User:Cloudy176/Department of bubbly negative numberbottles/Aarex hydra


 * revision as of 16:08, July 31, 2015 by Fluoroantimonic Acid

The Aarex hydra game is a one-player game played on trees labelled with any hydra. From the game arises a function \(\text{AH}(n)\) whose rate of growth is described by the ordinal \(\psi(\psi_I(0))\).

Rules
The game starts with any labelled tree \(T\), such that the root is marked with a special label (call it +) and every child of root has label 0. This tree is called a hydra.

At each step, we perform a transformation on the tree with two parameters: a leaf node \(a\) and a nonnegative integer \(n\). We alter the hydra using the following rules:


 * 1) If \(a\) has empty label, we proceed as in Kirby-Paris' game. Call the node's parent \(b\), and its grandparent \(c\). First we delete \(a\). If \(c\) exists (i.e. \(b\) is not the root), we make \(n\) copies of \(b\) and all its children and attach them to \(c\).
 * 2) If \(a\) has label \(u + \) (\(u\) is any hydra), we go down the tree looking for a node \(b\) with label \(v \leq u\) (which is guaranteed to exist, as nodes directly above root are all 0's). Consider \(b\) and subtree with root at \(b\), and call that subtree \(S\). Create a copy of \(S\), call it \(S'\). Within \(S'\), we relabel \(b\) with \(u\) and relabel \(a\) with \(0\). Back in the original tree, replace \(a\) with \(S'\).

Contributors
Googleaarex, Cloudy176, Fluoroantimonic Acid

Discussions

 * See this forum post for the proposal that caused the deletion of this page

Can you make this page and Array hydra page, please? AarexTiao 23:17, November 27, 2013 (UTC)

Just an honest question - do we have to delete all these pages? Are they doing anything bad? And, if so, can't we make bot to that? LittlePeng9 (talk) 13:49, December 22, 2013 (UTC)
 * That's just a cleaning-up our wiki, they don't mess up our work. Ikosarakt1 (talk ^ contribs) 14:10, December 22, 2013 (UTC)
 * I made this page. AarexTiao 15:26, December 22, 2013 (UTC)

Aarex, as you're a creator of this concept I'm asking you for some clarifications:
 * What hydras are exactly allowed? I guess ___... (an infinite chain of labels) is disallowed.
 * What is an empty label?
 * By rule 2, something like ()+7 is a correct label, even though labels must be hydras.
 * For u<v to make sense, as u and v are any labels, we need total ordering on set of all hydras.
 * Nothing is said about what happens if label is "limit" hydra, like (()). LittlePeng9 (talk) 22:19, December 23, 2013 (UTC)
 * ()+7 is a correct label, and 2 is too. The acts as \(\omega\), so if you have a limit hydra, you have to replace  with n (the number of steps). If you have no  label, solve the hydra in the label until you find an . Example:  You have the hydra () It will become ..., and then ...n Wythagoras (talk) 07:23, December 25, 2013 (UTC)

Nearly two years later, and still no complete definition has been provided. LittlePeng9 (talk) 18:04, August 23, 2015 (UTC)