1RB1LD 1RC1RB 1LC1LA 0RC0RD

1RB1LD_1RC1RB_1LC1LA_0RC0RD (bbch) is the current Blanking Busy Beaver BLB(4,2) and Beeping Busy Beaver BBB(4,2) champion, creating a blank tape after 32,779,477 steps and quasihalting after 32,779,478 steps. It is also a translated cycler with a preperiod of 32,779,478 steps, a period length of 1 step and a cycle offset of 1 symbol to the left. It was discovered and reported by Nick Drozd in 2021.

Analysis by Shawn Ligocki

Let

$$D(a,b)=0^{\infty }\;1^{a}\;0^{b}\;{\textrm {D>}}\;0^{\infty }$$

then:

$${\begin{array}{lc}D(a+3,&b)&\to &D(a,b+5)\\D(0,&b)&=&{\textrm {Blank}}\\D(1,&b)&\to &D(b+2,4)\\D(2,&b)&\to &D(b+3,4)\\\end{array}}$$

let $D(a)=D(a,4)$, then we can simplify to:

$${\begin{array}{lc}D(3k)&\to &{\textrm {Blank}}\\D(3k+1)&\to &D(5k+6)\\D(3k+2)&\to &D(5k+7)\\\end{array}}$$

Starting from $D(2)$ (at step 19) we get the trajectory:

$${\begin{array}{l}D(2)&\to &D(7)&\to &D(16)&\to &D(31)&\to &D(56)&\to &D(97)\\&\to &D(166)&\to &D(281)&\to &D(472)&\to &D(791)&\to &D(1322)\\&\to &D(2207)&\to &D(3682)&\to &D(6141)&\to &{\textrm {Blank}}\\\end{array}}$$

which has the remarkable luck of applying this Collatz-like map 14 times before reaching the blanking config (expected # of applications before halting is 3).

See also, previous analysis in 2021: https://www.sligocki.com/2021/07/17/bb-collatz.html

Relation to other machines

The map and trajectory are equivalent to that of the BB(5) champion. For all $k$, let $f(k)$ be the number such that $D(k)\to D(f(k))$, or ${\text{HALT}}$ if $D(k)\to {\textrm {Blank}}$, and let $g$ be the map simulated by the BB(5) champion. Then:

$${\begin{array}{lc}g(2(3k)+2)&=&g(6k+2)&=&{\text{HALT}}&=&f(3k)\\g(2(3k+1)+2)&=&g(6k+4)&=&10k+14&=&2f(3k+1)+2\\g(2(3k+2)+2)&=&g(6k+6)&=&10k+16&=&2f(3k+2)+2\\\end{array}}$$

So the size of this machine's BLB output is tied to the size of the BB(5) champion's output.