  In this paper, we consider classification problem of asymmetric gapped
Hamiltonians, which are given as the non-degenerate part of the Hamiltonians
introduced in [O1]. We consider the $C^1$-classification, which takes into
account the effect of boundaries. We show that the left and right degeneracies
of edge ground states are the complete invariant. As a corollary, we consider
the bulk-classification problem. We study Hamiltonians that1.are given by
translation invariant finite range interactions, 2.are gapped in the bulk,
3.are frustration-free, 4.have uniformly bounded ground state degeneracy on
finite intervals, and 5.have a unique bulk ground state. We show that for the
bulk-classification, any such Hamiltonians are equivalent.
