## FANDOM

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I have to change the hydra rules a bit for a easier analysis for me. It spawns n nests instead of one, so it only makes the hydra stronger. (n is the number of steps that has been done, including the step itself to make sure it's at least one)

First, we look to the hydra (+(0(1))) it spawns (+(0(0(0)))) in the normal version. (for reduction see below). In the other version, (0(1)) can be upper bounded by f(ε0), as it spawns n nests.

(0(1)(0)) expands to (0(1))(0(1))...(0(1))(0(1)), so it can be upper bounded by f(ε0+1).

Again, we get a table:

Hydra upper bound on ordinal
(0(1)) ε0
(0(1)(0)) ε0+1
(0(1)(0(0))) ε0+ω
(0(1)(0(1))) ε0*2
(0(1)(0(1))(0(1))) ε0*3
(0(1)(0(1)(0))) ε0*ω
(0(1)(0(1)(0(1)))) ε0^2
(0(1)(0(1)(0(1)(0(1))))) ε0^ε0
(0(1)(1)) ε1
(0(1)(1)(0)) ε1+1
(0(1)(1)(0(1)(1))) ε1*2
(0(1)(1)(1)) ε2
(0(1)(1)(1)(1)) ε3
(0(1(0))) εω
(0(1(0(1)))) εε0
(0(1(1))) ζ0
(0(1(1))(1)) ε(ζ0+1)
(0(1(1))(1(0(1(1))))) ε(ζ0*2)
(0(1(1))(1(1))) ζ1
(0(1(1)(0))) ζω
(0(1(1)(0(1)))) ζε0
(0(1(1)(1))) η0
(0(1(1(0)))) φ(ω,0)
(0(1(1(0(1))))) φ(ε0,0)
(0(1(1(0(1(1)))))) φ(ζ0,0)
(0(1(1(1)))) φ(1,0,0)

Ι believe it's enough, we already reached (0(1)(1)(1)(1)), the hydra for BH(3), at ε3.

We can say $$\text{BH}(3) < f_{\varepsilon_3}(6)$$.

## Reduction of (+(0(1)))

(+(0(1))) expands to (+(0(0(0)))), which expands to (+(0(0)(0)(0))). Using my hydra post, we'll see it reduces in less than $$f_{3}(5)$$ steps.

In general, it reduces in less than $$f_{\omega}(n+2)$$ steps, where n is the number of steps that has been preformed.

This will give a better upper bound.

## A better upper bound

Hydra at root node Ordinal
(0(1)) $$\omega$$
(0(1)(0)) $$\omega+1$$
(0(1)(0(0))) $$\omega2$$
(0(1)(0(0)(0))) $$\omega3$$
(0(1)(0(0(0)))) $$\omega^2$$
(0(1)(0(1))) $$\omega^2$$
(0(1)(0(1))(0(1))) $$\omega^22$$
(0(1)(1)) $$\omega^3$$
(0(1)(1)(0)) $$\omega^3+1$$
(0(1)(1)(0(0))) $$\omega^3+\omega$$
(0(1)(1)(0(1))) $$\omega^3+\omega^2$$
(0(1)(1)(0(1)(0))) $$\omega^32$$
(0(1)(1)(0(1)(0)(0))) $$\omega^4$$
(0(1)(1)(0(1)(0(0)))) $$\omega^\omega$$
(0(1)(1)(0(1)(0(0(0))))) $$\omega^{\omega^2}$$
(0(1)(1)(0(1)(0(1))) $$\omega^{\omega^2}$$
(0(1)(1)(0(1)(0(1)(0))) $$\omega^{\omega^3}$$
(0(1)(1)(0(1)(1))) $$\omega^{\omega^3}$$
(0(1)(1)(1)) $$\omega^{\omega^3+1}$$
(0(1)(1)(1)(0(1)(1)(0))) $$\omega^{\omega^3+1}2$$
(0(1)(1)(1)(0(1)(1)(0)(0))) $$\omega^{\omega^3+2}$$
(0(1)(1)(1)(0(1)(1)(0(0)))) $$\omega^{\omega^3+\omega}$$
(0(1)(1)(1)(0(1)(1)(0(1)))) $$\omega^{\omega^3+\omega^2}$$
(0(1)(1)(1)(0(1)(1)(0(1)(0)))) $$\omega^{\omega^32}$$
(0(1)(1)(1)(0(1)(1)(0(1)(0)(0)))) $$\omega^{\omega^4}$$
(0(1)(1)(1)(0(1)(1)(0(1)(0(0))))) $$\omega^{\omega^\omega}$$
(0(1)(1)(1)(0(1)(1)(0(1)(0(1))))) $$\omega^{\omega^{\omega^2}}$$
(0(1)(1)(1)(0(1)(1)(0(1)(0(1)(0))))) $$\omega^{\omega^{\omega^3}}$$
(0(1)(1)(1)(0(1)(1)(0(1)(1)))) $$\omega^{\omega^{\omega^3}}$$
(0(1)(1)(1)(0(1)(1)(0(1)(1)(0)))) $$\omega^{\omega^{\omega^3+1}}$$
(0(1)(1)(1)(0(1)(1)(1))) $$\omega^{\omega^{\omega^3+1}}$$
(0(1)(1)(1)(1)) $$\omega^{\omega^{\omega^3+1}+1}$$
(0(1(0))) $$\varepsilon_0$$

I don't now how to transform the last one before the last one into a bound, but using the last ordinal and the comparisons before I can say $$BH(3) < f_{\varepsilon_0}(4)$$