User blog:Wythagoras/Extension of the busy beaver function

Super-oracle
The first step in my extension is defining a super-oracle: an oracle that can diagonalize over any number of normal oracles. If the TM comes in the ASK status, the super-oracle does the following things: Numbering for TM#x: First the 1 state 0 oracle TMs, then 2 state 0 oracle TMs, 1 state 1 oracle TMs, 3 state 0 oracle TMs, 2 state 1 oracle TMs, 1 state 2 oracle TMs, etc.
 * It counts the number of non-blank symbols. Call it x.
 * It simultates TM#x.
 * If the TM halts, the super-oracle sets the turing machine in state YES. If it doesn't halt, the super-oracle sets the turing machine in state NO.

Next would be a TM with 1 oracle and 1 super-oracle, this BB function has grow rate \(\omega^CK_{\omega+1}\)

We can also have a TM with 2 super-oracles, this BB function has grow rate \(\omega^CK_{\omega2}\)

The limit of this idea is \(\omega^CK_{\omega^2}\)