  Differential difference algebras are generalizations of polynomial algebras,
quantum planes, and Ore extensions of automorphism type and of derivation type.
In this paper, we investigate the Gelfand-Kirillov dimension of a finitely
generated module over a differential difference algebra through a computational
method: Gr\"obner-Shirshov basis method. We develop the Gr\"obner-Shirshov
basis theory of differential difference algebras, and of finitely generated
modules over differential difference algebras, respectively. Then, via
Gr\"obner-Shirshov bases, we give algorithms for computing the Gelfand-Kirillov
dimensions of cyclic modules and finitely generated modules over differential
difference algebras.
