We are not going to write any more programs to show, case by case, that this or that particular function is Turing-computable, not just because it gets painfully tedious, but because we can now fairly easily establish that every μ-recursive function is Turing-computable and, conversely, every Turing-computable function is μ-recursive. This equivalence between our two different characterizations of computable functions is of key importance, and we'll be seeing its significance in the remaining chapters.

μ-Recursiveness entails Turing computability

Every μ-recursive function can be evaluated ‘by hand’, using pen and paper, prescinding from issues about the size of the computation. But we have tried to build into the idea of a Turing computation the essentials of any hand-computation. So we should certainly hope and expect to be able to prove:

Proof sketch We'll say that a Turing program is dextral (i.e. ‘right-handed’) if

1. i. in executing the program – starting by scanning the leftmost of some block(s) of digits – we never have to write in any cell to the left of the initial scanned cell (or scan any cell more than one to the left of that initial cell); and

2. ii. if and when the program halts, the final scanned cell is the same cell as the initial scanned cell.