Most variations on this definition also produce the same end-result. This is the essence of the Church-Turing thesis. However, the requirement that M always terminate (i.e. that the function it computes not be partial) may not be dropped so easily.

We may similarly define a function f taking words over A and returning words over A as computable if f is the function computed by some Turing machine M *which terminates for each input*.

It is not computable whether a Turing machine calculates a computable function.