  A long standing problem of Gian-Carlo Rota for associative algebras is the
classification of all linear operators that can be defined on them. In the
1970s, there were only a few known operators, for example, the derivative
operator, the difference operator, the average operator, and the Rota-Baxter
operator. A few more appeared after Rota posed his problem. However, little
progress was made to solve this problem in general. In part, this is because
the precise meaning of the problem is not so well understood. In this paper, we
propose a formulation of the problem using the framework of operated algebras
and viewing an associative algebra with a linear operator as one that satisfies
a certain operated polynomial identity. This framework also allows us to apply
theories of rewriting systems and Gr\"{o}bner-Shirshov bases. To narrow our
focus more on the operators that Rota was interested in, we further consider
two particular classes of operators, namely, those that generalize differential
or Rota-Baxter operators. As it turns out, these two classes of operators
correspond to those that possess Gr\"obner-Shirshov bases under two different
monomial orderings. Working in this framework, and with the aid of computer
algebra, we are able to come up with a list of these two classes of operators,
and provide some evidence that these lists may be complete. Our search has
revealed quite a few new operators of these types whose properties are expected
to be similar to the differential operator and Rota-Baxter operator
respectively.
  Recently, a more unified approach has emerged in related areas, such as
difference algebra and differential algebra, and Rota-Baxter algebra and
Nijenhuis algebra. The similarities in these theories can be more efficiently
explored by advances on Rota's problem.
