  As with nonlocal continuous and semi-discrete integrable systems, the study
of nonlocal discrete integrable systems is also of interest. In this paper,
local and nonlocal reductions of a fully discrete negative order
Ablowitz-Kaup-Newell-Segur equation are investigated. We give out the exact
solutions in double Casoratian form to the reduced nonlocal discrete
sine-Gordon equations by the bilinearization reduction method. Then, through
the continuum limits, nonlocal semi-discrete sine-Gordon equations and their
solutions are obtained. The dynamics of soliton solutions are analyzed and
illustrated by asymptotic analysis. The research ideas and methods in this
paper can be generalized to promote the studies on nonlocal discrete integrable
systems.
