User blog comment:LittlePeng9/Random Turing machines/@comment-5150073-20130412221541

For Kirby-Paris hydra game: using some logic and attempts to understand how your TM in works, I found some comparisons between your hydras and Chris Bird's nested array notation. Let n is the number of )'s before space and hydra. Then:

=> n+1 => n+2 => n+3 () => 2n+2 () => 2n+4 () => 2n+6 ()() => 4n+6 ()() => 4n+10 ()() => 4n+14 ()()() => 6n+18 ()()()() => 8n+22 ()()()()() => 10n+26 () => > 2^n () => > 2^^n () => > 2^^^n () => > 2^^^^n (()) => > {n,n,n} (()) => > {n,n,1,2} (()) => > {n,n,2,2} (()) => > {n,n,3,2} (()()) => > {n,n,n,2} (()()()) => > {n,n,n,3} (()()()()) => > {n,n,n,4} (()) => > {n,n,n,n} (()) => > {n,n,n,n,n} (()) => > {n,n,n,n,n,n} ((())) => > {n,n [2] 2} ((())) => > {n,n,2 [2] 2} ((())) => > {n,n,3 [2] 2} ((())()) => > {n,n,n [2] 2} ((())(())) => > {n,n [2] 3} ((())(())(())) => > {n,n [2] 4} ((())) => > {n,n [2] n} ((())) => > {n,n [2] n,n} ((())) => > {n,n [2] n,n,n} ((()())) => > {n,n [2][2] 2} ((()()())) => > {n,n [2][2][2] 2} ((())) => > {n,n [3] 2} ((()())) => > {n,n [3][3] 2} ((()()())) => > {n,n [3][3][3] 2} ((())) => > {n,n [4] 2} ((())) => > {n,n [5] 2} (((()))) => > {n,n [1,2] 2} (((()))) => > {n,n [2,2] 2} (((()))) => > {n,n [3,2] 2} (((()()))) => > {n,n [1,3] 2} (((()()()))) => > {n,n [1,4] 2} (((()))) => > {n,n [1,1,2] 2} (((()))) => > {n,n [1,1,1,2] 2} (((()))) => > {n,n [1,1,1,1,2] 2} ((((())))) => > {n,n [1 [2] 2] 2} ((((())))) => > {n,n [1 [3] 2] 2} (((((()))))) => > {n,n [1 [1,2] 2] 2} ((((((())))))) => > {n,n [1 [1 [2] 2] 2] 2}

Note: it works with sufficiently large n: for example, ) ((())) should give not {1,1 [2] 2} = 1 )'s, but )))))))))) ((())) would output around {10,10 [2] 2} )'s as I believe.