  Motivated by a theorem of Groves and Wilton, we propose the study of the
lattice of numberings of isomorphism classes of marked groups as a rigorous and
comprehensive framework to study global decision problems for finitely
generated groups. We establish the Rice and Rice-Shapiro Theorems for recursive
presentations, and establish similar results for co-recursive presentations. We
give an algorithmic characterization of finitely presentable groups in terms of
semi-decidability of two decision problems: the word problem and the marked
quotient problem, which we introduce. We explain how this result can be used to
define algorithmic generalizations of finite presentations. Finally, we discuss
how the Adian-Rabin Theorem provides incomplete answers in several respects.
