  One of the most prominent tools for abstract argumentation is the Dung's
framework, AF for short. It is accompanied by a variety of semantics including
grounded, complete, preferred and stable. Although powerful, AFs have their
shortcomings, which led to development of numerous enrichments. Among the most
general ones are the abstract dialectical frameworks, also known as the ADFs.
They make use of the so-called acceptance conditions to represent arbitrary
relations. This level of abstraction brings not only new challenges, but also
requires addressing existing problems in the field. One of the most
controversial issues, recognized not only in argumentation, concerns the
support cycles. In this paper we introduce a new method to ensure acyclicity of
the chosen arguments and present a family of extension-based semantics built on
it. We also continue our research on the semantics that permit cycles and fill
in the gaps from the previous works. Moreover, we provide ADF versions of the
properties known from the Dung setting. Finally, we also introduce a
classification of the developed sub-semantics and relate them to the existing
labeling-based approaches.
