User:Wythagoras/Rado's sigma function/BB(7,3)

\(\Sigma(7,3) > \{2,\{2,374676382,374676380\},374676380\}\) (Wythagoras, September 2016)

Ackermannian growth using four states
0 _ 2 r 0 0 1 1 r 0 0 2 1 r 1 1 1 2 l 3 1 2 2 l 2 1 _ _ r halt 2 2 _ l 2 2 1 1 l 2 2 _ 2 r 0 3 1 1 l 3 3 2 2 l 3 3 _ 1 r 0 This machine executes a function \(f(a_1,a_2,a_3,\cdots,a_n) \) of \(n\) variables, defined as We have \(f(1,\underbrace{0,0,\cdots,0,0}_{k},l) > 2 \uparrow^k l\) with induction.
 * \(f(a_1,a_2,a_3,\cdots,a_n)=f(a_1+2,a_2-1,a_3,\cdots,a_n)\) if \(a_2>0\).
 * \(f(a_1,\underbrace{0,0,\cdots,0,0}_{l},0,a_k,\cdots,a_n)=f(1,\underbrace{0,0,\cdots,0,0}_{l},a_1+k+2,a_k-1,\cdots,a_n)\) if \(a_k>0\).
 * \(f(a_1,\underbrace{0,0,\cdots,0,0}_{k})=a_1+k-1\).

Input with three states and three symbols
0 _ 1 l 1 0 1 2 r 0 0 2 1 r 2 1 _ _ r 0 1 1 2 l 1 1 2 1 r 1 2 _ 1 l halt 2 1 1 l 0 2 2 1 l 2 The output of this machine is 222...222211 with 374,676,381 2's, with the head on the first one. This becomes 1222...222211 with the head on the first one with state 3 of the above machine. This becomes 12....21...121 with 374,676,380 2's and 374,676,382 1's after a few (actually more than a trillion) steps. This gives

\(\Sigma(7,3) > \{2,\{2,374676382,374676380\},374676380\}\)