Define Bβ(x) as the smallest y which βB(x,y)=BB(x).It is finite for finite x.(because βB is monotonic,and has to start at 0 and get to BB(x) at infinity,and it can't change from BB(x)-1 to BB(x) at infinity-1(because you can't have infinity-1))It is uncomputable too,since by computing it we can find a non finite upper bound of βB,to replace the uncomputableness of βB(x,inf)
So yeah,two new interesting functions,of one is uncomputable(i found those functions when trying to induce a contradiction in BB(x)(using brute force,but forgot that the tape is infinite,thus proving that βB is computable at finite y)
Where should I submit them?Were them discovered before by other people?
What should they be called?(i called the βB function the busy little beaver function and the Bβ function the unbusy beaver function)