View source for 1RB0RA 1LC1LF 1RD0LB 1RA1LE 1RZ0LC 1RG1LD 0RG0RF

{{machine|1RB0RA_1LC1LF_1RD0LB_1RA1LE_1RZ0LC_1RG1LD_0RG0RF}}
{{TM|1RB0RA_1LC1LF_1RD0LB_1RA1LE_1RZ0LC_1RG1LD_0RG0RF}} is a halting [[BB(7)]] TM which runs for over <math>2 \uparrow^{12} 2 \uparrow^{12} 3</math> steps.

== Analysis by Shawn Ligocki ==
Consider general configurations matching the regex: <math>0^\infty \; 11 \; (1 \; (01)^*)^* \; 0011100 \; \text{A>} \; 0^\infty</math>

=== Low level rules ===
<pre>
              01 1 01^n 0011100 A> 00   -->                 1 01^n+2 0011100 A>
           01^3 11 01^n 0011100 A> 0^6  -->            1 01^n+5 1 01 0011100 A>
01^3 (1 01)^k+1 11 01^n 0011100 A> 0^6  -->  1 01^n+6 (1 01)^k 11 01 0011100 A>
 011 (1 01)^k   11 01^n 0011100 A> 0^2  -->  1 Z> 111 01^n+1 00 101^k+2
</pre>

=== Mid level rules ===
Let

<math display=block>
B(a; b, c, ..., z) = 0^\infty \; 111 \; (01)^{3z+1} \; 1 \; \cdots \; 1 (01)^{3c+1} \; 1 \; (01)^{3b+1} \; 1 \; (01)^0 \; 1 \; (01)^{3a+1} \; 0011100 \; \text{A>} \; 0^\infty
</math>

and let <code>B(a; [x]*k, y, ...)</code> = <math>B(a; \underbrace{x, \cdots, x}_k, y, ...)</math> (In other words, <code>[x]*k</code> represents k repeats of the value x in a config).

then

<pre>
B(a; b+1, ...) -> B(2a+4; b, ...)
B(a; [0]*k, 0, n+1, ...) -> B(0; [0]*k, a+2, n, ...)
B(a; [0]*k) -> Halt(3a + 2k + 9)

Start at step 8178: B(2, [1]*12)
</pre>

=== High level rule ===
Let

<math display=block>\begin{array}{l}
f_0(x)     & = & 2x + 4 \\
f_{k+1}(x) & = & f_k^{x+2}(0) \\
\end{array}</math>

then

<math display=block>
B(a; \underbrace{0, \cdots, 0}_k, n, ...) \to B(f_k^n(a); \underbrace{0, \cdots, 0}_k, n, ...)
</math>

=== Bound ===
Let

<math display=block>\begin{array}{l}
a_0 & = & 2 \\
a_{k+1} & = & f_k(a_k) \\
</math>

then

<math display=block>
B(a_0; \underbrace{1, \cdots, 1}_k) \to B(a_k, \underbrace{0, \cdots, 0}_k)
</math>

and so this TM halts with a sigma score of <math> 3 a_{12} + 33 </math>

Note that <math>f_k(x) = (2 \uparrow^k (x+4)) - 4</math> and so for <math>k \ge 2</math>,

<math display=block>
a_k + 4 > 2 \uparrow^k 2 \uparrow^k 3
</math>

and so this TM halts with sigma score <math>\sigma > 2 \uparrow^{12} 2 \uparrow^{12} 3</math>.

This bound is pretty tight: <math>\sigma < 2 \uparrow^{12} 2 \uparrow^{12} 4</math>.