First, programs are defined to be finite in length. Second, for a program to be said to compute a function, it must do so for all possible inputs. That’s by de

First, programs are defined to be finite in length. Second, for a program to be said to compute a function, it must do so for all possible inputs. That’s by definition, but there are good reasons why these definitions are useful: if we accept infinite or incomplete programs, everything would be considered computable. Anyway, your program can only compute BB for a finite number of inputs (depending on how far you bother to expand the “else …” part).