 5\).
+
Let <math>\mathbb{N}</math> denote the set of non-negative integers, and <math>T</math> the set of formal strings consisting of small letters in the Latin alphabet. For an <math>A \in T</math>, we denote by <math>\textrm{len}(A)</math> the length of <math>A</math>. For an <math>\alpha \in T</math> of length <math>1</math>, we denote by <math>\textrm{ord}(\alpha)</math> the positive integer corresponding to the ordinal numeral of α with respect to the usual ordering of small letters in the Latin alphabet. For example, we have <math>\textrm{ord}(a) = 1</math>, <math>\textrm{ord}(b) = 2</math>, and <math>\textrm{ord}(c) = 3</math>. For an <math>n \in \mathbb{N}</math>, we denote by <math>P(n)</math> the <math>(1+n)</math>-th prime number. For example, we have <math>P(0) = 2</math>, <math>P(1) = 3</math>, and <math>P(2) = 5</math>.
   
 
We define a total computable function
 
We define a total computable function
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\end{eqnarray*}
 
\end{eqnarray*}
 
in the following recursive way:
 
in the following recursive way:
# If \(\textrm{len}(A) = 0\), then set \((A) := 1\).
+
# If <math>\textrm{len}(A) = 0</math>, then set <math>(A) := 1</math>.
# Suppose \(\textrm{len}(A) \neq 0\).
+
# Suppose <math>\textrm{len}(A) \neq 0</math>.
## Denote by \(\alpha \in T\) the formal string of length \(1\) given as the rightmost letter of \(A\).
+
## Denote by <math>\alpha \in T</math> the formal string of length <math>1</math> given as the rightmost letter of <math>A</math>.
## Denote by \(B \in T\) the formal string given by removing the right most letter from \(A\).
+
## Denote by <math>B \in T</math> the formal string given by removing the right most letter from <math>A</math>.
## Set \((A) := (B)P(\textrm{len}(B))^{\textrm{ord}(\alpha)}\).
+
## Set <math>(A) := (B)P(\textrm{len}(B))^{\textrm{ord}(\alpha)}</math>.
The totality follows from the induction on \(\textrm{len}(A)\), and we have
+
The totality follows from the induction on <math>\textrm{len}(A)</math>, and we have
 
\begin{eqnarray*}
 
\begin{eqnarray*}
 
(abc) & = & (ab)P(\textrm{len}(ab))^{\textrm{ord}(c)} = (ab)P(2)^3 = (ab)5^3 = (a)P(\textrm{len}(a))^{\textrm{ord}(b)}5^3 \\
 
(abc) & = & (ab)P(\textrm{len}(ab))^{\textrm{ord}(c)} = (ab)P(2)^3 = (ab)5^3 = (a)P(\textrm{len}(a))^{\textrm{ord}(b)}5^3 \\
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which is compatible with the creator's intention.
 
which is compatible with the creator's intention.
   
Another possible alternative definition is given by replacing \(P\) by the enumeration of prime numbers with respect to one-based indexing, i.e. \(P(1) = 2\), \(P(2) = 3\), and \(P(3) = 5\), instead of changing rule 2.
+
Another possible alternative definition is given by replacing <math>P</math> by the enumeration of prime numbers with respect to one-based indexing, i.e. <math>P(1) = 2</math>, <math>P(2) = 3</math>, and <math>P(3) = 5</math>, instead of changing rule 2.
 
==== Examples ====
 
\((abc) = 2250\)
 
 
=== Current definition ===
 
 
After the issues in the original definition are pointed out in [[#Issues|Issues]] secition, the creator updated the definition following the alternative definition in [[#Alternative definition|Alternative definition]] section.
 
 
Several information here only applies to the current version of the notation, which is given after the issues on the original definition were pointed out.
 
 
The expressions in this notation are of this form:
 
 
(ABCDEFGHIJKLMN…)
 
 
Here the “ABCDEFGHIJKLMN…” are a sequence of small letters in the Latin alphabet. The wrapping braces are simply to distinguish the expressions from actual words. () is also a valid expression. An example of a valid expression is (abc).
 
 
Terminology because they make the definition easier to write:
 
<br />
 
 
# ord(A) for the letter A is defined as 1 if A = "a" 2 if A = "b" 3 if A = "c" 4 if A = "d" etc.
 
# len(A) for the expression A is defined as is the number of letters in A.
 
# P(A) for the integer A is defined as the Ath prime (zero-based index). So P(0) = 2.
 
 
Note that P(A) is ill-defined when A = -1. Every expression output a large number. To solve a (possibly empty) expression, we need some rules as follows:
 
 
 
# () = 1
 
# (#A) = (#)P(len(#))<sup>ord(A)</sup>
 
 
 
 
Here # denotes a substring of the current expression. It can also be empty. If there are multiple rules to apply to a single expression, the uppermost-numbered rule which is applicable and whose result is a valid expression will be applied. Although it is not clarified, "A" in rule 2 is a variable which means a single small letter rather than a valid expression, because the creator considers ord(A). Readers should be careful that the creator uses "A" also as variables for a valid expression and an integer, as the defitions of len and P show.
 
 
==== Issues ====
 
 
The issue on the weird explanation on the possibility of the application of two or more rules remains, as the creator just rephrased it as '''If there are multiple rules to apply to a single expression, the uppermost-numbered rule which is applicable and whose result is a valid expression will be applied.'''
 
 
==== Examples ====
 
(abc) = 2250
 
 
   
 
=== Growth rate ===
 
=== Growth rate ===
 
The limit function, also provided by [[User:P進大好きbot]], is given by the map assigning to each <math>n \in \mathbb{N}</math> the value <math>(A(n))</math> of the formal string <math>A(n)</math> of length <math>n</math> consisting of <math>z</math>. It is quite elementary to show an upperbound, thanks to Bertrand's postulate.
The following is the analysis by the creator in the original source:<ref name=":0" />
 
: ''It’s hard to compare the asymptotic growth rate of this notation with the other notations like [[Fast-growing hierarchy]], mainly because they input lists of numbers rather than letters. However, we can use [https://en.wikipedia.org/wiki/Fibonacci_word fibonacci word] function “S” to convert a number into a string of letters. After I tried some numbers, it looks like (S(N)) is likely asymptotic to f<sup>2</sup>(f<sup>2</sup>(N)) (double-exponential growth rate) in the FGH. That’s it for this part of the notation.''
 
However, there is no written explicit definition of S or f<sup>2</sup> in the source, the meaning is ambiguous.
 
 
Instead, a limit function of the notation is easily given by the map assigning to each \(n \in \mathbb{N}\) the value \((A(n))\) of the formal string \(A(n)\) of length \(n\) consisting of \(z\). There is no difficulty, of which the creator was afraid, in the formulation, and we do not need complicated coding like the Fibonacci word function. Also, it is quite elementary to show an upperbound, thanks to Bertrand's postulate.
 
   
 
{| class="wikitable"
 
{| class="wikitable"
 
! Proposition
 
! Proposition
 
|-
 
|-
| For any \(n \in \mathbb{N}\), \((A(n)) \leq 10^{4n(n+1)}\) holds with respect to the alternative definition.
+
| For any <math>n \in \mathbb{N}</math>, <math>(A(n)) \leq 10^{4n(n+1)}</math> holds with respect to the alternative definition.
 
|-
 
|-
 
|}
 
|}
   
; Proof
+
==== Proof ====
: We show the assertion by the induction on \(n\). If \(n = 0\), then we have \((A(n)) = () = 1 = 10^{4n(n+1)}\). Suppose \(n > 0\). We have
+
: We show the assertion by the induction on <math>n</math>. If <math>n = 0</math>, then we have <math>(A(n)) = () = 1 = 10^{4n(n+1)}</math>. Suppose <math>n > 0</math>. We have
 
\begin{eqnarray*}
 
\begin{eqnarray*}
 
& & (A(n)) = (A(n-1))P(\textrm{len}(A(n-1)))^{\textrm{ord}(z)} = (A(n-1))P(n-1)^{26} \\
 
& & (A(n)) = (A(n-1))P(\textrm{len}(A(n-1)))^{\textrm{ord}(z)} = (A(n-1))P(n-1)^{26} \\
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\end{eqnarray*}
 
\end{eqnarray*}
   
Similarly, it is possible to obtain a lowerbound of the value for the formal string of length \(n\) consisting of \(a\). In this sense, the asymptotic growth rate of the limit function can be easily estimated.
+
Similarly, it is possible to obtain a lowerbound of the value for the formal string of length <math>n</math> consisting of <math>a</math>. In this sense, the asymptotic growth rate of the limit function can be easily estimated.
 
Later, the creator updated the description.<ref name="current" /> Using the highest possible letters, the Fibonacci word function S is given by the following recursive definition according to the creator: S(0)='y',S(1)='z',S(n) = S(n-1)S(n-2). For example, S(2)='yz'. The creator claims that the function (S(n)) '''most likely''' has double-exponential growth rate, because they tested the ratios (S(n))/(S(n-1)) for values 1-14.<ref name="current" /> However, it does not solve the issue that f<sup>2</sup> is undefined. Since the creator replaced f<sup>2</sup> by f<sub>2</sub>, he or she might confounded f<sub>2</sub> with f<sup>2</sup> for some f.
 
 
Unfortunately, it is obvious that the resulting function (S(n)) is bounded by \((A(n))\), because \(\textrm{ord}(y) = 25 < 26 = \textrm{ord}(z)\). Therefore the creator's statement that it has double-exponential growth rate or is approximated to \(f_2(f_2(n))\) is a fake, as \((A(n))\) is bounded by \(10^{4n(n+1)}\). Note that the current version is essentially based on the alternative definition, and hence the computation of the upperbound completely works for the current version. As readers might have already known, the creator's "'''most likely'''" statements are usually fakes.
 
 
== See also ==
 
'''Unfortunately, this section is the target of the removal by the creator.'''<ref name="removal 1" /> Also, the creator somewhy repeating to remove a link to Category:Numbers by Nirvana Supermind.
 
* [[Rampant Array Notation]]
 
* [[Extensible Illion System]]
 
* [[Quick array notation]]
 
* [[Infra Notation]]
 
   
 
== Sources ==
 
== Sources ==
   
 
<references />
 
<references />
[[Category:Numbers by Nirvana Supermind]]
 
 
[[Category:Notations]]
 
[[Category:Notations]]
 
[[Category:Functions]]
 
[[Category:Functions]]

Revision as of 08:45, 9 February 2021

Alphabet notation is a notation created by Wikia user Nirvana Supermind.[1][2][3]. It is a based on recursion, and inputs a string of English letters. It has only this part currently:

  • Basic Alphabet Notation

The creator clarifies that he or she intends to create at least six other parts:[2]

  • Basic Cascading Alphabet Notation
  • Nested Basic Cascading Alphabet Notation
  • Two-level Cascading Alphabet Notation
  • Cascading Alphabet notation
  • Tetrational Alphabet Notation
  • Arrow Alphabet Notation


Note that the creator is using (single) capital letters to represent variables, and single small letters to mean the actual letters. For example, "A" in this article means a variable instead of the capital 'A', while "a" in this article means the actual letter 'a'.

Basic Alphabet Notation

Current definition

Several informations here only applies to the current version of the notation, which is given after the issues on the original definition were pointed out.

The expressions in this notation are of this form:

(ABCDEFGHIJKLMN…)

Here the “ABCDEFGHIJKLMN…” are a sequence of small letters in the Latin alphabet. The wrapping braces are simply to distinguish the expressions from actual words. () is also a valid expression. An example of a valid expression is (abc).

Terminology because they make the definition easier to write:

  1. ord(A) for the letter A is defined as 1 if A = "a" 2 if A = "b" 3 if A = "c" 4 if A = "d" etc.
  2. len(A) for the expression A is defined as is the number of letters in A.
  3. P(A) for the integer A is defined as the Ath prime (zero-based index). So P(0) = 2.

Note that P(A) is ill-defined when A = -1. Every expression output a large number. To solve a (possibly empty) expression, we need some rules as follows:


  1. () = 1
  2. (#A) = (#)P(len(#))ord(A)


Here # denotes a substring of the current expression. It can also be empty. If there are multiple rules to apply to a single expression, the uppermost-numbered rule which is applicable and whose result is a valid expression will be applied.

Examples

(abc) = 2250

Original definition

All information here only applies to the old version of the notation.

The expressions in this notation are of this form:

(ABCDEFGHIJKLMN…)

Here the “ABCDEFGHIJKLMN…” are a sequence of small letters in the Latin alphabet. The wrapping braces are simply to distinguish the expressions from actual words. () is also a valid expression. An example of a valid expression is (abc).

Terminology because they make the definition easier to write:

  1. ord(A) for the letter A is defined as 1 if A = "a" 2 if A = "b" 3 if A = "c" 4 if A = "d" etc.
  2. len(A) for the expression A is defined as is the number of letters in A.
  3. P(A) for the integer A is defined as the Ath prime (zero-based index). So P(0) = 2.

Note that P(A) is ill-defined when A = -1. Every expression is intended to output a large number. To solve a (possibly empty) expression, we need some rules as follows:


  1. () = 1
  2. (#A) = (#)P(len(#)+1)ord(A)


Here # denotes a substring of the current expression. It can also 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. Although it is not clarified, "A" in rule 2 is a variable which means a single small letter rather than a valid expression, because the creator considers ord(A). Readers should be careful that the creator uses "A" also as variables for a valid expression and an integer, as the defitions of len and P show.


Issues

The description "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."[2] is weird, because there is no valid expression to which two rules are applicable. Moreover, there are no more rules, while the creator expresses "two or more rules". But this is no problem with this, as the creator specified or rather than "and", meaning that the "two" is inclusive.

Moreover, (abc) is intended to coincide with 2250, according to the creator.[2] However, the actual value should be computed in the following way:

(abc) = (ab)P(len(ab)+1)ord(c) = (ab)P(2+1)3 = (ab)73 = (a)P(len(a)+1)ord(b)73
= (a)P(1+1)273 = (a)5273 = ()P(len()+1)ord(a)5273 = ()P(0+1)15273
= ()315273 = 1×315273 = 25725

Therefore the original definition is not compatible with the intended behaviour.

However, the new definition has fixed some of these issues, and is compatible with the intended value.


Examples

Intended Value

(abc) = 2250

Actual Value

(abc) = 25725

P進大好きbot's definition

User:P進大好きbot created a fixed definition of the notation, before the creator updated their definition. It is quite elementary to solve the issues of the old notation: We have only to remove the weird description of the application of two or more rules and change rule 2. In order to make the solution clearer, we explain the precise alternative formulation.

Let denote the set of non-negative integers, and the set of formal strings consisting of small letters in the Latin alphabet. For an , we denote by the length of . For an of length , we denote by the positive integer corresponding to the ordinal numeral of α with respect to the usual ordering of small letters in the Latin alphabet. For example, we have , , and . For an , we denote by the -th prime number. For example, we have , , and .

We define a total computable function \begin{eqnarray*} () \colon T & \to & \mathbb{N} \\ A & \mapsto & (A) \end{eqnarray*} in the following recursive way:

  1. If , then set .
  2. Suppose .
    1. Denote by the formal string of length given as the rightmost letter of .
    2. Denote by the formal string given by removing the right most letter from .
    3. Set .

The totality follows from the induction on , and we have \begin{eqnarray*} (abc) & = & (ab)P(\textrm{len}(ab))^{\textrm{ord}(c)} = (ab)P(2)^3 = (ab)5^3 = (a)P(\textrm{len}(a))^{\textrm{ord}(b)}5^3 \\ & = & (a)P(1)^2 5^3 = (a)3^2 5^3 = ()P(\textrm{len}())^{\textrm{ord}(a)}3^2 5^3 = ()P(0)^1 3^2 5^3 \\ & = & ()2^1 3^2 5^3 = 1 \cdot 2^1 3^2 5^3 = 2250, \end{eqnarray*} which is compatible with the creator's intention.

Another possible alternative definition is given by replacing by the enumeration of prime numbers with respect to one-based indexing, i.e. , , and , instead of changing rule 2.

Growth rate

The limit function, also provided by User:P進大好きbot, is given by the map assigning to each the value of the formal string of length consisting of . It is quite elementary to show an upperbound, thanks to Bertrand's postulate.

Proposition
For any , holds with respect to the alternative definition.

Proof

We show the assertion by the induction on . If , then we have . Suppose . We have

\begin{eqnarray*} & & (A(n)) = (A(n-1))P(\textrm{len}(A(n-1)))^{\textrm{ord}(z)} = (A(n-1))P(n-1)^{26} \\ & \leq & (A(n-1)) (2^n)^{26} < ((A(n-1)) 10^{8n} < 10^{4(n-1)n} \times 10^{8n} \\ & = & 10^{4n(n+1)}. \end{eqnarray*}

Similarly, it is possible to obtain a lowerbound of the value for the formal string of length consisting of . In this sense, the asymptotic growth rate of the limit function can be easily estimated.

Sources

  1. Nirvana Supermind. Alphabet Notation.  (Retrieved at UTC 12:00 13/01/2020)
  2. 2.0 2.1 2.2 2.3 Nirvana Supermind. Basic Alphabet Notation.  (Retrieved at UTC 12:00 13/01/2020)
  3. Nirvana Supermind. Basic Alphabet Notation.  (Retrieved at Wed, 13 Jan 2021 22:44:39 GMT)
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