User:MrSolis/Playground

5-state busy beaver winner (WIP Revamp)

The 5-state busy beaver (BB(5)) winner is 1RB1LC_1RC1RB_1RD0LE_1LA1LD_1RZ0LA (bbch). Discovered by Heiner Marxen and Jürgen Buntrock in 1989, this machine proved that $\operatorname {BB} (5)\geq 47176870{\phantom {}}$ and $\Sigma (5)\geq 4098$ at the time.

Analysis

Rules

Let $g(x):=0^{\infty }\;{\textrm {<A}}\,1^{x}\;0^{\infty }$. Then,

$${\begin{array}{lll}\\g(3x)&{\phantom {}}\xrightarrow {5x^{2}+19x+15} &g(5x+6),\\g(3x+1)&{\phantom {}}\xrightarrow {5x^{2}+25x+27} &g(5x+9),\\g(3x+2){\phantom {}}&\xrightarrow {6x+12} &0^{\infty }\;1\;{\textrm {Z>}}\;01\;001^{x+1}\;1\;0^{\infty }.\end{array}}$$

Proof

Consider the configuration $C(m,n):=0^{\infty }\;{\textrm {<A}}\;1^{m}\;001^{n}\;1\;0^{\infty }$. After one step this configuration becomes $0^{\infty }\;1\;{\textrm {B>}}\;1^{m}\;001^{n}\;1\;0^{\infty }{\phantom {}}$. We note the following shift rule:

$${\begin{array}{c}\\{\textrm {B>}}\;1^{a}\xrightarrow {a} 1^{a}\;{\textrm {B>}}{\phantom {}}\end{array}}$$

Using this shift rule, we get $0^{\infty }\;1^{m+1}\;{\textrm {B>}}\;001^{n}\;1\;0^{\infty }$ after $m$ steps. If $n=0$, then we get $0^{\infty }\;1^{m+4}\;{\textrm {<A}}\;1\;0^{\infty }$ four steps later. Another shift rule is needed here:

$$1^{3a}\;{\textrm {<A}}\xrightarrow {3a} {\textrm {<A}}\;001^{a}$$

In this instance, $\lfloor {\frac {m+4}{3}}\rfloor$ is substituted for $a$, which creates three different scenarios depending on the value of $m$ modulo 3. They are as follows:

1. If $m+4\equiv 0{\pmod {3}}$, then in $m+4$ steps we arrive at $0^{\infty }\;{\textrm {<A}}\;001^{(m+4)/3}\;1\;0^{\infty }$, which is the same configuration as $C(0,{\frac {m+4}{3}})$.
2. If $m+4\equiv 1{\pmod {3}}$, then in $m+3$ steps we arrive at $0^{\infty }\;1\;{\textrm {<A}}\;001^{(m+3)/3}\;1\;0^{\infty }$, which is five steps becomes ${\phantom {}}0^{\infty }\;{\textrm {<A}}\;111\;001^{(m+3)/3}\;1\;0^{\infty }$, or $C(3,{\frac {m+3}{3}})$.
3. If $m+4\equiv 2{\pmod {3}}$, then in $m+2$ steps we arrive at $0^{\infty }\;11\;{\textrm {<A}}\;001^{(m+2)/3}\;1\;0^{\infty }$, which in three steps halts with the configuration $0^{\infty }\;1\;{\textrm {Z>}}\;01\;001^{(m+2)/3}\;1\;0^{\infty }$, for a total of $2m+10$ steps from $C(m,n)$.

Returning to $0^{\infty }\;1^{m+1}\;{\textrm {B>}}\;001^{n}\;1\;0^{\infty }$, if $n\geq 1$, then in three steps it changes into $0^{\infty }\;1^{m+3}\;{\textrm {<D}}\;1\;001^{n-1}\;1\;0^{\infty }$. Here we can make use of one more shift rule:

$$1^{a}\;{\textrm {<D}}\xrightarrow {a} {\textrm {<D}}\;1^{a}$$

Doing so takes us to $0^{\infty }\;{\textrm {<D}}\;1^{m+4}\;001^{n-1}\;1\;{\phantom {}}0^{\infty }$ in $m+3$ steps, which after one step becomes the configuration $0^{\infty }\;{\textrm {<A}}\;1^{m+5}\;001^{n-1}\;1\;0^{\infty }$, or $C(m+5,n-1)$. To summarize:

$$C(m,n)\xrightarrow {2m+8} C(m+5,n-1){\text{ if }}n\geq 1.$$

We have $g(x)=C(x-1,0)$. As a result, if $x\equiv 0{\pmod {3}}$, we then get $C(0,{\frac {1}{3}}x+1)$ and the above rule is applied until we reach $C({\frac {5}{3}}x+5,0)$, equivalent to $g({\frac {5}{3}}x+6)$, in $\sum _{i=0}^{x/3}(2\times 5i+8)={\frac {5}{9}}x^{2}+{\frac {13}{3}}x+8{\phantom {}}$ steps for a total of ${\frac {5}{9}}x^{2}+{\frac {19}{3}}+15$ steps from $g(x)$ (with $g(0)$ we see the impossible configuration $C(-1,0)$, but it reaches $g(6)$ in 15 steps regardless). However, if $x\equiv 1{\pmod {3}}$, we then get $C(3,{\frac {x+2}{3}})$ which reaches $C(3+{\frac {5(x+2)}{3}})$, or $g({\frac {5x+22}{3}})$, in ${\frac {5}{9}}x^{2}+{\frac {47}{9}}x+{\frac {74}{9}}$ steps (${\frac {5}{9}}x^{2}+{\frac {65}{9}}x+{\frac {173}{9}}$ steps total).

The information above can be summarized as

$$g(x)\rightarrow {\begin{cases}g{\big (}{\frac {5}{3}}x+6{\big )}&{\text{if }}x\equiv 0{\pmod {3}}\\g{\big (}{\frac {5x+22}{3}}{\big )}&{\text{if }}x\equiv 1{\pmod {3}}\\0^{\infty }\;1\;{\textrm {Z>}}\;01\;001^{(x+1)/3}\;1\;0^{\infty }&{\text{if }}x\equiv 2{\pmod {3}}\end{cases}}$$

Substituting $x\leftarrow 3x$, $x\leftarrow 3x+1$, and $x\leftarrow 3x+2$ to each of these cases respectively gives us our final result.

Trajectory

The initial blank tape represents $g(0)$, and the rules are iterated 15 times before halting:

$${\begin{array}{lllllllllll}\\g(0)&\xrightarrow {15} &g(6)&\xrightarrow {73} &g(16)&\xrightarrow {277} \\g(34)&\xrightarrow {907} &g(64)&\xrightarrow {2757} &g(114)&\xrightarrow {7957} \\g(196)&\xrightarrow {22777} &g(334)&\xrightarrow {64407} &g(564)&\xrightarrow {180307} \\g(946)&\xrightarrow {504027} &g(1584)&\xrightarrow {1403967} &g(2646)&\xrightarrow {3906393} \\g(4416)&\xrightarrow {10861903} &g(7366)&\xrightarrow {30196527} &g(12284)&\xrightarrow {24576} &0^{\infty }\;1\;{\textrm {Z>}}\;01\;001^{4095}\;1\;0^{\infty }\end{array}}$$