  We propose a non-perturbative criterion to investigate whether supersymmetric
lattice gauge theories preserving partial SUSY can have the desired continuum
limit or not. Since the target continuum theories of the lattice models are
extended supersymmetric gauge theories including the topological field theory
(TFT) as a special subsector, the continuum limits of them should reproduce the
properties of the TFT. Therefore, whether the property of the TFT can be
recovered at the continuum limit becomes a non-perturbative criterion. Then we
accept it as a criterion. In this paper, among the topological properties, we
investigate the BRST cohomology on the two dimensional ${\mathcal N}=(4,4)$
CKKU lattice model without moduli fixing mass term. We show that the BRST
cohomology in the target continuum theory cannot be realized from the BRST
cohomology on the lattice. From this result, we obtain the possible implication
that the ${\mathcal N} = (4,4)$ CKKU model cannot recover the target continuum
theory if the non-perturbative effects are taken into account.
