View source for 1RB1RA 0RC1RC 1LD0LF 0LE1LE 1RA0LB ---0LC

{{machine|1RB1RA_0RC1RC_1LD0LF_0LE1LE_1RA0LB_---0LC}}
{{TM|1RB1RA_0RC1RC_1LD0LF_0LE1LE_1RA0LB_---0LC}} is one of a collection of 16 [[probviously]] halting tetrational [[BB(6)]] [[Cryptid]]s found by @mxdys and shared on Discord on 26 Jul 2024. They all have (almost) identical tape behavior and if they halt, they will all have the same sigma scores. However, they will have different step counts. If these halt they will have sigma score <math>> 10^{10^{10^7}}</math>.

== List of Cryptids ==
<pre>
1RB1RA_0RC1RC_1LD0LF_0LE1LE_1RA1LD_---0LC
1RB1RA_0RC1RC_1RD0LF_0LE1LE_1RA1LD_---0LC
1RB0RD_0RC1RC_1LD0LF_0LE1LE_1RA1LD_---0LC
1RB0RD_0RC1RC_1RD0LF_0LE1LE_1RA1LD_---0LC
1RB1RA_0RC1RC_1LD0LF_0LE1LE_1RA0LB_---0LC
1RB0RD_0RC1RC_1LD0LF_0LE1LE_1RA0LB_---0LC

1RB1RA_0RC1RF_1LD---_0LE1LE_1RA1LD_1LD0LF
1RB1RA_0RC1RF_1LD---_0LE1LE_1RA1LD_1RD0LF
1RB1RA_0RC1RF_1RD---_0LE1LE_1RA1LD_1LD0LF
1RB1RA_0RC1RF_1RD---_0LE1LE_1RA1LD_1RD0LF
1RB0RD_0RC1RF_1LD---_0LE1LE_1RA1LD_1LD0LF
1RB0RD_0RC1RF_1LD---_0LE1LE_1RA1LD_1RD0LF
1RB0RD_0RC1RF_1RD---_0LE1LE_1RA1LD_1LD0LF
1RB0RD_0RC1RF_1RD---_0LE1LE_1RA1LD_1RD0LF
1RB1RA_0RC1RF_1LD---_0LE1LE_1RA0LB_1LD0LF
1RB0RD_0RC1RF_1LD---_0LE1LE_1RA0LB_1LD0LF
</pre>

== Analysis by @mxdys ==
https://discord.com/channels/960643023006490684/1239205785913790465/1266406840103866399

<pre>
1RB1RA_0RC1RC_1LD0LF_0LE1LE_1RA1LD_---0LC

N(n,m) := 0^inf 1^5 A> 0^(2n+1) 1^m 01 0^inf

start from N(4,4)
N(2n,m+2)   --> N(3n+3,m)
N(2n+1,m+1) --> N(3n+4,m)
N(2n+4,0)   --> halt
N(2n+4,1)   --> N(4,6n+20)
N(2n+5,0)   --> N(4,6n+22)

example:
(4,4),(9,2),(16,1),(4,56),(9,54),(16,53),(27,51),(43,50),(67,49),(103,48),(157,47),(238,46),(360,44),...
</pre>This has been simulated out to 3 "resets" (rules which increase m): https://discord.com/channels/960643023006490684/1226543091264126976/1405047857970679840<pre>
N(4, 4)
N(4, 56)
N(4, 210273200)
N(4, 1.326953 * 10^24684623)
</pre>The number of iterations until the next reset is <math>> 6 \times 10^{24\,684\,622}</math> and would lead to an n value of <math>> 10^{10^{24\,684\,622}} > 10^{10^{10^7}}</math>. Currently there is no known way to simulate this in faster than linear time and to store the values in any way other than directly as bits, thus it is completely infeasible to simulate this to the next reset.

== Analysis by [[User:sligocki|Shawn Ligocki]] ==
https://discord.com/channels/960643023006490684/1239205785913790465/1267700760892674200

<pre>
1RB1RA_0RC1RC_1LD0LF_0LE1LE_1RA0LB_---0LC

C(a, b, c) = $ 1^2a+1 C> 0^2b 1^c 01 $

Level 1:
  C(a, b+2, c) -> C(a+3, b, c)
  C(a, 1, c+2) -> C(1, a+3, c)
  C(a, 0, c+1) -> C(1, a+1, c)

  C(a, 0, 0) -> C(1, 2, 2a+3)
  C(a, 1, 1) -> C(1, 2, 2a+7)
  C(a, 1, 0) -> Halt(2a+5)

Level 2:
  C(1, 2b,   c+1) -> C(1, 3b+2, c)
  C(1, 2b+1, c+2) -> C(1, 3b+4, c)

  C(1, 2b,   0) -> C(1, 2, 6b+5)
  C(1, 2b+1, 1) -> C(1, 2, 6b+9)
  C(1, 2b+1, 0) -> Halt(6b+7)


C(1, 0, 0)  @11
C(1, 2, 5)  @55
  ...
C(1, 17, 1)
C(1, 2, 57)
  ...
C(1, 70_091_065, 1)
C(1, 2, 210_273_201)
  ...
</pre>

Rules validated in https://github.com/sligocki/busy-beaver/blob/main/rust/src/validator.rs#L1042

Example growth:
<pre>
          0  C(1, 2, 5)  [0s]
          4  C(1, 2, 57)  [0s]
         45  C(1, 2, 210_273_201)  [0s]
    100_000  C(1, 10^17_602, 210_123_530)  [1s]
    200_000  C(1, 10^35_211, 209_973_408)  [4s]
    300_000  C(1, 10^52_820, 209_823_652)  [10s]
    400_000  C(1, 10^70_429, 209_673_652)  [17s]
    500_000  C(1, 10^88_039, 209_523_616)  [27s]
    600_000  C(1, 10^105_648, 209_373_548)  [38s]
    700_000  C(1, 10^123_257, 209_223_420)  [52s]
    800_000  C(1, 10^140_866, 209_073_477)  [68s]
    900_000  C(1, 10^158_475, 208_923_310)  [86s]
  1_000_000  C(1, 10^176_084, 208_773_236)  [107s]
</pre>

Interestingly, both of the 2 reset's I've been able to simulate to both follow the C(1, 2b+1, 1) -> C(1, 2, 6b+9) path.

== Equivalent TMs ==
These 6 TMs all have the exact identical tape behavior:
<pre>
1RB1RA_0RC1RC_1LD0LF_0LE1LE_1RA1LD_---0LC   1RA 1LD 1LD
1RB1RA_0RC1RC_1RD0LF_0LE1LE_1RA1LD_---0LC   1RA 1RD 1LD
1RB0RD_0RC1RC_1LD0LF_0LE1LE_1RA1LD_---0LC   0RD 1LD 1LD
1RB0RD_0RC1RC_1RD0LF_0LE1LE_1RA1LD_---0LC   0RD 1RD 1LD
1RB1RA_0RC1RC_1LD0LF_0LE1LE_1RA0LB_---0LC   1RA 1LD 0LB
1RB0RD_0RC1RC_1LD0LF_0LE1LE_1RA0LB_---0LC   0RD 1LD 0LB
</pre>

The one mxdys first shared: <code>1RB1RA_0RC1RC_1LD0LF_0LE1LE_1RA1LD_---0LC</code> is the most efficient and all the rest add extra steps during certain transitions. IIUC, the others would all have been pruned using Marxen's pruning algorithm. Specifically, these are all the options by changing:
<pre>
A2 -> {1RA,0RD}
C0 -> {1LD,1RD}
E1 -> {1LD,0LB}
</pre>

Except that you cannot have both <code>C0->1RD</code> and <code>E1->0LB</code>
They are all equivalent b/c:
<pre>
1RB1RA_0RC1RC_1LD0LF_0LE1LE_1RA1LD_---0LC
  vs. 1RB1RA_0RC1RC_1LD0LF_0LE1LE_1RA0LB_---0LC
    {E1->0LB, Bx->xRC, C0->1LD} == E1->1LD
  vs. 1RB1RA_0RC1RC_1RD0LF_0LE1LE_1RA1LD_---0LC
    {C0->1RD, Dx->xLE, E1->1LD} == C0->1LD
  vs. 1RB0RD_0RC1RC_1LD0LF_0LE1LE_1RA1LD_---0LC
    {A1->0RD, Dx->xLE, E0->1RA} == A1->1RA
</pre>
I believe that this means that either
<pre>
1RB0RD_0RC1RC_1RD0LF_0LE1LE_1RA1LD_---0LC   0RD 1RD 1LD
1RB0RD_0RC1RC_1LD0LF_0LE1LE_1RA0LB_---0LC   0RD 1LD 0LB
</pre>

will be the longest running depending on whether there are more E1 or C0 transitions in <code>1RB1RA_0RC1RC_1LD0LF_0LE1LE_1RA1LD_---0LC</code>'s complete [[transition history]].

[[Category:BB(6)]][[Category:Cryptids]]