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
0 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.