Fix an

alphabet A. A

language L (i.e. a

set of

words composed of letters of A) is called

recursive if there exists a

Turing machine which

accepts that language. That is, there

exists some Turing machine M such that when M is run on a word w, M(w)

terminates and produces the output

1 if w is in L, and M(w)

terminates and produces the output

if w is not in L.

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.