5.3.1 Finite Identification
A learner \(M\) \(\mathbf{FIN}\)-identifies a language \(L\) from text if, for every text \(t\) for \(L\), there exists a time \(n_0\) such that \(M\) outputs exactly one hypothesis, \(M\) outputs “?” (no guess) for \(n {\lt} n_0\) and outputs a single index \(e\) at time \(n_0\) with \(W_e = L\), never changing its mind. A class \(\mathcal{L}\) is \(\mathbf{FIN}\)-identifiable if a single learner \(\mathbf{FIN}\)-identifies every \(L \in \mathcal{L}\).
\(\mathbf{FIN}\) is the strongest identification criterion: the learner must produce the correct answer in finitely many steps with no subsequent revision. The class of \(\mathbf{FIN}\)-identifiable families is a strict subset of \(\mathbf{Ex}\)-identifiable families; for instance, the class of all finite languages is \(\mathbf{Ex}\)-identifiable (Example 5.3) but not \(\mathbf{FIN}\)-identifiable, since the learner cannot know when the last element has been seen.