of verifying the acyclicity of an argument would be to “track” its evaluation, e.g. in order to accept $a$ we need to accept $b$, to accept $b$ we need to accept $c$ and so on. This basic case becomes more complicated when disjunction is introduced. We then receive a number of such “paths”, with only some of them proving to be acyclic. Moreover, they might be conflicting one with each other, and we can have a situation in which all acyclic evaluations are blocked and a cycle is forced. Our approach to acyclicity is based on the idea of such “paths” that are accompanied by sets of arguments used to detect possible conflicts. Let us now introduce the formal definitions. Given an argument $s\in S$ and $x\in\{in, out\}$, by $min\_dec(x,s)$ we will denote the set of minimal two–valued interpretations that are decisively $x$ for $s$. By minimal we understand that both $v^{{\mathbf{t}}}$ and $v^{{\mathbf{f}}}$ are minimal w.r.t. set inclusion.

Let $A \subseteq S$ be a nonempty set of arguments. A **positive dependency function** on $A$ is a function $pd$ assigning every argument $a \in A$ an interpretation $v \in min\_dec(in, a)$ s.t. $v^t \subseteq A$ or $\mathcal{N}$ (null) iff no 