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)

Racheline, congratulations on discovering a TM that crushes Graham's number. I was very impressed. but I think that in reality BB(8,2) will beat Graham's number. But it is unrealistic for a human to build such a machine. Only to search in the wild. Unfortunately. Nevertheless, congratulations again and further success!!! --Konkhra (talk) 03:29, 18 August 2024 (UTC)

And how about an article about BB(14)?--Konkhra (talk) 03:30, 18 August 2024 (UTC)

IMHO, beyond Graham's number I think we should only list highlights (if anything). Like first TM bigger than blah for various notable blah. Otherwise the churn on this page will be high and the quality low. But I'm open to other points of view. If we do extend this much further, I think maybe we should have a more rigorous process for getting new results demonstrated before they are added. --sligocki (talk) 02:57, 18 August 2024 (UTC)

BB(51) epsilon_0 champion

Can you show the transitions of BB(51) on Large champions transitions page? --Jacobzheng (talk) 03:24, 3 September 2024 (UTC)

Here are the transitions and description from Racheline:

Σ(51) > (f_ε0)^8(f_ω^ω^3(a lot)) > f_ε0+1(8)
n0 = 21
n1 = 42
n2 = 46
M = (
    ((1, -1, 0), (0, -1, 1)),
    ((1, -1, 0), (1, 1, 2)),
    ((1, -1, 2), (0, -1, 4)),
    ((0, -1, 10), (1, -1, 4)),
    ((0, -1, 3), (0, -1, 5)),
    ((1, -1, 4), (1, 1, 6)),
    ((1, -1, 7), (0, -1, 15)),
    ((0, -1, 8), (1, -1, 7)),
    ((0, -1, 9), (1, -1, 7)),
    ((1, 1, 11), (1, -1, 9)),
    ((0, -1, 20), (0, -1, 17)),
    ((0, 1, 12), (1, 1, 11)),
    ((0, 1, 13), (1, 1, 13)),
    ((1, -1, 14), (1, 1, 12)),
    ((0, -1, 6), (1, -1, 10)),
    ((0, -1, 16), (1, -1, 16)),
    ((1, -1, 20), (1, -1, 15)),
    ((0, -1, 18), (1, -1, 18)),
    ((1, 1, 19), (1, -1, 17)),
    ((1, -1, 19), (1, -1, 20)),
    ((0, -1, n2 + 4), (0, -1, 21)),
    ((0, 1, n1 + 1), (0, -1, 9)),

    ((0, 1, n0 + 1), (1, 1, n0 + 2)),
    ((1, 1, n0 + 5), (0, -1, n0 + 3)),
    ((1, -1, n0 + 4), (0, -1, n0 + 3)),
    ((1, 1, n0 + 4), (1, 1, n0 + 1)),
    ((0, 1, n2 + 4), (0, 1, n0 + 6)),
    ((1, 1, n0 + 5), (1, -1, n0 + 14)),
    ((0, 1, n0 + 8), (0, 1, n0 + 9)),
    ((0, -1, 4), (1, 1, n0 + 7)),
    ((1, -1, n0 + 10), (1, -1, 0)),
    ((0, -1, n0 + 11), (1, -1, n0 + 12)),
    ((0, -1, n0 + 12), (1, -1, n0 + 10)),
    ((0, -1, n0 + 13), (0, -1, n0 + 14)),
    ((0, -1, n0 + 16), (1, -1, n0 + 12)),
    ((1, 1, n0 + 15), (1, 1, n0 + 17)),
    ((0, 1, n0 + 16), (1, 1, n0 + 7)),
    ((0, -1, n0 + 8), (1, 1, n0 + 15)),
    ((1, 1, n0 + 17), (1, 1, n0 + 18)),
    ((0, 1, n0 + 17), (1, 1, n0 + 19)),
    ((0, 1, n0 + 20), (1, -1, n0 + 21)),
    ((1, -1, n2 + 3), (1, 1, n0 + 19)),
    ((1, 1, n0 + 15), (0, -1, n0 + 21)),

    ((0, 1, n1 + 2), (1, 1, n1 + 2)),
    ((1, -1, n1 + 3), (1, 1, n1 + 1)),
    ((0, -1, n1 + 4), (1, -1, n1 + 4)),
    ((0, 1, n0 + 1), (1, -1, n1 + 3)),

    ((1, -1, n0 + 20), (1, -1, n2 + 1)),
    ((0, -1, n2 + 1), (0, -1, n2 + 3)),
    ((1, -1, n0 + 1), (1, -1, n2 + 2)),
    ((0, -1, n2 + 2), (1, 1, -1))
)
 
(start from state 30)

I haven't investigated it yet. Sligocki (talk) 14:27, 3 September 2024 (UTC)