TITLE: Busy Beaver - Proof for BB(2) = 4
QUESTION [8 upvotes]: Hi,
I need to prove the above claim.
I can show that $BB(2)\ge 4$ by building a turing machine,
but how can i show that $BB(2) \le 4$?
Searched a lot over the web, and saw that Rado proved it in 1963.
Thanks.

REPLY [2 votes]: Let allow me to write down this table of possible rules:
A0|0LA|1LA|0RA|1RA|0LB|1LB|0RB|1RB|0LH|1LH|0RH|1RH
A1|0LA|1LA|0RA|1RA|0LB|1LB|0RB|1RB|0LH|1LH|0RH|1RH
B0|0LA|1LA|0RA|1RA|0LB|1LB|0RB|1RB|0LH|1LH|0RH|1RH
B1|0LA|1LA|0RA|1RA|0LB|1LB|0RB|1RB|0LH|1LH|0RH|1RH
There will be indeed 20736 TMs, but you can easily eliminate many of them. You probably know that Robert Munafo in his site concluded that the first rule, for A0 must be 1RB, so, writing one and move to the right. So our table is:
A0|1RB
A1|0LA|1LA|0RA|1RA|0LB|1LB|0RB|1RB|0LH|1LH|0RH|1RH
B0|0LA|1LA|0RA|1RA|0LB|1LB|0RB|1RB|0LH|1LH|0RH|1RH
B1|0LA|1LA|0RA|1RA|0LB|1LB|0RB|1RB|0LH|1LH|0RH|1RH
This gives only 12^3 = 1728 TMs. We can reduce it further if we note that in all rules we can leave only one possibitily for the halting rule: 1RH (or 1LH, no matter for the result). So:
A0|1RB
A1|0LA|1LA|0RA|1RA|0LB|1LB|0RB|1RB|1RH
B0|0LA|1LA|0RA|1RA|0LB|1LB|0RB|1RB|1RH
B1|0LA|1LA|0RA|1RA|0LB|1LB|0RB|1RB|1RH
This leaves us with 729 TMs. Now we can note that in the rule B0, we can eliminate the possibilities 0RA, 1RA, 0RB, 1RB and 1RH, first four lead to infinite loop, the last one lead to 2 ones. Our table is:
A0|1RB
A1|0LA|1LA|0RA|1RA|0LB|1LB|0RB|1RB|1RH
B0|0LA|1LA|0LB|1LB
B1|0LA|1LA|0RA|1RA|0LB|1LB|0RB|1RB|1RH
324 TMs for now. Don't worry, we mustn't shall check them all, we need to explore 4 branches from 0LA, 1LA, 0LB, 1LB in rule B0. Now we can create the sub-ruleset for every branch, for example:
Branch B0|0LA
A1|0LA|1LA|0RA|1RA|0LB|1LB|0RB|1RB
B1|1RH
Note that one rule must be halting anyways, otherwise TM will works infinitely regardless to complexity of its behavior. In Branch B0|0LA, this rule must be B1|1RH, and in A1 we can eliminate the halting rule.
In every branch you will have 8 TMs, so there are 32 TMs to check. Simulate them on http://morphett.info/turing/turing.html to figure out their behavior.