User:Wythagoras/Rado's sigma function/Analysis of LittlePeng9's work/Hydra machines

LittlePeng9 has made some hydra Turing machines, which are the most fascinating Turing machines I've ever seen. He is definitely a genius TM programmer.

This page will give some bounds based on these machines and will show some tricks to reduce states.

3-color Kirby-Paris hydra
The original machine has 45 states. By adding 3 states and reducing 7 states, we can prove

\(\Sigma(41,3) > f_{\varepsilon_0}(374676379)\). Also,

\(\Sigma(39,3) > f_{\varepsilon_0}(5)\)

Transfinite Buchholz hydra
LittlePeng already gave a bound \(\Sigma(168,7) > f_{\vartheta(\varepsilon_{\Omega_\omega+1})}(683)\). We can reduce 14 states, and I found two times three states that are exact the same, saving 4 states. It gives \(\Sigma(150,7) > f_{\vartheta(\varepsilon_{\Omega_\omega+1})}(683)\)