Mother of Giants

Unsolved problem:

Does the Mother of Giants quasihalt? If so, how many steps does it take to quasihalt?

The Mother of Giants is a collection of adjacent Turing machines, some of which are Cryptids in the 5-state Beeping Busy Beaver problem that probviously quasihalt. They must all be proven to halt or not if we want to solve BBB(5).

The TMs are all the "children" of 1RB1LE_0LC0LB_0LD1LC_1RD1RA_---0LA where children means all the TMs created by filling in the undefined E0 transition.

Overview of the children

An overview of the 8 most interesting children.

Machine	Status
1RB1LE_0LC0LB_0LD1LC_1RD1RA_1RC0LA (bbch) and 1RB1LE_0LC0LB_0LD1LC_1RD1RA_1LC0LA (bbch)	Probviously quasihalting Cryptids
1RB1LE_0LC0LB_0LD1LC_1RD1RA_1LA0LA (bbch)	Probviously quasihalting Cryptid
1RB1LE_0LC0LB_0LD1LC_1RD1RA_0LC0LA (bbch)	Probviously quasihalting Cryptid
1RB1LE_0LC0LB_0LD1LC_1RD1RA_0RC0LA (bbch) (current champion) and 1RB1LE_0LC0LB_0LD1LC_1RD1RA_0LD0LA (bbch)	Quasihalt after $>10^{14\,006}$ steps.
1RB1LE_0LC0LB_0LD1LC_1RD1RA_1RD0LA (bbch)	Quasihalts after about $10^{12\,978}$ steps.
1RB1LE_0LC0LB_0LD1LC_1RD1RA_0RD0LA (bbch)	Quasihalts after about $10^{4\,079}$ steps.

1RB1LE_0LC0LB_0LD1LC_1RD1RA_0RC0LA and 1RB1LE_0LC0LB_0LD1LC_1RD1RA_0LD0LA

Both quasihalt after $>10^{14\,006}$ steps. Analysis by Shawn Ligocki:

let
  G(n, m) = 0^inf <D 0^n 1^m 0^inf
then:
  G(4k  , m+2) -> G(7k+ 7, m)
  G(4k+1, m+2) -> G(7k+ 8, m+1)
  G(4k+2, m+2) -> G(7k+ 8, m+1)
  G(4k+3, m+2) -> G(7k+14, m)
  G(n, 1) -> G(1, n+3)
  G(n, 0) -> Spin Out
If you compare this to my Collatz-like analysis of the 10^4079 machine (emailed on Mar 16), you'll see that it is extremely similar! The only difference is two of the constants on the right have changed. And likewise the 10^325 machine also had a similar Collatz-like analysis. So it appears that the 5x2 TM space has been able to simulate many parameterizations of these 2-stage style Collatz-like functions and we are noticing the especially "lucky" ones which run for quite a long time before quasihalting (but not toooo "lucky" or else my simulator will not run long enough to catch them).

The orbit for this TM starting on a blank tape is:

  0 : G(1, 1)
  1 : G(1, 4)
    2 : G(8, 3)
    3: G(21, 1)
  4 : G(1, 24)
    5 : G(8, 23)
    6 : G(21, 21)
    7 : G(43, 20)
    8 : G(84, 18)
    9 : G(154, 16)
    10 : G(274, 15)
    11 : G(484, 14)
    12 : G(854, 12)
    13 : G(1499, 11)
    14 : G(2632, 9)
    15 : G(4613, 7)
    16 : G(8079, 6)
    17 : G(14147, 4)
    18 : G(24766, 2)
    19 : G(43345, 1)
  20 : G(1, 43348)
    21 : G(8, 43347)
    22 : G(21, 43345)
    ...

    28_832 : G(2533...2210, 0)

    Spin out

So, it grew m up to >43k before spinning out (vs. 12k in the 10^4079 case).

1RB1LE_0LC0LB_0LD1LC_1RD1RA_0RD0LA

Quasihalts after about $10^{4\,079}$ steps. Analysis by Shawn Ligocki:

let
  G(n, m) = 0^inf <E 0^n 1^m 0^inf
then
  G(4k+1, m+2) -> G(7k+ 8, m+1)
  G(4k+2, m+2) -> G(7k+ 6, m+1)
  G(4k+3, m+2) -> G(7k+14, m)
  G(4k+4, m+2) -> G(7k+12, m)
  G(n, 1) -> G(1, n+3)
  G(n, 0) -> Spin Out
and for completeness (although we can never reach G(0, m) by applying these rules):
  G(0, m+2) -> G(7, m)
and finally, at step 4 we are in G(1, 1).

As before, iterating these relations leads to a sort of "2-stage" Collatz-like behavior:
  G(1, 1)
  G(1, 4)
    G(8, 3)
    G(19, 1)
  G(1, 22)
    G(8, 21)
    G(19, 19)
    G(42, 17)
    G(76, 16)
    G(138, 14)
    G(244, 13)
    G(432, 11)
    G(761, 9)
    G(1338, 8)
    G(2344, 7)
    G(4107, 5)
    G(7196, 3)
    G(12598, 1)
  G(1, 12601)
    G(8, 12600)
    G(19, 12598)
    ...
G(1876213840710521598391379605525487806521622108810778432527722867392307436766159159315654305525834552560172561652032118063458627370309987185176524224005388698700446967268349729419384342385261686600728253800107702390205303703109911013679543376784994065131479319486732208008605302317371513424001996538578227420511014587912085639154043098929108973505716971259211164084851171147584492195641828442403195419357046301200977722765015982127629619366127227507539382553155352425623517192820793588314301840633848042122283284527197514334793445174332133304252240350530614310123013246222482691057522680607329801073658994807307108563977792602674937915145127717601763644562450026084468482011253492787037376937107425102889344711991496275353115166560777221320784075353693443765935139868061950972649919070516499077698769361018135243624369764825602272332642411663431603264357025025605181511270774475664657506683051463739351416893336641567178700404415824985611548629898352736718482963400988142981278524634899806152389037871669098549259903261356156435822856442997107359268004720245053216395652317114834047445010107196444627059271727548915696497640596592913994488623239627564677236759609661035594688933526335837056500047460869283130575904150442715133699885285716657627307991036082003072952095275483832305837708309180971389464320818551264951669392266140549796563778468321806984538951296296552524443477377012382914243026356951988052699080778947387243967930857325467739175240274180541636551276654811833607927610176824006302410755537315645723335878886137585589884378754363497710819886157044977428594424751379562193743624891055837449756993817095474523342307530343858042369299814007843193782820421402610428397007107529266963233646286254595510683332294534735691214261351756352969840115522216379692717569241523917906031879093475856928361266209321993314289669996574460021380115709710519270157850694498748462253621222466800698577194648206832226152375452456786050873778773714111576650409478390819793529657854689649591879416169292957147115001498762307255727596635383777744252057, 0)
  Spin Out

See also

https://www.sligocki.com/2022/04/03/mother-of-giants.html for details.