# Is there a recursive problem encoding the Turing completeness of a model of computation?

Suppose we have a model of computation $$C$$ we want to show to be Turing complete. The usual strategy would be to emulate within $$C$$ any model of computation we already know to be Turing complete (e.g. an arbitrary Turing machine). On the other hand, a model of computation is Turing complete if and only if it can compute (the indicator function of) all recursive sets. Is there a special recursive set $$S$$ (or finite subset of recursive sets) so that it is enough for $$C$$ to compute $$S$$ to assure that $$C$$ is Turing complete? Informally, $$S$$ would be a set for whose computation branching, looping and memory are all needed in an unavoidable way. If such sets do exist, which are the simplest ones?

In a practical sense, the utility of these sets would be the following test: if your model of computation can solve this type of (e.g. numerical or geometric) problem, then it is Turing complete.