  In this paper, local and nonlocal complex reduction of a discrete and a
semi-discrete negative order Ablowitz-Kaup-Newell- Segur equations is studied.
Cauchy matrix type solutions, including soliton solutions and Jordan-block
solutions, for the resulting local and nonlocal complex discrete and
semi-discrete sine-Gordon equations are constructed. Dynamics of 1-soliton
solution are analyzed and illustrated.
