 Solution

No. The propertyis an index set: it depends only on the domain $W_n$ of the partial function computed, not on the syntax of its program. It is nontrivial. For example, an index whose domain is a singleton belongs to $\mathcal{S}$, whereas an index whose domain has two elements does not. The Rice theorem therefore shows that $\mathcal{S}$ is not a computable set, so no algorithm can decide the requested property for arbitrary $n$.

Invoking the Church–Turing thesis, this also rules out any effective procedure in the informal sense, because such a procedure would be represented by a computable function.