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A logic program with function symbols is called finitely ground if there is a finite propositional logic program whose stable models are exactly the same as the stable models of this program. Finite groundability is an important property for logic programs with function symbols because it makes feasible to compute such programs' stable models using traditional ASP solvers. In this paper, we introduce new decidable classes of finitely ground programs called poly-bounded and k-EXP-bounded programs, which, to the best of our knowledge, strictly contain all other decidable classes of finitely ground programs discovered so far in the literature. We also study the relevant complexity properties for these classes of programs. We prove that the membership complexities for poly-bounded and k-EXP-bounded programs are EXPTIME-complete and (k+1)-EXPTIME-complete, respectively.