  In this paper we introduce a working generalization of the theory of
Gr\"obner bases for algebras of partial difference polynomials with constant
coefficients. One obtains symbolic (formal) computation for systems of linear
or non-linear partial difference equations arising, for instance, as discrete
models or by the discretization of systems of differential equations. From an
algebraic viewpoint, the algebras of partial difference polynomials are free
objects in the category of commutative algebras endowed with the action by
endomorphisms of a monoid isomorphic to $\N^r$. Then, the investigation of
Gr\"obner bases in this context contributes also to the current research trend
consisting in studying polynomial rings under the action of suitable symmetries
that are compatible with effective methods. Since the algebras of difference
polynomials are not Noetherian ones, we propose in this paper a theory for
grading them that provides a Noetherian subalgebras filtration. This implies
that the variants of the Buchberger's algorithm we developed for difference
ideals terminate in the finitely generated graded case when truncated up to
some degree. Moreover, even in the non-graded case, we provide criterions for
certifying completeness of eventually finite Gr\"obner bases when they are
computed within sufficiently large bounded degrees. We generalize also the
concepts of homogenization and saturation, and related algorithms, to the
context of difference ideals. The feasibily of the proposed methods is shown by
an implementation in Maple that is the first to provide computations for
systems of non-linear partial difference equations. We make use of a test set
based on the discretization of concrete systems of non-linear partial
differential equations.
