Indistinguishability complexity of a subset
I assume $S&#39;$ is parametrized by $n$, i.e.~it is not defined only for a single $n$ but for infinitely many $n$ (otherwise statements like "polynomial time" do not make sense). If so, you are merely defining the fundamental problem of the complexity of deciding membership in a language (i.e. the problem of complexity theory). I am not sure what you mean by "uniform": if you mean that there could be a different algorithm for different $n$, then this is the usual non-uniform complexity.