Dollar function

Dollar function is a notation for large numbers made by googology wiki user .User:Wythagoras/Dollar function Retrieved 2021/07/06 The first version was posted at June 5, 2013.User blog:Wythagoras/Dollar function part 1
The notation is currently composed of 5 parts, which is as follows:
* Bracket notation
* Extended Bracket notation
* Linear Array Notation
* Dimensional Array Notation
* Nested Array Notation
As of July 6, 2021, it is written that the new definition is complete up to the Extended Bracket notation. It has several issues on the definition, but is considered to be quite notable in googology. For more details, see #Issues section annd #Notability section.
== Bracket notation ==
Bracket notation is the first part of Dollar function, which uses only normal brackets, defined as follows.User:Wythagoras/Dollar function/Bracket notation Retrieved 2021/07/06
Scan from the right to the left until you find a number. That number is the active number, also scan in levels if you come across them.
\(\bullet\) can be anything (that is well-defined at that point)
#\(a\$ = a\)
#\(a\$\bullet b = a+b\$\bullet\)
#\([\bullet b]_{\bullet_2} = [\bullet b-1]_{\bullet_2}[\bullet b-1]_{\bullet_2}...[\bullet b-1]_{\bullet_2}[\bullet b-1]_{\bullet_2}\) with \(a\) pairs of brackets.
#\(\bullet 0 = \bullet\) if \(\bullet\) isn't empty
#\([0] = a\)
Note: Brackets without level have level 1.
According to the analysis by Wythagoras, it works quite similiar to the Hardy Hierarchy and Kirby-Paris Hydra, and the limit of the growth rate is \(f_{\varepsilon_0}(a)\) as shown below.
: \(a\$[[0]_2] = [[...[[0]]...]] \approx f_{\varepsilon_0}(a)\)
However, the analysis does not make sense, for Bracket notation is ill-defined as we will explain in #Issues section.
We note that the original definition describes the third rule in the following different way:
: 3. \(a\$ \circ [b+1 \bullet] \circ = a\$ \circ [b \bullet][b \bullet]\ldots[b \bullet][b \bullet] \circ\) \(a [b \bullet]\)'s Where \(0\) and \(b\) are the less nested numbers
: \(\bullet\) is the remainder of the array and \(\circ\) contains only brackets
Here, the terminology "less nested number" is ambiguous. The reader should be careful that the same symbol in a single equality stands for a single term in usual mathematics, but the \(\circ\) in the rough description is perhaps intended not to follow this rule.
== Extended Bracket notation ==
Extended Bracket notation is the extended version of Bracket Notation, and the second part of Dollar Function. It is defined as follows.User:Wythagoras/Dollar function/Extended Bracket notation Retreved 2021/07/06
Scan from the right to the left until you find a number. That number is the active number, also scan in levels if you come across them.
\(\bullet\) can be anything (that is well-defined at that point)
#\(a\$ = a\)
#\(a\$\bullet b = a+b\$\bullet\)
#\([\bullet b]_{\bullet_2} = [\bullet b-1]_{\bullet_2}[\bullet b-1]_{\bullet_2}...[\bullet b-1]_{\bullet_2}[\bullet b-1]_{\bullet_2}\) with a pairs of brackets.
#\(\bullet 0 = \bullet\) if \(\bullet\) isn't empty
#\([0] = a\)
#\([[0]_{\bullet b}\bullet_2]_{\bullet b-1} = [[[[...[[0]\bullet_2]_{\bullet b-1}...\bullet_2]_{\bullet b-1}\bullet_2]_{\bullet b-1}\bullet_2]_{\bullet b-1}\bullet_2]_{\bullet b-1}\) with \(a+1\) pairs of brackets in total and if there isn't a bracket with level \(\bullet b-1\), add it, find the nearest bracket with a lower level and add it directly inside it.
Note: Brackets without level have level 1.
According to the analysis by Wythagoras, it is similiar to \(\Omega\)s in FGH and the extended Buchholz Hydra, and the analysis is shown to the level up to \(\vartheta(\Omega_\Omega)\) as shown below as of July 6, 2021.
: \([[0]_{[0]_2}]\) has level \(\vartheta(\Omega_\Omega)\)
However, the analysis does not make sense, for Extended Bracket notation is ill-defined by the same reasons as the ill-definedness of Bracket notation. Moreover, the \(\vartheta\) function is unspecified, and hence the estimation is debatable even if we fix the definition. Moreover, the source includes the inappropriate expression \(\psi(\psi_I(0))\), whose definition is unspecified and which has many know common mistakes. For more details, see also the countable limit of Extended Buchholz's function#Warning.
== Issues ==
There are several issues on the definition pointed out by p進大好きbot:A difference page of the talk page of this article.
# A notation is a function, and a function is a pair of the domain and the assignment. However, there does not seem to be a written definition of dollar function. Therefore it can be considered as an ill-defined notation. Although many googologists do not care about such ill-definedness due to the lack of the definition of the domain, this issue causes another issue, which we will explain in the next line.
# There is no definition of \(a\) and \(b\) appearing in the computation rule of Bracket notation. Perhaps they should be regarded as integers, non-negative integers, or positive integers, because \(b\) admits decrements.
## If \(b\) is intended to be any integer, then the process does not seem to be well-founded. Therefore the notation is ill-defined even if we ignore the lack of the definition of the domain.
## If \(b\) is intended to be any non-negative integer, then the result of \([0]_0\) includes an invalid expression \([-1]_0\). Therefore the notation is ill-defined even if we ignore the lack of the definition of the domain.
## If \(b\) is intended to be any positive integer, then the result of \([1]_2\) includes an invalid expression \([0]_2\). Therefore the notation is ill-defined even if we ignore the lack of the definition of the domain.
# The third and fifth rules refer to the value of \(a\), which is not uniquely determined by the input expression in the left hand side. Perhaps the creator forgot to insert \(a \$\).
# There is no rule to relate an expression with a subscript to an expression without a subscript, and to relate an expression with a subscript to an expression with another subscript. Perhaps the creator forgot to set rules to remove or decrement subscripts.
## One hint "Brackets without level have level 1" is given, but it just means that expressions like \([0]\) are of level \(1\).
## Even if we interprete that level of an expression with a subscript is the subscript, the coincidence of the level of two distinct expressions does not ensure the coincidence of the values of the expressions, as the expressions \([0]\) and \([1]\) of level \(1\) are perhaps intended to have two distinct values. Therefore there is no rule to remove the subscript from \([0]_0\) or \([0]_1\).
## Similarly, we have no rule to interprete \([0]_2\) as an expression with smaller subscripts.
# The scanning process is quite ambiguous, because it refers to the horizontal location of a letter in a string admitting subscripts.
In addition, the description "\(\bullet\) can be anything (that is well-defined at that point)" refers to the well-definedness of a notation itself, and includes a circular logic or an unformalised predicate. As a consequence, Bracket notation can be considered as an ill-defined notation.
== Notability ==
According to Denis Maksudov, dollar function was the first notation in this community working in a similar way to the Hardy hierarchy and Hydra.A difference page of the talk page of this article. Therefore it is considered to be quite notable in googology from the point of view on the historical significance.
== Sources ==
Category:Notations
Category:Functions
Category:Incomplete