  We consider a theory of noncommutative Gr\"obner bases on decreasingly
filtered algebras whose associated graded algebras are commutative. We transfer
many algorithms that use commutative Gr\"obner bases to this context. As an
important application, we implement very efficient algorithms to compute the
Ext groups over the Steenrod algebra $\mathscr{A}$ at the prime $2$.
Especially, the cohomology of the Steenrod algebra $Ext_{\mathscr{A}}^{*,
*}(\mathbb{F}_2, \mathbb{F}_2)$, which plays an important role in algebraic
topology, is calculated up to total degree of 261, including the ring structure
in this range.
