I am curious about the NP-completeness (or if not, an efficient algorithm) for the following generalization of the subset sum problem:
In subset sum, we are given a number $t$ and a collection $S$ of integers with $|S|=n$, and ask if we can use a subset $S'\subseteq S$ to sum up to $t$. We can generalize the problem by extending the operation allowed: instead with only addition, we can permit addition together with multiplication and parenthesization.
It seems with the extended case, the usual reduction technique of encoding 3SAT in the problem breaks down, as parenthesization together with multiplication is difficult to handle (on the other hand, it seems like multiplication itself is easier to handle, as it can be expressed as a summation of identical elements).
Intuitively this generalized problem looks much harder; however, I haven't managed to find a way to prove its NP-completeness. I'm wondering if it can be indeed proven to be NP-complete, and what sort of reduction technique could be used in this problem.