User blog:Wythagoras/A far better bound for TREE(3)

1. {}

2. []

3. []

4. ][][

5. (][[])

6. (][())

7. (][)

8. (][)

9. (][)

10. (][)

11.

12. ((([])))

13. (([]()))

(I should have let this reduce to [][][], but I don't want to do whole analysis again)

14. (([]))

15. (([])([]))

16. (([])[][][])

17. (([])[][]())

18. (([])[][])

19. (([])[][])

20. (([])[][])

21. (([])[][])

22. (([])[]tree(8))

tree(8). (([])tree(tree(8)))

tree(tree(8)). (([])tree(tree(8)))

tree(tree(8))+1. (([])tree(tree(7))([])([]))

tree(tree(8))+2. (([])tree(tree(7))([])[][][])

tree(tree(8))*2. (([])tree(tree(7))([]))

tree(tree(8))*2. (([])tree(tree(7))[]tree(tree(8)))

tree2(tree(8)). (([])tree(tree(7)))

tree2(tree(8)). (([])tree(tree(7))([])tree2(tree(8)))

tree2(tree2(tree(8))). (([])tree(tree(7)))

tree3(tree(tree(7))). ((((...((([])))...))))

...

Structures
However, I don't think this is optimal. I guess we can reach SVO*2 with ([][]), and then we reach ε(SVO+1) at. But for now I don't see any way to do that, so here is my improved bound:

\(\text{TREE}(3) > f_{\vartheta(\Omega^\omega)+\varepsilon_1}(f_{\vartheta(\Omega^\omega)+\varepsilon_1}(f_{\vartheta(\Omega^\omega)+\varepsilon_1}(f_{\vartheta(\Omega^\omega)+\varepsilon_1}(\text{tree}(\text{tree}(7)))))\)