Y sequence

Y sequence is a difference sequence system introduced by a Japanese googologist Yukito. It is intended to be an extension of hyper primitive sequence system, which is a also an extension of primitive sequence system. Although it has not been formalised yet, the expression \((1,3)\) in Y sequence is expected to correspond to the limit of Bashicu matrix system version 2.3, and its novel idea has given a new direction in googology.

= Explanation =

An expression in Y sequence, which is also called a Y sequence, is a finite array \(a\) satisfying one of the following three conditions: For example, \((1)\), \((1,2,1,2)\), and \((1,3)\) are Y sequences, while \((0,1)\), \((2)\), or \((1,\omega+1)\) are not Y sequences.
 * 1) \(a\) is the empty sequence \(\).
 * 2) \(a\) is a non-empty sequence of positive numbers whose leftmost entry is \(1\).
 * 3) \(a\) is the sequence \((1,\omega)\).

We denote by \(\mathbb{Y}\) the set of Y sequences, and by \(\mathbb{N}_{>0}\) the set of positive integers. Assume that the expansion rule were formalised into a well-defined map \begin{eqnarray*} \textrm{Expand} \colon \mathbb{Y} \times \mathbb{N}_{>0} & \to & \mathbb{Y} \\ (a,n) & \mapsto & \textrm{Expand}(a,n). \end{eqnarray*} We define a partial computable function \begin{eqnarray*} \textrm{Y}[ \ ] \colon \mathbb{Y} \times \mathbb{N}_{>0} & \to & \mathbb{N} \\ (a,n) & \mapsto & \textrm{Y}a[n] \end{eqnarray*} in the following recursive way: Assume \(((1,\omega),n) \in \textrm{dom}([ \ ])\) for any \(n \in \mathbb{N}_{>0}\). Let \(f\) denote the total computable function \(n \mapsto \textrm{Y}(1,\omega)[n]\).
 * 1) If \(a = \), then \(\textrm{Y}a[n] = n\).
 * 2) If \(a\) is a non-empty sequence of positive integers whose leftmost entry is \(1\), then \(\textrm{Y}a[n] = \textrm{Y} \textrm{Expand}(a,n)[n]\).
 * 3) If \(a = (1,\omega)\), then \(\textrm{Y}a[n] = \textrm{Y}(1,n)[n]\).

Yukito named \(f^{2000}(1)\) Y sequence number. Since \((1,3)\) is supposed to correspond to the limit of Bashicu matrix system version 2.3, Y sequence number is supposed to be significantly greater than Bashicu matrix number with respect to the version. We note that the well-definedness of Bashicu matrix number is unknown, and hence the statement might not make sense even if \(\textrm{Expand}\) will be fully defined.

= Expansion =

Although Y sequence has not been formalised yet, Yukito has explained expansions for several examples. Here is a list of known expansions of Y sequences originally given by Yukito.

Needless to say, the table does not unqiuely characterise the expansion rule. Indeed, Yukito officially keeps the expansions of several Y sequences to be undecided.

= Alternative Formalisations =

Through the examples of expansions, several googologists are trying to find a formal rule (partially or essentially) consistent with the original expansions. The formalisations by others are not regarded as an official defintion of Y sequence, and should be distinguished from the original Y sequence. Sometimes several googologists introduce their formalisation as "the definition of Y sequence" or something like that, it does not mean that they are allowed to express so by Yukito or even that those are compatible with the original explanation by Yukito.

Yukito stated that Y sequence is the name only for the difference sequence system which he himself will complete, and he did not want others to name their own difference sequence systems Y sequence version 1.1 or something like that. Therefore others tend to call their own difference sequence systems distinct names unless they directly ask Yukito permissions.

= References =