Aarex Hydra going to have round brackets only. Every game have a tree, contains empty or round brackets with other tree inside, starting in step 1.
Let n be the step number. Each time the process completed, increase the step number by 1.
- If there is () in the end, remove the last ().
- If there is (#()) in the end, follow the rule:
- Define A0 as (#()).
- If there is a round bracket before Ax (where x is the maximum value which Ax is defined), define that as An+1.
- Else if there is round bracket containing all A0, A1, ... Ax-1, Ax inside, define that as Ax.
- Find m as the minimum value which A0 > Am, and call Am as B. If not, define B as whole tree.
- Replace innermost A0 with Am, n times. If Am is not equal to (#), do not use it and replace innermost A0 with (#)Am instead, n times.
- Finally, remove the last () in innermost A0.
- Else, follow the process in the tree in the last ().
Let AH(n) be n, where n is the final steps when the tree is empty after the starting tree (((...()...))) with n ()s inside out.
In analysis, a tree in the left will be empty after amount of steps in the right.
- Empty - 0
- () - 1
- ()() - 2
- ()()() - 3
- ()()()... with n ()s - n
- (()) - w
- We will analyse the whole tree only.
- (())() - w+1
- (())()() - w+2
- (())()() - w+3
- (())()()()... with n ()s - w+n
- (())()(()) - w2
- (())()(())() - w2+1
- (())()(())()(()) - w3
- (())()(())()(())()(()) - w4
- (())()(())()(())()(())()(()) - w5
- (())()(())()(())()... with n (())()s - wn
- (())(()) - w^2
- (())(())() - w^2+1
- (())(())()(()) - w^2+w
- (())(())()(())()(()) - w^2+w2
- (())(())()(())()(())()(()) - w^2+w3
- (())(())()(())()(())()(())... with n ()(())s - w^2+wn
- (())(())()(())(()) - w^2*2
- (())(())()(())(())() - w^2*2+1
- (())(())()(())(())()(()) - w^2*2+w
- (())(())()(())(())()(())(()) - w^2*3
- (())(())()(())(())()(())(())()(())(()) - w^2*4
- (())(())()(())(())()(())(())()(())(())()(())(()) - w^2*5
- (())(())()(())(())()(())(())()... with n (())(())()s - w^2*n
- (())(())(()) - w^3
- (())(())(())() - w^3+1
- (())(())(())()(()) - w^3+w
- (())(())(())()(())(()) - w^3+w^2
- (())(())(())()(())(())(()) - w^3*2
- (())(())(())(()) - w^4
- (())(())(())(())(()) - w^5
- (())(())(())(())(())(()) - w^6
- (())(())(())... with (())s - w^n
- (()()) - w^w
- (()())() - w^w+1
- (()())()(()) - w^w+w
- (()())()(())(()) - w^w+w^2
- (()())()(())(())(()) - w^w+w^3
- (()())()(())(())(())... with n (())s - w^w+w^n
- (()())()(()()) - w^w*2
- (()())()(()())()(()) - w^w*2+w
- (()())()(()())()(()()) - w^w*3
- (()())()(()())()(()())()(()()) - w^w*4
- (()())()(()())()(()())()(()())()(()()) - w^w*5
- (()())()(()())()(()())()... with n (()())()s - w^w*n
- (()())(()) - w^(w+1)
- (()())(())()(()())(()) - w^(w+1)2
- (()())(())(()) - w^(w+2)
- (()())(())(())(()) - w^(w+3)
- (()())(())(())(())(()) - w^(w+4)
- (()())(())(())(())... with n (())s - w^(w+n)
- (()())(())(()()) - w^(w2)
- (()())(())(()())(()) - w^(w2+1)
- (()())(())(()())(())(()()) - w^(w3)
- (()())(()()) - w^w^2
- (()())(()())(()()) - w^w^3
- (()())(()())(()())(()()) - w^w^4
- (()())(()())(()())... with n (()())s - w^w^n
- (()()()) - w^w^w
- (()()())()(()()()) - w^w^w*2
- (()()())(())(()()()) - w^(w^w*2)
- (()()())(()())(()()()) - w^w^(w*2)
- (()()())(()()()) - w^w^w^2
- (()()())(()()())(()()()) - w^w^w^3
- (()()())(()()())(()()())(()()()) - w^w^w^4
- (()()())(()()())(()()())... with n (()()())s - w^w^w^n
- (()()()()) - w^w^w^w
- (()()()())(()()()()) - w^w^w^w^2
- (()()()()()) - w^w^w^w^w
- (()()()()()()) - w^w^w^w^w^w
- (()()()()()()()) - w^w^w^w^w^w^w
- (()()()...) with n ()s - w^^n
- (()(())) - e0
- (()(()))()(()(())) - e0*2
- (()(()))(())(()(())) - w^(e0*2)
- (()(()))(()())(()(())) - w^w^(e0*2)
- (()(()))(()()())(()(())) - w^w^w^(e0*2)
- (()(()))(()()()...)(()(())) with n ()s - w^w^...^w^w^(e0*2) with n 'w^'s
- (()(()))(()(())) - e1
- (()(()))(()(()))()(()(()))(()(())) - e1*2
- (()(()))(()(()))()(()(()))(()(())) - w^(e1*2)
- (()(()))(()(()))(()(())) - e2
- (()(()))(()(()))(()(()))(()(())) - e3
- (()(()))(()(()))(()(()))(()(()))(()(())) - e4
- (()(()))(()(()))(()(()))... with n (()(()))s - e(n-1)
- (()(())()) - ew
- (()(())())(()(())) - e(w+1)
- (()(())())(()(()))(()(())()) - e(w*2)
- (()(())())(()(())()) - e(w^2)
- (()(())()()) - e(w^w)
- (()(())()()()) - e(w^w^w)
- (()(())()()()()) - e(w^w^w^w)
- (()(())()()()...) with n ()s - e(w^^n)
- (()(())()(())) - ee0
This is the mix of Buchholz hydra and my hydra. In this hydra, we use Buchholz rules, but all Kirby-Paris hydra rules were replaced to all my hydra rules.
- If there is (0) in the end, remove the last (0).
- If there is (n) in the end, follow the rule:
- Define A0 as (n).
- If there is a round bracket before Ax (where x is the maximum value which Ax is defined), define that as An+1.
- Else if there is round bracket containing all A0, A1, ... Ax-1, Ax inside, define that as Ax.
- Define Bx as a number before tree in Ax.
- Find m as the minimum value which B0 > Bm, and call Bm as C. If not, define C as whole tree.
- Replace innermost A0 with (n-1 Am), n times.
- Finally, replace the last (n) with (n-1) in innermost A0.
- If there is (#()) in the end, follow the rule:
- Define A0 as (#()).
- If there is a round bracket before Ax (where x is the maximum value which Ax is defined), define that as An+1.
- Else if there is round bracket containing all A0, A1, ... Ax-1, Ax inside, define that as Ax.
- Find m as the minimum value which A0 > Am, and call Am as B. If not, define B as whole tree.
- Replace innermost A0 with Am, n times. If Am is not equal to (#), do not use it and replace innermost A0 with (#)Am instead, n times.
- Finally, remove the last () in innermost A0.
- Else, follow the process in the tree in the last (0).
There is also nested Buchholz-Aarex Hydra, and I will replace numbers with trees and use Nested Buchholz Hydra rules. We simply replace (nA) with (A)_n in this extension and replace numbers with amount of ()s.