  In this paper we revisit Rota's Classification Problem on classifying
algebraic identities for linear operator. We reformulate Rota's Classification
Problem in the contexts of rewriting systems and Gr\"obner-Shirshov bases,
through which Rota's Classification Problem amounts to the classification of
operators, given by their defining operator identities, that give convergent
rewriting systems or Gr\"obner-Shirshov bases. Relationship is established
between the reformulations in terms of rewriting systems and that of
Gr\"obner-Shirshov bases. We provide an effective condition that gives
Gr\"obner-Shirshov operators and obtain a new class of Gr\"obner-Shirshov
operators.
