BB(2)

The 2-state 2-symbol Busy Beaver problem BB(2) was solved by Tibor Radó using pencil and paper and announced in his seminal Busy Beaver paper, On Non-Computable Functions.

Deciders

The only deciders needed to prove BB(2) are cycler and translated cycler deciders. All 26 infinite TMs listed below are either Translated Cyclers or Cyclers.

Champions

S(2) = 6 and there are 5 shift champions in TNF:

• 1RB1LB_1LA1RZ (bbch) leaves 4 ones (the ones champion)
• 1RB0LB_1LA1RZ (bbch) leaves 3 ones
• 1RB1RZ_1LB1LA (bbch) leaves 3 ones
• 1RB1RZ_0LB1LA (bbch) leaves 2 ones
• 0RB1RZ_1LA1RB (bbch) leaves 2 ones

Σ(2) = 4 and there is one unique ones champion in TNF:

• 1RB1LB_1LA1RZ (bbch) runs for 6 steps (a shift champion)

Enumeration

In TNF-1RB there are exactly 41 2-state, 2-symbol TMs, of which 15 halt and 26 never halt. The last group can be divided to 3 cyclers and 23 translated cyclers.

Halting TMs

The following TMs are ordered by the number of steps before halting and by the number of ones left in the tape (represented by the left and right numbers, respectively).

1RB1LB_1LA1RZ Halt 6 4
1RB1RZ_1LB1LA Halt 6 3
1RB0LB_1LA1RZ Halt 6 3
1RB1RZ_0LB1LA Halt 6 2
1RB1LA_1LA1RZ Halt 5 3
1RB1LA_0LA1RZ Halt 5 2
1RB1RZ_1LB1RA Halt 4 2
1RB1RB_1LA1RZ Halt 4 2
1RB1RZ_1LB0RA Halt 4 1
1RB0RB_1LA1RZ Halt 4 1
1RB1RZ_1LA--- Halt 3 2
1RB---_1LB1RZ Halt 3 2
1RB1RZ_0LA--- Halt 3 1
1RB---_0LB1RZ Halt 3 1
1RB---_1RZ--- Halt 2 2

Cyclers

The following TMs are ordered by the period and preperiod (represented by the left and right numbers, respectively).

1RB0RB_0LA--- Cycler 4 0
1RB1RB_0LA--- Cycler 2 1
1RB---_0LB1RB Cycler 2 1

Translated Cyclers

1RB1RA_1LA--- Translated Cycler
1RB1RA_0LA--- Translated Cycler
1RB1LB_0LA--- Translated Cycler
1RB0RA_1LA--- Translated Cycler
1RB0RA_0LA--- Translated Cycler
1RB0LB_0LA--- Translated Cycler
1RB0LA_1LA--- Translated Cycler
1RB0LA_0LA--- Translated Cycler
1RB---_1RB--- Translated Cycler
1RB---_1RA--- Translated Cycler
1RB---_1LB1RB Translated Cycler
1RB---_1LB1LB Translated Cycler
1RB---_1LB0RB Translated Cycler
1RB---_1LB0LB Translated Cycler
1RB---_1LB0LA Translated Cycler
1RB---_0RB--- Translated Cycler
1RB---_0RA--- Translated Cycler
1RB---_0LB1RA Translated Cycler
1RB---_0LB1LB Translated Cycler
1RB---_0LB0RB Translated Cycler
1RB---_0LB0RA Translated Cycler
1RB---_0LB0LB Translated Cycler
1RB---_0LB0LA Translated Cycler