Let $\mathbb K$ be a characteristic-$0$ field, $R=\mathbb K[x_1,\ldots,
x_n]$ be a polynomial ring, and $W=\mathbb K[x_1,\ldots,x_n,\partial_1,\ldots \partial_n]$ be the Weyl
algebra. As usual $W$ acts on $R$ as,
\begin{equation}
  \label{eq:1}
  x_i \bullet f=x_i f,\quad \partial_i \bullet f=\frac{\partial
    f}{\partial x_i},\quad \forall f \in R
\end{equation}
Let $\mathcal D=\{D_1, \ldots, D_k\}$ be a finite subset of $W$. Denote
$\mathcal D \bullet R$ be the $\mathbb K$-linear space,
\begin{equation}
  \label{eq:2}
  \mathcal D \bullet R\equiv {\rm span}_{\mathbb K}\{ D_i \bullet f
|1\leq i\leq k,\ \forall f\in R\}
\end{equation}
The $\mathbb K$-linear quotient space $R/(\mathcal D \bullet R)$ is
the object of our interest. We want a practical way (1) to
calculate $\dim_{\mathbb K} R/(\mathcal D \bullet R)$, if it is finite,
 (2) to determine if a given polynomial $F\in R$ is in $(\mathcal D
   \bullet R)$ or not.

Note that if $D_i$'s contain no differential operator, i.e. if
$D_i\in R$, $1\leq i \leq k$, then both questions can be solved by
the (commutative) Gr\"obner basis computation of a polynomial
ideal. For the general case, is there a way to solve them by computing Gr\"obner
basis for an ideal of the Weyl algebra?