paper [@inproc:adf], only three of them – conflict–free, model and grounded (initially referred to as well–founded) – are still used (issues with the other formulations can be found in [@tofix:newadf; @inproc:adm; @report:strass]). Moreover, the treatment of cycles and their handling by the semantics was not sufficiently developed. In this section we will address all of those issues. Before we continue, let us first motivate our choice on how to treat cycles. The opinions on support cycles differ between the available frameworks, as we have shown in Section \[sec:bip\]. Therefore, we would like to explore the possible approaches in the context of ADFs by developing appropriate semantics.

The classification of the sub–semantics that we will adopt in this paper is based on the inside–outside intuition we presented in the introduction. Appropriate semantics will receive a two–element prefix $xy-$, where $x$ will denote whether cycles are permitted or not on the “inside” and $y$ on the “outside”. We will use $x, y \in \{a,c\}$, where $a$ will stand for *acyclic* and $c$ for *cyclic* constraints. As the conflict–free (and naive) semantics focus only on what we can accept, we will drop the prefixing in this case. Although the model, stable and grounded 