This thesis explores the left orderings of finitely generated torsion-free abelian groups and finitely generated torsion-free nilpotent groups through the lens of computational complexity theory. We demonstrate that both types of groups admit regular left orderings. In chapters 2 and 3, the preliminaries of orderable groups and formal languages are introduced. In Chapter 4, we establish that the set of regular left orderings of $(\mathbb{Z}^n,+)$ is dense within the space of left orderings, $\mathrm{LO}(\mathbb{Z}^n)$. This result is achieved by approaching the problem from a geometric perspective and applying techniques from lattice theory. In Chapter 5, we extend this finding to finitely generated torsion-free nilpotent groups to obtain an analogous result, that is, we show that the set of regular left orderings of a finitely generated torsion-free nilpotent group is also dense within its space of orderings.