  Weighted gradual semantics provide an acceptability degree to each argument
representing the strength of the argument, computed based on factors including
background evidence for the argument, and taking into account interactions
between this argument and others. We introduce four important problems linking
gradual semantics and acceptability degrees. First, we reexamine the inverse
problem, seeking to identify the argument weights of the argumentation
framework which lead to a specific final acceptability degree. Second, we ask
whether the function mapping between argument weights and acceptability degrees
is injective or a homeomorphism onto its image. Third, we ask whether argument
weights can be found when preferences, rather than acceptability degrees for
arguments are considered. Fourth, we consider the topology of the space of
valid acceptability degrees, asking whether "gaps" exist in this space. While
different gradual semantics have been proposed in the literature, in this
paper, we identify a large family of weighted gradual semantics, called
abstract weighted based gradual semantics. These generalise many of the
existing semantics while maintaining desirable properties such as convergence
to a unique fixed point. We also show that a sub-family of the weighted gradual
semantics, called abstract weighted (L^p,\lambda,\mu)-based gradual semantics
and which include well-known semantics, solve all four of the aforementioned
problems.
