Talk:Champions

Larger champions

What are the known two-symbol champions beyond BB(16)? Vielhaber, Chacón, and Ceballos's paper "Friedman's 'Long Finite Sequences': The End of the Busy Beaver Contest" gives a 2450-state two-symbol busy beaver halting after at least n(4) steps, where n is Friedman's block subsequence function, but this is a very large jump up in state count from 14 states for f_ω+1(65536). (This paper has some mistakes with their (symbol,state count) notation for TMs, for example referring to Aaronson's and Yedidia's machine as a (7910,2) machine rather than a (2,7910) machine, so maybe the machine whose state count I wrote here is the wrong one.) C7X (talk) 22:00, 17 August 2024 (UTC)

Today i found some bounds on BB(20) and BB(21), though i'm not sure if they're new. $\Sigma (20)>f_{\omega +2}^{2}(21)$ due to 1LR0LR_0RJ1RG_0RD1LC_1RQ1RE_1LO0RQ_1LH1LF_0LC1LH_0LI0LF_1RD0LF_---0RK_0LM1LL_0LL1LM_1LE1LN_0LF0LM_0RP1LO_0LF0RQ_1RB1RQ_0RS1LA_1LT1RS_0LP0RR (bbch) and $\Sigma (21)>f_{\omega ^{2}}^{2}(4\uparrow \uparrow 341)$ due to 0LI0LF_0RJ1RG_0RD1LC_1RH1RE_1LO0RH_1LA1LF_0LC1LA_1RB1RH_1RD0LF_1LP0RK_0LM1LL_0LL1LM_1LE1LN_0RQ0LM_0RP1LO_1LR0RH_1LF1RQ_---0LS_1LH0LT_1LH1LU_0LE1LR (bbch).

The old googology wiki claims some other bounds, two of which are already implied by the bounds mentioned above, but $\Sigma (85)>f_{\varepsilon _{0}}(1907)$ and the bounds for more than 2 symbols seem to still be the best known.

Racheline (talk) 23:08, 17 August 2024 (UTC)