For a computational problem, can the set of instances be finite? A computational problem consists of a set of instances. In most cases the set of instances is infinite. Like the set of all graphs, e.g.:


  Given a graph $G$, is there a Hamiltonian circuit in $G$?


Can the set of instances be finite and a problem still be non-trivial?
A finite problem can be decided in O(1) time: we can simply compare the input to a finite (bounded) list of known cases, and emit the output consequently.

Indeed, for that we do not need a Turing powerful system: a finite-state automaton suffices.

So, the problem is indeed trivial.

Co-finite (finite complement) problems are similarly trivial.