  In [7], Kastler gave a brute-force proof of the fact, announced by Connes,
that the Wodzicki residue of the inverse square of the Dirac operator D is
proportional to the Einstein-Hilbert action of general relativity. In this
paper, we obtain the Wodzicki residue of the inverse square of the
pseudodiffential pertubations of the Dirac operator. And we give the proof of
the Kastler-Kalau-Walze type theorems for the pseudodiffential pertubations of
the Dirac operator on 4-dimensional compact manifolds with boundary.
