m (Undo revision 312930 by Nirvana Supermind (talk) Reverting the fake removement and the math tag ignoring the talk page) Tag: Undo 
(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) Tag: Source edit 

Line 17:  Line 17:  
== Basic Alphabet Notation == 
== Basic Alphabet Notation == 

+  === Current definition === 

−  Since the current version shares one issue with the original version, we explain the original definition first. 

+  
⚫  
+  
⚫  
+  
⚫  
+  
⚫  
+  
⚫  
⚫  
+  
⚫  
⚫  
⚫  
+  
⚫  
+  
+  
⚫  
⚫  
+  
+  
+  
⚫  
+  
⚫  
⚫  
=== Original definition === 
=== Original definition === 

−  +  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: 

Line 48:  Line 76:  
==== Issues ==== 
==== Issues ==== 

−  The description "If there are two or more distinct rules to apply to a single expression, the uppermostnumbered 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'''". 
+  The description "If there are two or more distinct rules to apply to a single expression, the uppermostnumbered 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. 
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: 

Line 54:  Line 82:  
: = (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. 
+  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 ==== 

Line 63:  Line 94:  
(abc) = 25725 
(abc) = 25725 

−  === 
+  === P進大好きbot's definition === 
−  It is quite elementary to solve the issues 
+  [[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 
+  Let <math>\mathbb{N}</math> denote the set of nonnegative 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 

Line 75:  Line 106:  
\end{eqnarray*} 
\end{eqnarray*} 

in the following recursive way: 
in the following recursive way: 

−  # If 
+  # If <math>\textrm{len}(A) = 0</math>, then set <math>(A) := 1</math>. 
−  # Suppose 
+  # Suppose <math>\textrm{len}(A) \neq 0</math>. 
−  ## Denote by 
+  ## 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 
+  ## Denote by <math>B \in T</math> the formal string given by removing the right most letter from <math>A</math>. 
−  ## Set 
+  ## Set <math>(A) := (B)P(\textrm{len}(B))^{\textrm{ord}(\alpha)}</math>. 
−  The totality follows from the induction on 
+  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 \\ 

Line 88:  Line 119:  
which is compatible with the creator's intention. 
which is compatible with the creator's intention. 

−  Another possible alternative definition is given by replacing 
+  Another possible alternative definition is given by replacing <math>P</math> by the enumeration of prime numbers with respect to onebased indexing, i.e. <math>P(1) = 2</math>, <math>P(2) = 3</math>, and <math>P(3) = 5</math>, instead of changing rule 2. 
−  === 
+  === Growth rate === 
+  It's hard to compare the asymptotic growth rate of this notation with the other notations like [[Fastgrowing hierarchy]], mainly because the limit function of them can be easily computed, while this one cannot as it inputs strings rather than numbers. The creator used the of {{wFibonacci word}} to convert numbers to strings. 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(n1)S(n2) 

−  After the issues in the original definition are pointed out in [[#IssuesIssues]] secition, the creator updated the definition following the alternative definition in [[#Alternative definitionAlternative definition]] section. 

⚫  
⚫  
⚫  An alternative 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. 

⚫  
−  
⚫  
−  
⚫  
−  
⚫  
⚫  
−  
⚫  
⚫  
⚫  
−  
⚫  
−  
−  
⚫  
⚫  
−  
−  
−  
−  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 uppermostnumbered 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 ==== 

−  
⚫  
−  
⚫  
⚫  
−  
−  
−  === Growth rate === 

−  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 [[Fastgrowing 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)) (doubleexponential 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. 

−  
⚫  
{ class="wikitable" 
{ class="wikitable" 

! Proposition 
! Proposition 

 
 

−   For any 
+   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 ==== 

−  : We show the assertion by the induction on 
+  : 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(n1))P(\textrm{len}(A(n1)))^{\textrm{ord}(z)} = (A(n1))P(n1)^{26} \\ 
& & (A(n)) = (A(n1))P(\textrm{len}(A(n1)))^{\textrm{ord}(z)} = (A(n1))P(n1)^{26} \\ 

Line 149:  Line 145:  
\end{eqnarray*} 
\end{eqnarray*} 

−  Similarly, it is possible to obtain a lowerbound of the value for the formal string of length 
+  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 descripttion. 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(n1)S(n2). 

−  
⚫  For example, S(2)=' 

−  
== Sources == 
== Sources == 

<references /> 
<references /> 

−  [[Category:Numbers by Nirvana Supermind]] 

[[Category:Notations]] 
[[Category:Notations]] 

[[Category:Functions]] 
[[Category:Functions]] 
Revision as of 00:37, 14 January 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
 Twolevel 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'.
Contents
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:
 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 (zerobased index). So P(0) = 2.
Note that P(A) is illdefined 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(#))^{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 uppermostnumbered 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:
 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 (zerobased index). So P(0) = 2.
Note that P(A) is illdefined 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
 (#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 uppermostnumbered 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 uppermostnumbered 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)7^{3} = (a)P(len(a)+1)^{ord(b)}7^{3}
 = (a)P(1+1)^{2}7^{3} = (a)5^{2}7^{3} = ()P(len()+1)^{ord(a)}5^{2}7^{3} = ()P(0+1)^{1}5^{2}7^{3}
 = ()3^{1}5^{2}7^{3} = 1×3^{1}5^{2}7^{3} = 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 nonnegative 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:
 If , then set .
 Suppose .
 Denote by the formal string of length given as the rightmost letter of .
 Denote by the formal string given by removing the right most letter from .
 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 onebased indexing, i.e. , , and , instead of changing rule 2.
Growth rate
It's hard to compare the asymptotic growth rate of this notation with the other notations like Fastgrowing hierarchy, mainly because the limit function of them can be easily computed, while this one cannot as it inputs strings rather than numbers. The creator used the of Fibonacci word to convert numbers to strings. 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(n1)S(n2)
For example, S(2)='zy'. The creator claims that the function (S(n)) has doubleexponential growth rate, because they tested the ratios (S(n))/(S(n1)) for values 114.^{[3]}
An alternative 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(n1))P(\textrm{len}(A(n1)))^{\textrm{ord}(z)} = (A(n1))P(n1)^{26} \\ & \leq & (A(n1)) (2^n)^{26} < ((A(n1)) 10^{8n} < 10^{4(n1)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
 ↑ Nirvana Supermind. Alphabet Notation. (Retrieved at UTC 12:00 13/01/2020)
 ↑ ^{2.0} ^{2.1} ^{2.2} ^{2.3} Nirvana Supermind. Basic Alphabet Notation. (Retrieved at UTC 12:00 13/01/2020)
 ↑ ^{3.0} ^{3.1} Nirvana Supermind. Basic Alphabet Notation. (Retrieved at Wed, 13 Jan 2021 22:44:39 GMT)