Fix an alphabet
A. A language
L (i.e. a set
s composed of letters of A) is called recursive
if there exists a Turing machine
that language. That is, there exist
s 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.