  Minimal models of a Boolean formula play a pivotal role in various reasoning
tasks. While previous research has primarily focused on qualitative analysis
over minimal models; our study concentrates on the quantitative aspect,
specifically counting of minimal models. Exact counting of minimal models is
strictly harder than #P, prompting our investigation into establishing a lower
bound for their quantity, which is often useful in related applications. In
this paper, we introduce two novel techniques for counting minimal models,
leveraging the expressive power of answer set programming: the first technique
employs methods from knowledge compilation, while the second one draws on
recent advancements in hashing-based approximate model counting. Through
empirical evaluations, we demonstrate that our methods significantly improve
the lower bound estimates of the number of minimal models, surpassing the
performance of existing minimal model reasoning systems in terms of runtime.
