Green's machines

In 1964, Milton W. Green hand-crafted a family of fast-growing Turing Machines with n-states, 2-symbols for all $n\ge 4$. This family grows roughly as fast as the Ackermann function:$$B{B}_{2n}\approx 3{\uparrow }^{n-2}3$$

Over the years, these Turing Machines lost champion status. The last machines to lose champion status were BB(9), BB(10), and BB(11). In August 2024, Racheline discovered the better bounds $BB(9)>10\uparrow \uparrow 1820$ and $BB(10)>10\uparrow \uparrow \uparrow \uparrow \uparrow 4$.

Machines

# States	TM	Sigma Score
4	1LB1RZ_0LC1LC_0LD0LC_1RD1RA (bbch)	7
5	1LD1LB_1LZ1LA_0LB1LD_0LE0LD_1RE1RC (bbch)	13
6	1LB1RZ_0LC1LC_0LD0LC_1LE1RA_0LF0LE_1RF1RD (bbch)	35
7	1LD1LB_1LZ1LA_0LB1LD_0LE0LD_1LF1RC_0LG0LF_1RG1RE (bbch)	22,961
8	1LB1RZ_0LC1LC_0LD0LC_1LE1RA_0LF0LE_1LG1RD_0LH0LG_1RH1RF (bbch)	$\frac{7\cdot 3^{93}-3}{2}$
9	1LD1LB_1LZ1LA_0LB1LD_0LE0LD_1LF1RC_0LG0LF_1LH1RE_0LI0LH_1RI1RG (bbch)	$>3\uparrow \uparrow 31$
10	1LB1RZ_0LC1LC_0LD0LC_1LE1RA_0LF0LE_1LG1RD_0LH0LG_1LI1RF_0LJ0LI_1RJ1RH (bbch)	$>3\uparrow \uparrow 3\uparrow \uparrow 4$
11	1LD1LB_1LZ1LA_0LB1LD_0LE0LD_1LF1RC_0LG0LF_1LH1RE_0LI0LH_1LJ1RG_0LK0LJ_1RK1RI (bbch)	$>3\uparrow \uparrow \uparrow 3\uparrow \uparrow \uparrow 2$
12	1LB1RZ_0LC1LC_0LD0LC_1LE1RA_0LF0LE_1LG1RD_0LH0LG_1LI1RF_0LJ0LI_1LK1RH_0LL0LK_1RL1RJ (bbch)	$>3\uparrow \uparrow \uparrow 3\uparrow \uparrow \uparrow 4$
13	1LD1LB_1LZ1LA_0LB1LD_0LE0LD_1LF1RC_0LG0LF_1LH1RE_0LI0LH_1LJ1RG_0LK0LJ_1LL1RI_0LM0LL_1RM1RK (bbch)	$>3\uparrow \uparrow \uparrow \uparrow 3\uparrow \uparrow \uparrow \uparrow 2$
14	1LB1RZ_0LC1LC_0LD0LC_1LE1RA_0LF0LE_1LG1RD_0LH0LG_1LI1RF_0LJ0LI_1LK1RH_0LL0LK_1LM1RJ_0LN0LM_1RN1RL (bbch)	$>3\uparrow \uparrow \uparrow \uparrow 3\uparrow \uparrow \uparrow \uparrow 4$
15	1LD1LB_1LZ1LA_0LB1LD_0LE0LD_1LF1RC_0LG0LF_1LH1RE_0LI0LH_1LJ1RG_0LK0LJ_1LL1RI_0LM0LL_1LN1RK_0LO0LN_1RO1RM (bbch)	$>3\uparrow \uparrow \uparrow \uparrow \uparrow 3\uparrow \uparrow \uparrow \uparrow \uparrow 2$

External Links

• Shawn Ligocki. Green's Machines. 2023. https://www.sligocki.com/2023/10/19/greens-machines.html