Hyper-Robin notation

Hyper-Robin notation is a notation to generate large numbers created by Googology Wiki user Binary198.Binary198, A mathematical journey retrieved at 2021-10-30 0:30 It is ill-defined due to the lack of the definition of the domain, the lack of the precise quantification of variables, overlapping partial specialisation, ambiguous use of an ellipsis, and use of invalid expressions, as we will explain later in #Issues section.
Binary198 has expressed that they do not plan on fixing the notation, but are glad that it has been noticed by others (see Talk:Abellion).
== Definition ==
The actual definition of Hyper-Robin notation is given in the following way:
: f{0}(n) = 1
: f{1}(n) = n
: f{2}(n) = 2n
: f{n}(n) = n↑2
: f{n+1}(n) = n↑3, not (n+1)(n+1)
: f{n+2}(n) = n↑4
: f{2n}(n) = n↑(n+2)
: f{3n}(n) = n↑(2n+2)
: ...
: f{x, y}(n) = (2)(f{x}(n↑y))↑f{x}(n↑y) −1 n
== Issues ==
Since the definition is given in a personal website which is not peer-reviewed, there are many issues is the notation:
# The lack of the definition of the domain: The domain of f is not clarified. Therefore it is ambiguous what tuple of arguments can be inputs of f. In this article, we guess that f{a}(n) is supposed to be defined for any sequence a of non-nagetive integers of length ∈{1,2} and any non-negative integer n.
# The lack of the precise quantification of variables: The variables n, x, and y are not appropriately quantified with specific range. Therefore it is ambiguous what conditions are assumed in each computation rule. In this article, we guess that n, x, and y are supposed to run through all non-negative integers.
# Overlapping partial specialisation: We can apply both the third and the fifth rule can be applied to f{2}(1), but the result depends on which rule is applied: The third rule returns 2×1 = 2, while the fifth rule returns 1↑3 = 1.
# Ambiguous use of an ellipsis: The rule is decribed with ambiguous ellipses. For example, this rule does not have information of expressions like f3(4).
As a conclusion, this notation is ill-defined.
Moreover, this notation is used to define Abellion as f{100}{10}, but this expression is invalid in this notation. Indeed, the second argument requires parentheses rather than braces. Even if {10} is a typo of (10), the value of f{100}(10) is unclear because of the issue on the ambiguity of the ellipses.
== Sources ==
Category:Notations
Category:Incomplete